The Brown measure of the free multiplicative Brownian motion
Bruce K. Driver Brian C. Hall Todd Kemp




Department of Mathematics, University of California San
Diego, La Jolla, CA 92093, USA
bdriver@ucsd.edu

Department of Mathematics, University of Notre Dame, Notre Dame,
IN 46556, USA
bhall@nd.edu

Kemp's research supported in part by NSF CAREER Award DMS-1254807 and NSF Award DMS-1800733
Department of Mathematics, University of California San Diego, La Jolla, CA 92093, USA
tkemp@math.ucsd.edu



The free multiplicative Brownian motion b_t is the large-N limit of the
Brownian motion on 𝖦𝖫(N;ℂ), in the sense of ∗-distributions. The natural candidate for the large-N limit of the
empirical distribution of eigenvalues is thus the Brown measure of b_t. In
previous work, the second and third authors showed that this Brown measure is
supported in the closure of a region Σ_t that appeared in work of Biane.
In the present paper, we compute the Brown measure completely. It has a
continuous density W_t on Σ_t, which is strictly
positive and real analytic on Σ_t. This density has a simple form in
polar coordinates:

    W_t(r,θ)=1/r^2w_t(θ),

where w_t is an analytic function determined by the geometry of the region
Σ_t.

We show also that the spectral measure of free unitary Brownian motion u_t
is a “shadow” of the Brown measure of
b_t, precisely mirroring the relationship between Wigner's semicircle law
and Ginibre's circular law. We develop several new methods, based on
stochastic differential equations and PDE, to prove these results.



The Brown measure of the free multiplicative Brownian motion
    Todd Kemp
    Received: date / Revised version: date
============================================================







§ INTRODUCTION




 §.§ ∗-Distribution and Brown measure


Two of the most fundamental models in random matrix theory are the
Gaussian unitary ensemble and the Ginibre ensemble. Each is a
Gaussian random matrix; the Ginibre ensemble has i.i.d. complex Gaussian
entries, with variance 1/N, while the Gaussian unitary ensemble is the
Hermitian part of the Ginibre ensemble. For our purposes, it is natural to
think of these two ensembles as endpoints of Brownian motion on Lie
algebras. Indeed, the Lie algebra gl(N;ℂ) of the general
linear group consists of all N× N complex matrices. For each fixed t,
the Brownian motion Z_t on this space (with an appropriately scaled time
parameter) is distributed as the Ginibre ensemble, scaled by √(t).
Similarly, the space of Hermitian matrices is equal to iu(N), where
u(N) is the Lie algebra of the unitary group, namely the space of
skew-Hermitian matrices. For each fixed t, the Brownian motion X_t on
this space is distributed as the Gaussian unitary ensemble, scaled by
√(t).

Among the earliest results in random matrix theory are the discovery of the
large-N limits of the empirical eigenvalue distributions of these ensembles.
That is to say, the (random) counting measure 1/N∑_j=1^N
δ_λ_j of the eigenvalues {λ_j} of each ensemble
has an almost-sure limit which is a deterministic measure. The eigenvalues of
X_t are real, and their limit empirical eigenvalue distribution is the
semicircle law on the interval [-2√(t),2√(t)] (cf. <cit.>);
the eigenvalues of Z_t are complex, and their limit empirical eigenvalue
distribution is the uniform probability measure on the disk of radius
√(t) (cf. <cit.>). We note for later reference a simple but
intriguing link between these two limiting distributions: the push-forward
under the “real part” map of the uniform
measure on the disk is the semicircular distribution on an interval.

It is convenient to recast the empirical eigenvalue distribution in an
analytic form. For an N× N matrix A=A_N, the function

    L_A(λ)      =1/2Nlog([(A-λ)^∗(A-λ
    )])
         =1/N∑_j=1^Nlog(|λ-λ_j|
    )

is subharmonic on ℂ. Actually, L_A is harmonic on the
complement of the spectrum, and -∞ at the eigenvalues. The
(distributional) Laplacian of L is therefore a positive measure; in fact it
is (up to a factor of 2π) equal to the empirical eigenvalue distribution
of the matrix A. Hence, if one can compute an appropriate “large-N limit” of the functions L_A_N, the Laplacian
of the limiting function will provide a natural candidate for the limiting
empirical eigenvalue distribution.

Free probability theory affords a medium in which to identify abstract limits
of random matrix ensembles themselves. The limits are constructed as operators
in a tracial von Neumann algebra (𝒜,τ), and the limit is with
respect to ∗-distribution. If A_N is a sequence of N× N
random matrices, an operator a∈𝒜 is said to be a limit
in ∗-distribution of A_N if, for each polynomial p in two
noncommuting variables, we have

    lim_N→∞1/Ntrace[p(A_N,A_N^∗
    )]=τ p(a,a^∗)]

almost surely. For X_t and Z_t, Voiculescu <cit.> showed
that the large-N limits can be identified as certain free stochastic
processes, namely the free additive Brownian motion x_t and the
free circular Brownian motion c_t. These limiting process are no
longer random: they are one-parameter families of operators with freely
independent increments, and ∗-distributions that can be described
elegantly in the combinatorial framework of free probability.

In particular, x_t is a self-adjoint operator, and it therefore has a
spectral resolution E^x_t: a projection-value measure so that
x_t=∫_ℝλ E^x_t(dλ). The composition
μ_x_t:=τ∘ E^x_t is a probability measure on ℝ
called the spectral measure of x_t (in the state τ). The
statement that X_t has the semicircle law as its limit empirical
eigenvalue distribution is equivalent to the statement that μ_x_t is semicircular.

On the other hand, since the operator c_t is not normal, there is no
spectral theorem to yield a spectral measure. There is, however, a substitute:
the Brown measure, introduced in <cit.>. For any operator a∈𝒜, define a function L_a on ℂ by

    L_a(λ)=τlog(| a-λ|
    )],

where |a-λ| is the self-adjoint operator ((a-λ)^∗(a-λ))^1/2. The quantity L_a(λ) is the logarithm of the
Fuglede–Kadison determinant <cit.> of a-λ. It is finite
outside the spectrum of a but may become -∞ at points in the
spectrum. If 𝒜 is the space of all N× N matrices and
τ=1/Ntrace, then L_a agrees with the function in
(<ref>). In a general tracial von Neumann algebra (𝒜,τ),
the function L_a is subharmonic.

The Brown measure of a is then defined in terms of the
distributional Laplacian of L_a:

    μ_a:=1/2πΔ L_a.

If a is self-adjoint, the Brown measure of a coincides with the spectral
measure. If a=c_t is the free circular Brownian motion at time t, its
Brown measure μ_c_t is equal to the uniform probability measure on the
disk of radius √(t).

By regularizing the right-hand side of (<ref>), one can construct the
Brown measure μ_a as a weak limit,

    dμ_a(λ)=1/4π
    Δ_λlim_ε→0^+τlog[(a-λ)^∗(a-λ)+ε
    )] dλ,

where Δ_λ is the Laplacian with respect to λ and
dλ is the Lebesgue measure on the plane. (See <cit.>
and <cit.>.) It is not hard to see that the Brown measure of
a is determined by the ∗-moments of a, but the dependence is
singular: if a sequence of operators a_n converges in ∗-distribution
to a, the Brown measures μ_a_n need not converge to μ_a.



 §.§ Brownian motions on 𝖴(N), 𝖦𝖫(N;ℂ),
and their large-N limits


As we have noted, the Gaussian unitary ensemble and the Ginibre ensemble can
be described in terms of Brownian motions on the Lie algebras u(N)
and gl(N;ℂ). Specifically, the Brownian motions are
induced by a choice of inner product on these finite-dimensional vector
spaces; in both cases, we use the inner product

    ⟨ X,Y⟩ _N=NRetrace(X^∗Y).

(The factor of N in the definition produces the scaling of 1/N in the
variances of the two ensembles.) It is natural to consider the counterpart
Brownian motions on the Lie groups 𝖴(N) and 𝖦𝖫
(N;ℂ). In general, if G⊂𝖬_N(ℂ) is a
matrix Lie group with Lie algebra 𝔤⊂𝖬
_N(ℂ), there is a simple relationship between the Brownian motion
B_t on G and the Brownian motion A_t on 𝔤 (the latter
being the standard Brownian motion determined by an inner product on
𝔤). It is known as the rolling map, and it can be written
as a Stratonovitch stochastic differential equation (SDE):

    dB_t=B_t∘ dA_t,     B_0=I.



The solution of this SDE is a diffusion process on G whose increments
(computed in the left multiplicative sense) are independent and whose
generator is half the Laplacian on G determined by the left-invariant
Riemannian metric induced by the given inner product on 𝔤. Thus,
its distribution at each time is the heat kernel on the group. For
computational purposes, it is useful to write the SDE in Itô form; the
result depends on the structure of the group. Letting U_t=U_t^N denote
the Brownian motion on 𝖴(N) and B_t=B_t^N the Brownian
motion on 𝖦𝖫(N;ℂ), the corresponding Itô SDEs are

    dU_t     =iU_t dX_t-1/2U_t dt
    
    dB_t     =B_t dZ_t.



It is then natural to investigate the large-N limits of these random matrix
processes. The candidate large-N limits are the free stochastic
processes generated by the analogous free SDEs:

    du_t     =iu_t ds_t-1/2u_t dt
    
    db_t     =b_t dc_t.

(For the theory of free stochastic calculus, see <cit.>.) In
1997, Biane <cit.> introduced these processes, the
free unitary Brownian motion u_t, and the free
multiplicative Brownian motion b_t. (He called b_t=Λ_t, and
wrote a slightly different but equivalent free SDE for it.) The main result of
<cit.> was the theorem that the process u_t is indeed the
large-N limit in ∗-distribution of the unitary Brownian motion
U_t=U_t^N. Since U_t and u_t are unitary (hence normal)
operators, this also means that the empirical eigenvalue distribution of
U_t^N converges to the spectral measure of u_t.

Biane also computed the spectral measure ν_t of u_t. We now record
this result, since it relates closely to the results of the present paper. Let
f_t denote the holomorphic function on ℂ∖{1} defined
by

    f_t(λ)=λ e^t/21+λ/1-λ.

Then f_t has a holomorphic inverse χ_t in the open unit disk, and
χ_t extends continuously to the closed unit disk. Biane showed that

    χ_t=ψ_u_t/1+ψ_u_t

where ψ_u_t(z)=τ(1-zu_t)^-1]-1 is the (recentered)
moment-generating function of u_t. From this (and other SDE computations)
he determined the following result.

 The spectral
measure ν_t of the free unitary Brownian motion u_t is supported in
the arc

    {.  e^iϕ| |ϕ|<ϕ_max(t):=1/2√((4-t)t)+cos^-1(1-t/2)}

for t<4, and is fully supported on the circle for t≥4. The measure
ν_t has a continuous density κ_t, which is real analytic on the
interior of its support arc, given by

    κ_t(e^iϕ)=1/2π1-|χ_t(e^iϕ)| ^2/| 1-χ_t(e^iϕ)| ^2.




See, for example, p. 275 in <cit.>. In the same papers
<cit.> in which he introduced u_t, Biane considered
the free multiplicative Brownian motion b_t as well (for example,
computing its norm). He conjectured that it should be the large-N limit of
the Brownian motion B_t=B_t^N on 𝖦𝖫(N;ℂ), in
∗-distribution. This was proved by Kemp in <cit.>, with
complementary estimates of moments given in <cit.>. The goal of
the present paper is to fully determine the Brown measure of the free
multiplicative Brownian motion b_t, giving the full complex analog of
Theorem <ref>.



 §.§ The Brown measure of b_t


The main result of this paper is a formula for the Brown measure μ_b_t
of the free multiplicative Brownian motion b_t. We expect that
μ_b_t coincides with the large-N limit of the empirical eigenvalue
distribution of the Brownian motion B_t^N on 𝖦𝖫(N;ℂ
). Techniques that can be used to prove results of this sort—that a
limiting eigenvalue distribution agrees with a Brown measure—have been
developed in the context of the general circular law, in which the entries are
independent and identically distributed but not necessarily Gaussian. Analysis
of this model began with the work of Girko <cit.> and continued with
results of Bai <cit.>, Götze and Tikhomirov <cit.>, and ending with
the definitive version of the circular law established by Tao and Vu
<cit.>. All of these works compute the empirical eigenvalue distribution by
taking the Laplacian of the quantity L_A in (<ref>). They then
consider the  limiting eigenvalue distribution of (A-λ)^∗(A-λ) for each λ, from which they compute (essentially) the
Brown measure of the limiting object, which in this case is uniform on a disk.
Then to prove convergence of the eigenvalue distribution of the random matrices
to this uniform measure, they develop techniques for controlling the
singularities of the logarithm function in (<ref>) at infinity and (especially) at zero. 


























A previous result <cit.> of the second and third authors showed that
μ_b_t is supported on the closure of a certain region Σ_t
introduced by Biane in <cit.>; see Figures <ref> and
<ref>. (We reprove that result in the present paper by a
different method; see Theorem <ref> in Section <ref>.)
The proof in <cit.> is based on Biane's “free Hall
transform” 𝒢_t, introduced in <cit.>
. This transform was conjectured by Biane to be the the large-N limit of the
generalized Segal–Bargmann transform of Hall <cit.>, a
conjecture that was verified independently by Cébron <cit.> and the
authors of the present paper <cit.>. A key idea in Biane's work is
Gross and Malliavin's probabilistic interpretation <cit.> of the transform
in <cit.>.

For each λ outside Σ_t, <cit.> uses
𝒢_t to construct an “inverse”
 of b_t-λ and (b_t-λ)^2. These inverses are not
necessarily bounded operators, but live in the noncommutative L^2 space,
that is, the completion of 𝒜 with respect to the inner product
⟨ a,b⟩ :=τ(a^∗b). We then strengthen the
standard result that the Brown measure is supported on the spectrum of the
operator to show that existence of an L^2 inverse of (b_t-λ)^2
guarantees that λ is outside the support of μ_b_t. We note,
however, that the methods of <cit.> do not give any information about the
distribution of the Brown measure μ_b_t inside the region Σ
_t.



 §.§ Connection to Physics


The eigenvalue distribution of Brownian motion in 𝖦𝖫(N;ℂ),
in the large-N limit, has been studied in the physics literature, first by
Gudowska-Nowak, Janik, Jurkiewicz, and Nowak <cit.> and then by
Lohmayer, Neuberger, and Wettig <cit.>. At least in the case of
<cit.>, the motivation for considering this model is a connection to
two-dimensional Yang–Mills theory. Yang–Mills quantum field theory is a key
part of the standard model in particle physics and the two-dimensional case
can be treated in a mathematically rigorous fashion. Two-dimensional
Yang–Mills theory is a much-studied model, in part as a toy model of the
four-dimensional theory and in part because of its connections to string
theory <cit.>.

Yang–Mills theory with structure group G describes a random connection on a
principal G-bundle. One typically studies the theory through the associated
Wilson loop functionals, given by the expectation value of the trace
of the holonomy around a loop. Assume at first that G is compact, that the
spacetime manifold is the plane, and that the loop is a simple closed curve in
the plane. Then the distribution of the holonomy is described by Brownian
motion in G, with time-parameter proportional to the area enclosed by the
curve. (See works of Driver <cit.> and Gross–King–Sengupta
<cit.> and the references therein.)

Of particular importance is the case G=𝖴(N), with N tending to
infinity; the resulting theory is called the “master
field.” The master field in the plane is therefore built
around the large-N limit of Brownian motion in 𝖴(N), i.e., the
free unitary Brownian motion u_t. (See works of Singer <cit.>,
Anshelevich–Sengupta <cit.>, and Lévy <cit.>.) In this context,
the change in behavior of u_t at t=4—in which the support of the
spectral measure wraps all the way around the circle—is called a
“topological phase transition.” (See also
<cit.>, <cit.>, <cit.>, and <cit.> for recent
progress constructing a rigorous large-N Yang–Mills theory on surfaces
other than the plane.)

Although Yang–Mills theory is typically constructed when the structure group
G is compact, it requires only a small step of imagination to consider also
the case G=𝖦𝖫(N;ℂ). Thus, in <cit.>, Brownian
motion in 𝖦𝖫(N;ℂ) is considered as a sort of
“complex Wilson loop” computation. It is of
interest to determine whether the large-N limit—namely the free
multiplicative Brownian motion b_t—still has a topological phase
transition at t=4.

The papers <cit.> and <cit.> both derive, using nonrigorous
methods, the region into which the eigenvalues of B_t^N cluster in the
large-N limit. Both papers find this domain to be precisely the region
Σ_t considered in the present work. Since Σ_t wraps around
the origin precisely at time t=4, the authors conclude that indeed the
topological phase transition persists after the change from 𝖴(N) to
𝖦𝖫(N;ℂ). The paper <cit.> also considers a
two-parameter extension of the Brownian motion of the sort considered in
<cit.>, and finds that the eigenvalues cluster into
the domain denoted Σ_s,t in <cit.>. A rigorous version of these
results—specifically, that the Brown measure of the relevant free Brownian
motion is supported in Σ_t or Σ_s,t—was then obtained by the second and third authors in <cit.>.

We emphasize that the papers <cit.>, <cit.>, and <cit.> are
concerned only with the region into which the eigenvalues cluster.
Nothing is said there about how the eigenvalues are distributed in
the region. By contrast, in the present work, we not only prove (again) that
the Brown measure of b_t is supported in Σ_t, we
actually compute the Brown measure (Theorem <ref>).
Furthermore, we not only see the same transition at t=4 for the
𝖦𝖫(N;ℂ) case as for the 𝖴(N) case, we actually
find a direct connection (Proposition <ref>)
between the Brown measure of b_t and the spectral measure of u_t.



 §.§ Subsequent work


Since the first version of this paper appeared on the arXiv, three subsequent
works have appeared that use the techniques developed here to analyze Brown
measures of other operators. First, work of Ho and Zhong <cit.> has
extended the results of the present paper to the case of a free multiplicative
Brownian motion with an arbitrary unitary initial condition. This means that
they compute the Brown measure of ub_t, where u is a unitary element
freely independent of b_t. Ho and Zhong also compute the Brown measure of
x_0+c_t, where c_t is a free circular Brownian motion and x_0 is
a self-adjoint element freely independent of c_t. Second, Demni and Hamdi
<cit.> have analyzed the support of the Brown measure of u_tP, where
u_t is the free unitary Brownian motion and P is a projection freely
independent of u_t. Last, Hall and Ho <cit.> have computed the Brown
measure of x_0+ix_t, where x_t is the free additive Brownian motion
and x_0 is a self-adjoint element freely independent of x_t.

The reader may also consult the expository article <cit.> by the
second author, which provides a nontechnical introduction to the techniques
used in the present paper.



§ STATEMENT OF MAIN RESULTS




 §.§ A formula for the Brown measure


In this paper, we compute the Brown measure μ_b_t of the free
multiplicative Brownian motion b_t, using completely different methods
from those in <cit.>. To state our main result, we need to briefly describe
the regions Σ_t. For each t>0, consider the holomorphic function
f_t on ℂ∖{1} defined by (<ref>). It is easily
verified that if |λ| =1 then |
f_t(λ)| =1. There are, however, other points where
| f_t(λ)| =1. We then define

    F_t={λ∈ℂ||λ|≠1, | f_t(λ)| =1.  }

and

    E_t=F_t.




For each t>0, we define Σ_t to be the connected
component of the complement of E_t containing 1.


We will show (Theorem <ref>) that Σ_t may also be
characterized as

    Σ_t={.  λ∈ℂ| T(λ
    )<t}  ,

where the function T is defined in (<ref>). Each region Σ_t
is invariant under the maps λ↦1/λ and λ↦λ̅. If we consider a ray from the origin with angle θ, if
this ray intersects Σ_t at all, it does so in an interval of the form
1/r_t(θ)<r<r_t(θ) for some r_t(θ)>1. (See Figures
<ref> and <ref>.) See Section <ref>
for more information.














We are now ready to state our main result.


For all t>0, the Brown measure μ_b_t of b_t is
absolutely continuous with respect to the Lebesgue measure on the plane and
supported in the domain Σ_t. In Σ_t, the density
W_t of μ_b_t with respect to the Lebesgue measure is strictly
positive and real analytic, with the following form in polar coordinates:

    W_t(r,θ)=1/r^2w_t(θ)

for a certain even function w_t. This function may be computed as

    w_t(θ)=1/4π(  2/t+∂/∂θ2r_t(θ)sinθ/r_t(θ)^2+1-2r_t(θ
    )cosθ)

where r_t(θ) is the larger of the two radii where the ray with angle
θ intersects the boundary of Σ_t.


Since Σ_t is invariant under λ↦λ̅, the
function r_t(θ) is an even function of θ, from which it is
easy to check that the second term on the right-hand side of of
(<ref>) is also an even function of θ. Although we will
customarily let r_t(θ) denote the larger of the the two
radii, we note that

    r↦2rsinθ/r^2+1-2rcosθ

is invariant under r↦1/r. Thus, the value of w_t does not actually
depend on which radius is used. It is noteworthy that the one nonexplicit part
of the formula for w_t, namely the second term on the right-hand side of
(<ref>), is computable entirely in terms of the geometry of the
region Σ_t. According to Proposition <ref>, w_t
can also be computed as a logarithmic derivative along the boundary of
Σ_t of the function f_t in (<ref>).














It follows from (<ref>) that the function T equals t on the
boundary of Σ_t. It is then possible to use implicit differentiation
in the equation T(λ)=t to compute dr_t(θ)/dθ as a
function of r_t(θ) and θ. We may then use this computation to
rewrite (<ref>) in a form that no longer involves a derivative with
respect to θ, as follows.


The function w_t in Theorem <ref> may also be
computed in the form

    w_t(θ)=1/2π tω(r_t(θ),θ).

Here

    ω(r,θ)=1+h(r)α(r)cosθ+β(r)/β(r)cosθ+α(r),

where

    h(r)=rlog(r^2)/r^2-1;  α(r)=r^2+1-2rh(r);  β(r)=(r^2+1)h(r)-2r.




Thus, to compute w_t(θ), we evaluate ω/(2π t) on the
boundary of Σ_t and then parametrize the boundary by the angle
θ; see Figure <ref>. Using Proposition <ref>,
we can derive small- and large-t asymptotics of w_t(θ) as follows:

    w_t(θ)     ∼1/π t,   t small;
    
    w_t(θ)     ∼1/2π t,   t large.

See Section <ref> for details.















The following simple consequences of Theorem <ref> helps explain the
significance of the factor of 1/r^2 in the formula (<ref>) for
W_t.
















The Brown measure μ_b_t of b_t has the
following properties.



  * μ_b_t is invariant under the maps
λ↦1/λ and λ↦λ̅.


  * Let Ξ_t denote the image of Σ
_t∖(-∞,0) under the complex logarithm map, using the standard
branch cut along the negative real axis. We write points z∈Ξ_t as
(ρ,θ). Then for points in Ξ_t, the pushforward of μ_b_t by the logarithm map has density ω_t(ρ,θ) given by

    ω_t(ρ,θ)=w_t(θ),

independent of ρ.




As we have stated above, the region Σ_t is invariant under the maps
λ↦1/λ and λ↦λ̅. The invariance of
μ_b_t under λ↦λ̅ follows from the fact that
w_t is even. Then in polar coordinates, the map λ↦1/λ
is (r,θ)↦(1/r,-θ). Now, we may compute μ_b_t in
polar coordinates as

    dμ_b_t=(  1/r^2w_t(θ))  r dr dθ
    =w_t(θ) 1/rdr dθ.

If s=1/r and ϕ=-θ, then r=1/s and

    1/rdr=s(  -1/s^2 )  ds=-1/s ds,

so that dμ_b_t=w_t(ϕ) 1/sds dϕ. Similarly, if
ρ=log r, then dρ=(1/r) dr, so that dμ_b_t=w_t
(θ) dρ dθ.


Plots of w_t(θ) are shown in Figure <ref>. Note that for
t<4, not all angles θ actually occur in the domain Σ_t.
Thus, for t<4, the function w_t(θ) is only defined for θ in
a certain interval (-θ_max(t),θ_max(t))—where, as shown in
Section <ref>, θ_max(t)=cos^-1(1-t/2). Plots
of W_t for t=1 and t=4 are then shown in Figures <ref>,
<ref>, and <ref>. Actually, when t=1, the
function w_t is almost constant (see Figure <ref>). Thus,
the variation in W_t in Figure <ref> comes almost entirely
from the variation in the factor of 1/r^2 in (<ref>).

We also observe that by Point <ref> of Corollary
<ref>, half the mass of μ_b_t is contained in the
unit disk and half in the complement of the unit disk. Thus, although the
density W_t becomes large near the origin in, say, Figures
<ref> and <ref>, it is not correct
to say that most of the mass of μ_b_t is near the origin.







































 §.§ A connection to free unitary Brownian motion


It follows easily from Theorem <ref> that the distribution of the
argument of λ with respect to μ_b_t has a density given by

    a_t(θ)=2log[r_t(θ)]w_t(θ),

where, as in Theorem <ref>, we take r_t(θ) to be the outer
radius of the domain (with r_t(θ)>1). After all, the Brown measure in
the domain is computed in polar coordinates as (1/r^2)w_t(θ
)r dr dθ. Integrating with respect to r from 1/r_t(θ) to
r_t(θ) then gives the claimed density for θ.

Recall from Theorem <ref> that the limiting eigenvalue distribution
ν_t for Brownian motion in the unitary group was determined by Biane. We
now claim that the distribution in (<ref>) is related to Biane's
measure ν_t by a natural change of variable. To each angle θ
arising in the region Σ_t, we associate another angle ϕ by the
formula

    f_t(r_t(θ)e^iθ)=e^iϕ,

where f_t is as in (<ref>). (Recall that, by Definition
<ref>, the boundary of Σ_t maps into the unit circle under
f_t.) We then have the following remarkable direct connection between the
Brown measure of b_t and Biane's measure ν_t.


If θ is distributed according to the density
in (<ref>) and ϕ is defined by (<ref>), then ϕ
is distributed as Biane's measure ν_t.


















We may think of this result in a more geometric way, as follows. Define a map

    Φ_t:Σ_t→ S^1

by requiring (a) that Φ_t should agree with f_t on the boundary of
Σ_t, and (b) that Φ_t should be constant along each radial
segment inside Σ_t, as in Figure <ref>. (This
specification makes sense because f_t has the same value at the two
boundary points on each radial segment.) Explicitly, Φ_t may computed
as

    Φ_t(λ)=f_t(r_t(λ)e^iλ).

Then Proposition <ref> gives the following result, which
may be summarized by saying that the distribution ν_t of free
unitary Brownian motion is a “shadow” of
the Brown measure of b_t.


The push-forward of the Brown measure of b_t
under the map Φ_t is Biane's measure ν_t on S^1. Indeed, the
Brown measure of b_t is the unique measure μ on Σ_t with the following two properties: (1) the push-forward of μ
by Φ_t is ν_t and (2) μ is absolutely continuous with respect
to Lebesgue measure with a density W having the form

    W(r,θ)=1/r^2g(θ)

in polar coordinates, for some continuous function g.


Now, the results of <cit.> and <cit.> already indicate a
relationship between the free unitary Brownian motion u_t (whose spectral
measure is ν_t) and the free multiplicative Brownian motion b_t
(whose Brown measure we are studying in this paper). It is nevertheless
striking to see such a direct relationship between μ_b_t and ν_t. Indeed, Proposition <ref> precisely mirrors the
relationship between the semicircle law and the circular law. If c_t is a
circular random variable of variance t, and x_t is semicircular of
variance t, then the distribution of x_t (the semicircle law on the
interval [-2√(t),2√(t)]) is the push-forward of the Brown measure of
c_t (the uniform probability measure on the disk 𝔻
(√(t)) of radius √(t)) under a similar “shadow
map”: first project the disk onto its upper boundary circle
via (x,y)↦(x,√(t-x^2)), and then use the conformal map z↦
z+t/z from ℂ∖𝔻(√(t)) onto
ℂ∖-2√(t),2√(t)]. (The net result of these
two operations is (x,y)↦2x.) Since, as described in the introduction,
u_t and b_t are the “Lie group”
 versions of the “Lie algebra” operators
x_t and c_t, it is pleasing that this shadow relationship between
their Brown measures persists.



 §.§ The structure of the formula


We now explain the significance of the two terms on the right-hand side of the
formula (<ref>) for w_t. Following the general construction of
Brown measures in (<ref>), the density of the Brown measure is
computed as 1/4πΔ s_t(λ), where Δ is the
Laplacian with respect to λ and where

    s_t(λ)=lim_ε→0^+τlog
    ((b_t-λ)^∗(b_t-λ)+ε)].

It is then convenient to work in polar coordinates (r,θ). In these
coordinates, we may write Δ as

    Δ=1/r^2(  (  r∂/∂ r)
    ^2+∂^2/∂θ^2)  .




For each t>0, the function s_t(λ) is
real analytic for λ∈Σ_t and also for λ∈
(Σ_t)^c. At each boundary point, s_t(λ) and its
first derivatives with respect to λ approach the same value from the
inside of Σ_t as from the outside of Σ_t. For
λ inside Σ_t, we have

    (  r∂/∂ r)  ^2s_t=2/t.

For λ inside Σ_t, we also have that ∂ s_t
/∂θ is independent of r with t and θ fixed.
Indeed, ∂ s_t/∂θ(λ) is the unique function
on Σ_t that is independent of r and agrees with the angular
derivative of log(|λ-1| ^2) as we approach
∂Σ_t.


The formula (<ref>)—along with (<ref>)—accounts for
the first term on the right-hand side of (<ref>). Then since the
angular derivative of log(|λ-1| ^2) is
computable as

    ∂/∂θlog(|λ-1|
    ^2)=2Imλ/|λ-1| ^2
    =2rsinθ/r^2+1-2cosθ

as in (<ref>), we can recognize the second term on the right-hand
side of (<ref>) as the θ-derivative of ∂
s_t/∂θ. Thus, Theorem <ref>, together
with the formula (<ref>) for the Laplacian in polar coordinates,
accounts for the formula (<ref>) for w_t.



 §.§ Deriving the formula


We now briefly indicate the method we will use to compute the Brown measure
μ_b_t. Following the general construction of the Brown measure in
(<ref>), we consider the function S defined by

    S(t,λ,ε)=τlog((b_t-λ)^∗(b_t
    -λ)+ε)]

for λ∈ℂ and ε>0, where b_t is the free
multiplicative Brownian motion and τ is the trace in the von Neumann
algebra in which b_t lives. It is easily verified that as ε
decreases with t and λ fixed, S(t,λ,ε) also
decreases. Hence, the limit

    s_t(λ)=lim_ε→0^+S(t,λ,ε)

exists, possibly with the value -∞.

The general theory developed by Brown <cit.> shows that s_t(λ) is
a subharmonic function of λ for each fixed t, so that the Laplacian
(in the distribution sense) of s_t(λ) with respect to λ is a
positive measure. If this measure happens to be absolutely continuous with
respect to the Lebesgue measure, then the density W(t,λ) of the Brown
measure is computed in terms of the value of s_t(λ), as follows:

    W(t,λ)=1/4πΔ s_t(λ).

See also Chapter 11 in <cit.> and Section 2.3 in <cit.> for general
information on Brown measures.

The first major step toward proving Theorem <ref> is the following result.


The function S in (<ref>) satisfies the
following PDE:

    ∂ S/∂ t=ε∂ S/∂ε(  1+(|λ| ^2-ε
    )∂ S/∂ε-a∂ S/∂ a
    -b∂ S/∂ b)  ,  λ=a+ib,

with the initial condition

    S(0,λ,ε)=log(|λ-1| ^2
    +ε).




We emphasize that S(t,λ,ε) is only defined for
ε>0. Although, as we will see, lim_ε→
0^+S(t,λ,ε) is finite, ∂ S/∂ε
develops singularities in this limit. Thus, it is not correct to
formally set ε=0 in (<ref>) to obtain ∂
s_t/∂ t=0. (Actually, it will turn out that s_t(λ)
is independent of t for as long as λ remains outside
Σ_t, but not after this time; see Section
<ref>.)

After verifying this equation (Section <ref>), we will use the
Hamilton–Jacobi formalism to analyze the solution (Section <ref>). In
the remaining sections, we will then analyze the limit of the solution as
ε tends to zero and compute the Laplacian in (<ref>). The
expository article <cit.> of the second author provides an
introduction to the methods used in the present paper.

By way of comparison, we mention that a similar PDE was used in Biane's paper
<cit.>. There he studies the spectral measure μ_t of
x_0+x_t, the free additive Brownian motion with a nonconstant initial
distribution x_0 freely independent from x_t. Biane studies the Cauchy
transform G of μ_t:

    G(t,z)=∫_ℝμ_t(dx)/z-x,  Im(z)>0,

and shows that G satisfies the complex inviscid Burger's equation

    ∂ G(t,z)/∂ t=-G(t,z)∂ G/∂ z.

The measure μ_t may then be recovered, up to a constant, as
lim_ε→0^+ImG(t,x+iε).

In our paper, we similarly use a first-order, nonlinear PDE whose solution in
a certain limit gives the desired measure. We note, however, that the PDE
(<ref>) is not actually the main source of information about μ_t
in <cit.>. By contrast, our analysis of the Brown measure of
the free multiplicative Brownian motion b_t is based entirely on the
PDE in Theorem <ref>.

Finally, we mention that for the case of the circular Brownian motion c_t,
a PDE similar to the one in Theorem <ref> appeared in work of Burda,
Grela, Nowak, Tarnowski, and Warchoł <cit.>.



§ COMPARISON WITH THE EIGENVALUE DISTRIBUTION OF B_T^N



























































As mentioned in Section <ref>, the Brown measure of the free
multiplicative Brownian motion b_t is a natural candidate for the limiting
empirical eigenvalue distribution of the Brownian motion B_t^N in
𝖦𝖫(N;ℂ). We may express this idea formally as a conjecture.


For all t>0, the empirical eigenvalue
distribution of B_t^N converges almost surely as N→∞ to
the Brown measure μ_b_t of b_t.


While natural, Conjecture <ref> is technically
difficult to approach. It is by now well known that the logarithmic
singularity in the definition of the Brown measure can result in failure of
convergence of the empirical distribution of eigenvalues to the Brown measure
of the limit (in ∗-distribution) of the random matrix ensembles. Suppose,
for example, T_N is an N× N matrix with all 0 entries except for
1's just below the diagonal. Then all of T_N's eigenvalues are 0, and
hence the empirical eigenvalue distribution is a point-mass at 0 for each
N. However, in ∗-distribution, T_N converges to a Haar unitary u,
whose Brown measure is the uniform probability measure on the unit circle.
(See, for example, Section 2.6 of <cit.>.)

In <cit.>, Śniady proved that convergence to the Brown measure is
“generic”, in the sense that a small
(vanishing in the limit) independent Gaussian perturbation of the original
ensemble will always yield convergence to the Brown measure. (The required
size of the perturbation was more recently explored in <cit.>.) A main
step in Śniady's proof is to show (in our language) that the empirical
eigenvalue distribution of a complex Brownian motion Z_t on gl
(N;ℂ), with any deterministic initial condition, converges to the
appropriate Brown measure. That is to say, there is enough regular noise in
such matrix diffusions to kill any pseudo-spectral discontinuities. It is
natural to expect that the same should hold true for our geometric matrix
diffusion B_t.

In support of Conjecture <ref>, we first note that for
small t, the distribution of B_t^N—namely, the time-t heat kernel
measure on GL(N;ℂ), based at the identity—is
approximately Gaussian. Thus, B_t^N is distributed, for small t,
similarly to the Ginibre ensemble, shifted by the identity and scaled by
√(t). Thus, Conjecture <ref> leads us to expect
that that μ_b_t will be close, for small t, to the uniform
probability measure on a disk of radius √(t). This expectation is
confirmed by the asymptotics in Section <ref>.

We now offer several numerical tests of the conjecture. First, Figure
<ref> directly compares the density W_t with t=2
to the distribution of eigenvalues of B_t^N with t=2 and N=2,000.
Next, in light of Conjecture <ref> and Point
<ref> of Corollary <ref>, we expect that the
limiting distribution of the logarithms of the eigenvalues of
B_t^N will be constant in the horizontal direction. This expectation is
confirmed by simulations, as in the right-hand side of Figure
<ref>.

Furthermore, Conjecture <ref> predicts that the
large-N distribution of the arguments of the eigenvalues will be given by
the density a_t in (<ref>). Furthermore, Conjecture
<ref> and Proposition <ref>
predict that, if {λ_j}_j=1^N are the eigenvalues of B_t
^N, then for large N, the empirical distribution of {Φ_t
(λ_j)}_j=1^N will be given by Biane's measure ν_t. Both of
these predictions are confirmed by simulations; see Figures
<ref> and <ref>.



§ PROPERTIES OF Σ_T


We now verify some important properties of the regions Σ_t in
Definition <ref>. Define

    T(λ)={[ |λ-1| ^2log(|λ|
    ^2)/|λ| ^2-1,                         |λ|≠1;                     |λ-1| ^2,                        |λ| =1 ].  .

Note that the function

    x↦log(x)/x-1

has a removable singularity at x=1, with a limiting value of 1 at x=1.
Thus, T(λ) is a real analytic function on all of ℂ
∖{0}. Since, also,

    lim_x→0log(x)/x-1=+∞,

we see that T(λ)→+∞ as λ→0. By
checking the three cases |λ| >1, |λ| =1, and |λ| <1, we may verify
that T(λ)≥0 for all λ, with equality only if λ=1.


For all t>0, the region Σ_t may be
expressed as

    Σ_t={.  λ∈ℂ| T(λ
    )<t}

and the boundary of Σ_t may be expressed as

    ∂Σ_t={.  λ∈ℂ|
    T(λ)=t}  .




Thus, each fixed λ∈ℂ will be outside Σ
_t until t=T(λ) and will be inside Σ_t for all
t>T(λ). We may therefore say that T(λ) is the time
that the domain Σ_t gobbles up λ. See Figures
<ref> and <ref>.


For each t>0, the region Σ_t has the
following properties.



  * For t≤4, we have |λ|≤cos^-1(1-t/2) for all λ∈Σ_t, with equality precisely for the points on the unit circle with
cosθ=1-t/2.


  * Consider the ray from the origin with angle
θ; if t≤4, assume |θ| <cos^-1
(1-t/2). Then this ray intersects Σ_t precisely in an open interval
of the form 1/r_t(θ)<r<r_t(θ) for some r_t(θ)>1.


  * The boundary of Σ_t is smooth for all
t>0 with t≠4. When t=4, the boundary of Σ_t is smooth except
at λ=-1, near which it looks like the transverse intersection of two
smooth curves.


  * The region Σ_t is invariant under λ↦1/λ and
under λ↦λ̅.


  * The region Σ_t coincides with the one defined by Biane in
<cit.>.



For Point <ref>, see Figure <ref>.





























We now begin working toward the proofs of Theorems <ref> and
<ref>.


For λ∈ℂ with |λ|≠1, we have | f_t(λ)| =1 if and
only if T(λ)=t.



Since f_t(0)=0, we must have λ≠0 if | f_t
(λ)| is going to equal 1. For nonzero λ, we compute
that

    log(| f_t(λ)| )       =log|λ| +Re(  t/21+λ/1-λ) 
         =log|λ| +t/21-|λ| ^2/|λ-1| ^2.

Thus, for nonzero λ, the condition | f_t(λ
)| =1 is equivalent to

    0=log|λ| +t/21-|λ| ^2/|λ-1| ^2.

When |λ|≠1, this condition simplifies to

    t=|λ-1| ^2log(|λ|
    ^2)/|λ| ^2-1=T(λ),

as claimed.


We now state some important properties of the function r_t occurring in
the statement of Theorem <ref>; the proof is given on p.
setFtproof.


Consider a real number t>0 and an angle θ∈
(-π,π], where if t≤4, we require |θ|
<cos^-1(1-t/2). Then there exist exactly two radii r≠1 for which
| f_t(re^iθ)| =1, and these radii have the form
r=r_t(θ) and r=1/r_t(θ) with r_t(θ)>1. Furthermore,
r_t(θ) depends analytically on θ and if t≤4, then
r_t(θ)→1 as θ→±cos^-1(1-t/2).

If t≤4 and θ∈(-π,π] satisfies |θ|≥cos^-1(1-t/2), then there are no radii r≠1 with |
f_t(r)| =1.


Using the proposition, we can now compute the sets F_t and E_t
=F_t that enter into the definition of Σ_t. (Recall
(<ref>) and (<ref>).)















For t≤4, the set F_t consists of points of the form
r_t(θ)e^iθ and (1/r_t(θ))e^iθ for -cos
^-1(1-t/2)<θ<cos^-1(1-t/2). In this case, the closure of F_t
consists of F_t together with the points e^iθ on the unit circle
with cosθ=1-t/2. There are two such points when t<4 and one such
point when t=4, namely -1.

For t>4, the set F_t consists of points of the form r_t
(θ)e^iθ and (1/r_t(θ))e^iθ, where θ ranges
over all possible angles, and this set is closed.


See Figure <ref>.













We now set out to prove Proposition <ref>. In the proof, we will
always rewrite the equation | f_t(λ)| =1, for
|λ|≠1, as T(λ)=t (Lemma
<ref>).


Let us write the function T in (<ref>) in polar
coordinates. Then for each θ, the function r↦ T(r,θ) is
strictly decreasing for 0<r<1 and strictly increasing for r>1. For each
θ, the minimum value of T(r,θ), achieved at r=1, is
2(1-cosθ), and we have

    lim_r→0T(r,θ)=lim_r→+∞T(r,θ
    )=+∞.





We will show in Proposition <ref> that the function T(λ)
is the limit of another function t_∗(λ_0,ε_0) as
ε_0 goes to zero. Explicitly, this amounts to saying that
T(r,θ)=g_θ(δ), where g is defined in (<ref>) and
δ=r+1/r. Now, δ is decreasing for 0<r<1 and increasing for
r>1. Thus, the claimed monotonicity of T follows if g_ θ(δ)
an increasing function δ for each θ, which we will show in the
proof of Proposition <ref>.

For the convenience of the reader, we briefly outline how the argument goes in
the context of the function T(r,θ). We note that

    T(r,θ)=(r^2+1-2rcosθ)log(r^2)/r^2-1,

where if we assign log(r^2)/(r^2-1) the value 1 at r=1, then T is
analytic except at r=0. We then compute that, after simplification,

    ∂ T/∂ r=2[  -2r+(1+r^2)cosθ/(r^2-1)^2]  log(r^2)+r^2+1-2rcosθ/r^2-1
    2/r.

We then claim that for all θ, we have ∂ T/∂ r>0 for
r>1 and ∂ T/∂ r<0 for r<1. Note that for each fixed r,
the right-hand side of (<ref>) depends linearly on cosθ. Thus,
if, for a fixed r, if ∂ T/∂ r is positive when cosθ=1 and when cosθ=-1, it will be positive for all θ.
Specifically, we may say that

    ∂ T/∂ r(r,θ)≥min(  ∂
    T/∂ r(r,0),∂ T/∂ r(r,π))  .

It is now an elementary (if slightly messy) computation to check that the
right-hand side of (<ref>) is strictly positive for all r>1. A
similar argument then shows that ∂ T/∂ r is negative for all
θ and all 0<r<1.

We conclude that for each θ, the function r↦ T(r,θ) is
decreasing for 0<r<1 and increasing for r>1. The minimum value therefore
occurs at r=1, and this value is the value of r^2+1-2cosθ at
r=1, namely 2(1-cosθ). Finally, we can easily see that for r
approaching zero, we have

    T(r,θ)∼-log(r^2)→+∞

and for r approaching infinity, we have

    T(r,θ)∼log(r^2)→+∞,

as claimed.


The minimum value of
T(r,θ), achieved at r=1, is 2-2cosθ. This value is always
less than t, as can be verified separately in the cases t>4 (all θ)
and t≤4 (|θ| <cos^-1(1-t/2)). Thus, Lemma
<ref> tells us that the equation T(r,θ)=t has exactly one
solution for r with 0<r<1 and exactly one solution for r>1. Since, as is
easily verified, T(1/r,θ)=T(r,θ), the two solutions are
reciprocals of each other, and we let r_t(θ) denote the solution with
r>1. Since ∂ T/∂ r is nonzero for all r≠1, the implicit
function theorem tells us that r_t(θ) depend analytically on
θ.

Now, if t≤4 and θ approaches ±cos^-1(1-t/2), the minimum
value of 2-2cosθ—achieved at r=1—approaches 2-2(1-t/2)=t. It
should then be plausible that r_t(θ) will approach r=1. To make
this claim rigorous, we need to show that T(r,θ) increases rapidly
enough as r increases from 1 that the T(r,θ)=t is achieved close to
r=1. To that end, let g(r) denote the function on the right-hand side of
(<ref>), which is continuous everywhere and strictly positive for
r>1. Then for r>1, we have

    T(r,θ)-(2-2cosθ)≥ G(r):=∫_1^rg(s) ds.

Now, G(r) is continuous and strictly increasing for r≥0, with G(1)=0.
Thus, G it has a continuous inverse function satisfying G^-1(0)=1.

For ε>0, choose δ>0 so that G^-1(R)<1+ε when
0<R<δ. Then take θ sufficiently close to ±cos^-1(1-t/2)
that 2-2cosθ is within δ of t. Then

    G^-1(t-(2-2cosθ))<1+ε,

which is to say that there is an R with 1<R<1+ε such that

    ∫_1^Rg(s) ds=t-(2-2cosθ).

From (<ref>) we can then see that T(R,θ)>t. Thus, r_t
(θ) will satisfy

    1<r_t(θ)<R<1+ε.

We have therefore shown that r_t(θ)→1 as θ→±cos^-1(1-t/2).

Finally, if t≤4 and θ∈(-π,π] satisfies |θ|≥cos^-1(1-t/2), the minimum value of T(r,θ),
achieved at r=1, is 2-2cosθ≥ t. Thus, there are no values of
r≠1 where T(r,θ)=t.


We are now ready for the proofs of our main results about Σ_t.

We first claim that the set
E_t=F_t is precisely the set where T(λ)=t. To see
this, first note that F_t is, by Lemma <ref>, the set of
λ with |λ|≠1 where T(λ)=t.
Then by Corollary <ref>, the closure of F_t is obtained by
adding in the points on the unit circle (zero, one, or two such points,
depending on t) where cosθ=1-t/2. But these points are easily seen
to be the points on the unit circle where T(λ)=t.

Using Corollary <ref>, we see that the complement of the set
E_t={λ|T(λ)=t} has two connected components
when t<4 and three connected components when t≥4. Since T(1)=0<t, we
have T(λ)<t on the entire connected component of E_t^c
containing 1, which is, by definition, the region Σ_t. The remaining
components of E_t^c are the unbounded component and (for t≥4) the
component containing 0. Since T(λ) tends to +∞ at zero and at
infinity, we see that T(λ)>t on these regions, so that T(λ)<t
precisely on Σ_t.

It is also clear from Corollary <ref> that the boundary of the
region Σ_t (i.e., the connected component of E_t^c containing
1) contains the entire set E_t= {λ|T(λ)=t}  .


Point <ref>
follows easily from Corollary <ref>. For Point
<ref>, we note that by Proposition <ref>, we
have T(r,θ)<t for 1/r_t(θ)<r<r_t(θ), and T(r,θ
)≥ t for 0<r≤1/r_t(θ) and for r≥ r_t(θ). Thus, by
Theorem <ref>, the ray with angle θ intersects
Σ_t precisely in the claimed interval.

For Point <ref>, we have already shown that ∂
T/∂ r is nonzero except when r=1. When r=1, we know from
(<ref>) that

    T(r,θ)=|λ-1| ^2=2-2cosθ.

Thus, when r=1, we have ∂ T/∂θ=2sinθ, which is
nonzero except when θ=0 or θ=π. Thus, the gradient of
T(λ) is nonzero except when λ=0 (where T(λ) is
undefined), when λ=1, and when λ=-1. Since 0 is never in
Σ_t and 1 is always in Σ_t, the only possible singular point
in the boundary of Σ_t is at λ=-1. Since T(r,θ
)=2-2cosπ=4 when r=1 and θ=π, the point λ=-1 belongs to
the boundary of Σ_4.

Meanwhile, the Taylor expansion of T to second order at λ=-1 is
easily found to be T(λ)≈4+(Reλ
+1)^2/3-(Imλ)^2. By the Morse lemma, we can then
make a smooth change of variables so that in the new coordinate system,

    T(u,v)=4+u^2-v^2=4+(u+v)(u-v).

Thus, near λ=-1, the set T(λ)=4 is the union of the curves
u+v=0 and u-v=0.

The invariance of Σ_t under λ↦1/λ and under
λ↦λ̅ follows from the easily verified invariance of
T(λ) under these transformations. Finally, we verify that the domain
Σ_t, as we have defined it, coincides with the one originally
introduced by Biane in <cit.>. Let us start with the case t<4.
According to the discussion at the bottom of p. 273 in <cit.>, the
boundary of Biane's domain Σ_t consists in this case of two analytic
arcs. The interior of one arc lies in the open unit disk and the interior of
the other arc lies in the complement of the closed unit disk, while the
endpoints of both arcs lie on the unit circle. The first arc is then computed
by applying a certain holomorphic function χ(t,·) to the support of
Biane's measure ν_t in the unit circle. Now, χ(t,·) satisfies
f_t(χ(t,z))=z on the closed unit disk. (Combine the identity involving
κ on p. 266 of <cit.> with the definition of χ on p.
273.) We see that the interior of the first arc consists of points with
|λ|≠1 but | f_t(λ
)| =1. This arc must, therefore, coincide with the arc of points
with radius 1/r_t(θ). The second arc is obtained from the first by
the map λ↦1/λ and therefore coincides with the points of
radius r_t(θ). We can now see that the boundary of Biane's domain
coincides with the boundary of the domain we have defined. A similar analysis
applies to the cases t>4 and t=4, using the description of the boundary of
Σ_t in those cases at the top of p. 274 in <cit.>.




§ THE PDE FOR S


In this section, we will verify the PDE for S in Theorem <ref>.
The claimed initial condition (<ref>) holds because b_0=1. We
now proceed to verify the equation (<ref>) itself.

Let (c_t)_t≥0 denote a free circular Brownian motion. The rules of
free stochastic calculus, in “stochastic
differential” form, are as follows; see <cit.>. If g_t and h_t are processes adapted to
c_t, then

    dc_t g_t dc_t^∗     =dc_t^∗ g_t dc_t=τ
    (g_t) dt
    
    dc_t g_t dc_t     =dc_t^∗ g_t dc_t^∗=0
    
    dc_t dt       =dc_t^∗ dt=0
    τ(g_t dc_t h_t)       =τ(g_t dc_t^∗ h_t)=0.

In addition, we have the following Itô product rule: if a_t^1
,…,a_t^n are processes adapted to c_t, then

    d(a_t^1⋯ a_t^n)       =∑_j=1^n(a_t^1⋯ a_t
    ^j-1) da_t^j (a_t^j+1⋯ a_t^n)
         +∑_1≤ j<k≤ n(a_t^1⋯ a_t^j-1) da_t^j
     (a_t^j+1⋯ a_t^k-1) da_t^k (a_t^k+1⋯ a_t^n).



We let b_t be the free multiplicative Brownian motion, which satisfies the
free stochastic differential equation

    db_t=b_t dc_t,   b_0=1.

Throughout the rest of this section, we will use the notation

    b_t,λ:=b_t-λ.




We have

    ∂/∂ tτ(b_t,λ^∗b_t,λ
    )^n]=n∑_j=0^n-1τ(b_t,λ^∗b_t,λ
    )^j]τ b_tb_t^∗(b_t,λb_t,λ^∗
    )^n-j-1].




Note that second factor on the right-hand side of (<ref>) has
(b_t,λb_t,λ^∗)^n-j, with the adjoint on the
second factor.


Note that db_t,λ=db_t=b_t dc_t and db_t,λ^∗=db_t^∗=dc_t^∗ b_t^∗. We compute dτ
(b_t,λ^∗b_t,λ)^n] by moving the d inside the trace
and then applying the product rule in (<ref>) and
(<ref>). By (<ref>), the terms arising from
(<ref>) will not contribute. Furthermore, by (<ref>), the
only terms from (<ref>) that contribute are those where one d
goes on a factor of b_t,λ and one goes on a factor of b_t,λ^∗.

By choosing all possible factors of b_t,λ and all possible factors
of b_t,λ^∗, we get n^2 terms. In each term, after putting
the d inside the trace, we can cyclically permute the factors until, say,
the db_t,λ factor is at the end. There are then only n
distinct terms that occur, each of which occurs n times. By
(<ref>), each distinct term is computed as

    τ(b_t,λ^∗b_t,λ)^jdc_t^∗ b_t
    ^∗b_t,λ(b_t,λ^∗b_t,λ)^n-j-2b_t,λ^∗b_t dc_t]
         =τ b_t^∗b_t,λ(b_t,λ^∗b_t,λ)^n-j-2b_t,λ^∗b_t]τ(b_t,λ^∗b_t,λ)^j] dt
         =τ(b_t,λ^∗b_t,λ)^j]τ b_t
    b_t^∗(b_t,λb_t,λ^∗)^n-j-1] dt.

(The reader who doubts the validity of using the cyclic invariance of the
trace when some factors are differentials may compute each term by
first using (<ref>) and then using the cyclic
invariance of the trace, with the same result.) Since each distinct term
occurs n times, we obtain

    dτ(b_t,λ^∗b_t,λ)^n]=n∑_j=0^n-1
    τ(b_t,λ^∗b_t,λ)^j]τ b_t
    b_t^∗(b_t,λb_t,λ^∗)^n-j-1] dt

as claimed.



The function S in (<ref>) satisfies

    ∂ S/∂ t=ετ(b_t,λ^∗b_t,λ+ε)^-1]τ b_tb_t^∗(b_t,λb_t,λ^∗+ε)^-1].




Of course, since b_t=b_t,λ+λ, we can rewrite (<ref>)
in a way that involves only b_t,λ and not b_t.


We note that the definition of S actually makes sense for all ε∈ℂ with Re(ε)>0, using the standard
branch of the logarithm function. We note that for |ε| >| z| , we have

    1/z+ε     =1/ε(  1-(  -z/ε)  )  
         =1/ε[  1-z/ε+z^2
    /ε^2-z^3/ε^3+⋯]  .

Integrating with respect to z gives

    log(z+ε)=logε+∑_n=1^∞(-1)^n-1/n(  z/ε)  ^n.

Thus, for |ε| >‖ b_t^∗
b_t‖ , we have

    τlog(b_t,λ^∗b_t,λ+ε)]=logε+ ∑_n=1^∞(-1)^n-1/nε^n
    τ(b_t,λ^∗b_t,λ)^n].



Assume for the moment that it is permissible to differentiate (<ref>)
term by term with respect to t. Then by Lemma <ref>, we have

    ∂ S/∂ t=∑_n=1^∞(-1)^n-1/ε^n∑_j=0^n-1τ(b_t,λ^∗b_t,λ)^j]τ b_tb_t^∗(b_t,λb_t,λ^∗)^n-j-1].

Now, by <cit.>, the map t↦ b_t is continuous
in the operator norm topology; in particular, ‖ b_t‖
is a locally bounded function of t. From this observation, it is easy to see
that the right-hand side of (<ref>) converges locally uniformly in t.
Thus, a standard result about interchange of limit and derivative (e.g.,
Theorem 7.17 in <cit.>) shows that the term-by-term differentiation
is valid.

Now, in (<ref>), we let k=j and l=n-j-1, so that n=k+l+1. Then k
and l go from 0 to ∞, and we get

    ∂ S/∂ t=ε(  1/ε
    ∑_k=0^∞(-1)^k/ε^kτ(b_t,λ^∗b_t,λ)^k])  (  1/ε∑
    _l=0^∞(-1)^l/ε^lτ b_tb_t^∗(b_t,λb_t,λ^∗)^l])  .

(We may check that the power of ε in the denominator is k+l+1=n
and that the power of -1 is k+l=n-1.) Thus, moving the sums inside the
traces and using (<ref>), we obtain the claimed form of ∂
S/∂ t.

We have now established the claimed formula for ∂ S/∂ t for
ε in the right half-plane, provided |ε| is sufficiently large, depending on t and λ. Since,
also, S(0,λ,ε)=log(|λ-1|
^2+ε), we have, for sufficiently large |ε| ,

    S(t,λ,ε)=log(|λ-1| ^2
    +ε)+∫_0^tετ(b_s,λ^∗b_s,λ+ε)^-1]τ b_sb_s^∗(b_s,λb_s,λ^∗+ε)^-1] ds.

We now claim that both sides of (<ref>) are well-defined,
holomorphic functions of ε, for ε in the right
half-plane. This claim is easily established from the standard power-series
representation of the inverse:

    (A+ε+h)^-1     =(A+ε)^-1(1+h(A+ε)^-1
    )^-1
         =(A+ε)^-1∑_n=0^∞(-1)^nh^n(A+ε)^-n,

and a similar power-series representation of the logarithm. Thus,
(<ref>) actually holds for all ε in the right
half-plane. Differentiating with respect to t then establishes the claimed
formula (<ref>) for dS/dt for all ε in the right half-plane.



We have the following formulas for the derivatives
of S with respect to ε and λ:

    ∂ S/∂ε     =τ(b_t,λ^∗b_t,λ+ε)^-1]
    ∂ S/∂λ     =-τ b_t,λ^∗(b_t,λ^∗b_t, λ+ε)^-1]
    ∂ S/∂λ̅     =-τ b_t,λ(b_t,λ^∗b_t,λ+ε)^-1].





We use the formula for the derivative of the trace of a logarithm (Lemma 1.1
in <cit.>):

    d/duτlog(f(u))]=τ[  f(u)^-1df/du]
    .

(We emphasize that there is no such simple formula for the derivative of
log(f(u)) without the trace, unless df/du commutes with f(u).) The
lemma easily follows from this formula.


We are now ready for the verification of the differential equation for S.

We note that

    b_t,λ(b_t,λ^∗b_t,λ+ε)=(b_t,λb_t,λ^∗+ε)b_t,λ.

Multiplying by (b_t,λ^∗b_t,λ+ε)^-1 on the
right and (b_t,λb_t,λ^∗+ε)^-1 on the left
gives a useful identity:

    (b_t,λb_t,λ^∗+ε)^-1b_t,λ
    =b_t,λ(b_t,λ^∗b_t,λ+ε)^-1.

Replacing b_t,λ by its adjoint gives another version of the
identity:

    b_t,λ^∗(b_t,λb_t,λ^∗+ε
    )^-1=(b_t,λ^∗b_t,λ+ε)^-1b_t,λ
    ^∗.

Note that in both (<ref>) and (<ref>), both side have the same
pattern of starred and unstarred variables, always with the two outer
variables being the same and the middle one being different.

We also claim that

    τ(b_t,λ^∗b_t,λ+ε)^-1]=τ(b_t,λb_t,λ^∗+ε)^-1].

To verify this identity for large ε, we replace z by
b_t,λ^∗b_t,λ in the series (<ref>) and note that by
the cyclic invariance of the trace,

    τ(b_t,λ^∗b_t,λ)^n]=τ(b_t,λb_t,λ^∗)^n].

The result for general ε follows by an analyticity argument as in
the proof of Lemma <ref>.

We start from the formula for ∂ S/∂ t in Lemma
<ref>. Noting that

    b_tb_t^∗     =(b_t,λ+λ)^∗(b_t,λ+λ)
         =b_t,λb_t,λ^∗+λ b_t,λ^∗
    +λ̅b_t,λ+|λ| ^2,

we expand the second factor on the right-hand side of (<ref>) as

    τ b_tb_t^∗(b_t,λb_λ^∗+ε
    )^-1]       =τ b_t,λb_t,λ^∗(b_t,λb_t,λ^∗+ε)^-1]
         +λτ b_t,λ^∗(b_t,λb_t,λ^∗+ε)^-1]
         +λ̅τ b_t,λ(b_t,λb_t,λ^∗+ε)^-1]
         +|λ| ^2τ(b_t,λb_t,λ^∗+ε)^-1].

We then simplify the first term by writing b_t,λb_t,λ^∗=b_t,λb_t,λ^∗+ε-ε. In the middle
two terms, we use (<ref>), (<ref>), and cyclic invariance of the
trace. Using also (<ref>), we get

    τ b_tb_t^∗(b_t,λb_λ^∗+ε
    )^-1]       =1+(|λ| ^2-ε)τ(b_t,λ^∗b_t,λ+ε)^-1]
         +λτ b_t,λ^∗(b_t,λ^∗b_t,λ+ε)^-1]
         +λ̅τ b_t,λ(b_t,λ^∗b_t,λ+ε)^-1].



Thus,

    ∂ S/∂ t=ετ(b_t,λ^∗b_t,λ+ε)^-1](all the terms in (<ref>
    )).

All terms on the right-hand side of (<ref>) are expressible using Lemma
<ref> in terms of derivatives of S, and the claimed
differential equation follows.




§ THE HAMILTON–JACOBI METHOD




 §.§ Setting up the method


The equation (<ref>) is a first-order, nonlinear PDE of
Hamilton–Jacobi type. (The reader may consult, for example, Section 3.3 in
the book of Evans <cit.>, but we will give a brief self-contained
account of the theory in the proof of Proposition <ref>.) We
consider a Hamiltonian function obtained from the right-hand side of
(<ref>) by replacing each partial derivative with momentum variable,
with an overall minus sign. Thus, we define

    H(a,b,ε,p_a,p_b,p_ε)=-ε p_ε(1+(a^2+b^2)p_ε-ε p_ε-ap_a-bp_b).

We then consider Hamilton's equations for this Hamiltonian. That is to say, we
consider this system of six coupled ODEs:

    da/dt     =∂ H/∂ p_a;    db/dt=∂ H/∂ p_b;     dε/dt
    =∂ H/∂ p_ε;
    dp_a/dt     =-∂ H/∂ a;  dp_b/dt=-∂ H/∂ b;  dp_ε/dt
    =-∂ H/∂ε.

As convenient, we will let

    λ(t)=a(t)+ib(t).



The initial conditions for a, b, and ε are arbitrary:

    a(0)=a_0;   b(0)=b_0;  ε(0)=ε_0,

while those for p_a, p_b, and p_ε are determined by
those for a, b, and ε as follows:

    p_a(0)=2(a_0-1)p_0;   p_b(0)=2b_0p_0;   p_ε(0)=p_0,

where

    p_0=1/|λ_0-1| ^2+ε_0
    =1/(a_0-1)^2+b_0^2+ε_0.

The motivation for (<ref>) is that the momentum variables
p_a, p_b, and p_ε will correspond to the derivatives of
S along the curves (a(t),b(t),ε(t)); see (<ref>).
Thus, the initial momenta are simply the derivatives of the initial value
(<ref>) of S, evaluated at (a_0,b_0,ε_0).

For future reference, we record the value H_0 of the Hamiltonian at time
t=0.


The value of the Hamiltonian at t=0 is

    H_0=-ε_0p_0^2.





Plugging t=0 into (<ref>) and using (<ref>
) gives

    H_0=-ε_0p_0(1+(a_0^2+b_0^2)p_0-ε_0
    p_0-2a_0(a_0-1)p_0-2b_0^2p_0),

which simplifies to

    H_0=-ε_0p_0(1-p_0(a_0^2-2a_0+b_0^2+ε
    _0)).

But using the formula (<ref>) for p_0, we see that a_0
^2-2a_0+b_0^2+ε_0 equals 1/p_0-1, from which
(<ref>) follows.


The main result of this section is the following; the proof is given on p.
HJproof.


Assume λ_0≠0 and ε_0>0. Suppose a
solution to the system (<ref>) with initial conditions
(<ref>) and (<ref>) exists with
ε(t)>0 for 0≤ t<T. Then we have

    S(t,λ(t),ε(t))       =log(|λ_0-1|
    ^2+ε_0)-ε_0t/(|λ
    _0-1| ^2+ε_0)^2
         +log|λ(t)| -log|λ
    _0|

for all t∈0,T). Furthermore, the derivatives of S with respect to
a, b, and ε satisfy

    ∂ S/∂ε(t,λ(t),ε(t))     
    =p_ε(t);
    ∂ S/∂ a(t,λ(t),ε(t))       =p_a
    (t);
    ∂ S/∂ b(t,λ(t),ε(t))       =p_b(t).




Note that S(t,λ,ε) is only defined for ε>0.
Thus, (<ref>) and (<ref>) only make sense as long as the
solution to (<ref>) exists with ε(t)>0.

Since our objective is to compute Δ s_t(λ)=∂^2
s_t/∂ a^2+∂^2s_t/∂^2b^2, the formula
(<ref>) for the derivatives of S will ultimately be of as great
importance as the formula (<ref>) for S itself. We emphasize that
we are not using the Hamilton–Jacobi method to construct a solution
to (<ref>); the function S(t,λ,ε) is already defined
in (<ref>) in terms of free probability and is known (Theorem
<ref>) to satisfy (<ref>). Rather, we are using the
Hamilton–Jacobi method to analyze a solution that is already known
to exist.

We begin by briefly recapping the general form of the Hamilton–Jacobi method.


Fix an open set U⊂ℝ^n, a
time-interval [0,T], and a function H(𝐱,𝐩). Consider a
function S(t,𝐱) satisfying

    ∂ S/∂ t=-H(𝐱,∇_𝐱S),  𝐱∈ U, t∈0,T].

Consider a pair (𝐱(t),𝐩(t)) with 𝐱(t)∈ U,
𝐩(t)∈ℝ^n, and t∈0,T_1] with T_1≤
T. Assume this pair satisfies Hamilton's equations:

    dx_j/dt=∂ H/∂ p_j(𝐱(t),𝐩
    (t));  dp_j/dt=-∂ H/∂ x_j(𝐱
    (t),𝐩(t))

with initial conditions

    𝐱(0)=𝐱_0;  𝐩(0)=(∇_𝐱
    S)(0,𝐱_0).

Then we have

    S(t,𝐱(t))=S(0,𝐱_0)-H(𝐱_0,𝐩_0
    ) t+∫_0^t𝐩(s)·d𝐱/ds ds

and

    (∇_𝐱S)(t,𝐱(t))=𝐩(t).




Again, we are not trying to construct solutions to (<ref>), but rather
to analyze a solution that is already assumed to exist.


Take an arbitrary (for the moment) smooth curve 𝐱(t) and note that

    d/dtS(t,𝐱(t))       =∂ S/∂ t
    (t,𝐱(t))+∂ S/∂ x_j(t,𝐱
    (t))dx_j/dt
         =-H(𝐱(t),(∇_𝐱S)(t,𝐱(t)))+(∇
    _𝐱S)(t,𝐱(t))·d𝐱/dt,

where we use the Einstein summation convention. Let us use the notation

    𝐩(t)=(∇_𝐱S)(t,𝐱(t));

that is p_j(t)=∂ S/∂ x_j(t,𝐱(t)). Then
(<ref>) may be rewritten as

    d/dtS(t,𝐱(t))=-H(𝐱(t),𝐩(t))+𝐩
    (t)·d𝐱/dt.

If we can choose 𝐱(t) so that 𝐩(t) is somehow
computable, then the right-hand side of (<ref>) would be known and
we could integrate to get S(t,𝐱(t)).

To see how we might be able to compute 𝐩(t), we try
differentiating:

    dp_j/dt     =d/dt∂ S/∂ x_j
    (t,𝐱(t))
         =∂^2S/∂ t∂ x_j(t,𝐱(t))+∂^2S/∂ x_k∂ x_j(t,𝐱(t))dx_k
    /dt.

Now, from (<ref>), we have

    ∂^2S/∂ t∂ x_j     =∂^2
    S/∂ x_j∂ t
         =-∂/∂ x_jH(𝐱,∇_𝐱S)
         =-∂ H/∂ x_j(𝐱,∇_𝐱
    S)-∂ H/∂ p_k(𝐱,∇_𝐱
    S)∂^2S/∂ x_j∂ x_k.

Thus, (<ref>) becomes (suppressing the dependence on the path)

    dp_j/dt=-∂ H/∂ x_j+(  dx_k/dt-∂ H/∂ p_k)  ∂^2S/∂
    x_k∂ x_j.



If we now take 𝐱(t) to satisfy

    dx_j/dt=∂ H/∂ p_j,

the second term on the right-hand side of (<ref>) vanishes, and we
find that 𝐩(t) satisfies

    dp_j/dt=-∂ H/∂ x_j.

With this choice of 𝐱(t), (<ref>) becomes

    d/dtS(t,𝐱(t))=-H(𝐱_0,𝐩_0)+𝐩
    (t)·d𝐱/dt,

because H is constant along the solutions to Hamilton's equations.

Note that not all solutions (𝐱(t),𝐩(t)) to Hamilton's
equations (<ref>) and (<ref>) will arise by the above method. After
all, we are assuming that 𝐩(t)=(∇_𝐱S)(t,𝐱
(t)), from which it follows that the initial conditions (𝐱
_0,𝐩_0) will be of the form in (<ref>).

On the other hand, suppose we take a pair (𝐱_0,𝐩_0) as
in (<ref>). Let us then take 𝐱(t) to be the solution to

    dx_j/dt=∂ H/∂ p_j(𝐱(t),(∇
    _𝐱S)(t,𝐱(t))),  𝐱(0)=𝐱_0,

where since S is a fixed, “known”
 function, this ODE for 𝐱(t) will have unique solutions for as
long as they exist. If we set 𝐩(t)=(∇_𝐱
S)(t,𝐱(t)), then 𝐩(0)=𝐩_0 as in
(<ref>) and (<ref>) says that the pair (𝐱
(t),𝐩(t)) satisfies the first of Hamilton's equations. Applying
(<ref>) with this choice of 𝐱(t) shows that the pair
(𝐱(t),𝐩(t)) also satisfies the second of Hamilton's
equations. Thus, (𝐱(t),𝐩(t)) must be the unique
solution to Hamilton's equations with the given initial condition
(𝐱_0,𝐩_0).

We conclude that for any solution to Hamilton's equations with
initial conditions of the form (<ref>), the formula (<ref>)
holds. Since, also, H is constant along solutions to Hamilton's equations,
we may replace H(𝐱(t),𝐩(t)) by H(𝐱
_0,𝐩_0) in (<ref>), at which point, integration with
respect to t gives (<ref>). Finally, (<ref>)
holds by the definition of 𝐩(t).


We are now ready for the proof of Theorem <ref>.

We apply Proposition <ref> with n=3 and the open set U consisting of triples (a,b,ε) with ε>0. 
The PDE (<ref>) is of the
type in (<ref>), with H given by (<ref>). The initial
conditions (<ref>) are obtained by differentiating the
initial condition S(0,λ,ε)=log(|λ
-1| ^2+ε).

We let 𝐱(t)=(a(t),b(t),ε(t)) and 𝐩
(t)=(p_a(t),p_b(t),p_ε(t)). For the case of the Hamiltonian
(<ref>), a simple computation shows that

    𝐩·d𝐱/dt     =𝐩·∇_𝐩
    H
         =2H+ε p_ε
         =2H_0+ε p_ε.

Thus, the general formula (<ref>) becomes, in this case,

    S(t,𝐱(t))=S(0,𝐱_0)+H(𝐱_0,𝐩_0
    ) t+∫_0^tε(s)p_ε(s) ds.

But we also may compute that

    d/dtlog(  √(a^2+b^2))        =1/21/a^2+b^2(aȧ+bḃ)
         =1/21/a^2+b^2(  a∂ H/∂ p_a
    +b∂ H/∂ p_b) 
         =ε p_ε.

Thus,

    ∫_0^tε(s)p_ε(s) ds=log|λ
    (t)| -log|λ_0| .

If we now plug in the value of S(0,𝐱_0)=S(0,λ_0
,ε_0) and use Lemma <ref> along with the definition
(<ref>) of p_0, we obtain (<ref>). Finally,
(<ref>) is just the general formula (<ref>),
applied to the case at hand.




 §.§ Constants of motion


We now identify several constants of motion for the system (<ref>),
from which various useful formulas can be derived. Throughout the section, we
assume we have a solution to (<ref>) with the initial conditions
(<ref>) and (<ref>), defined on a
time-interval of the form 0≤ t<T. We continue the notation λ
(t)=a(t)+ib(t).


Along any solution of (<ref>), following
quantities remain constant:



  * The Hamiltonian H,


  * The “angular
momentum” in the (a,b) variables, namely ap_b-bp_a, and


  * The argument of λ, assuming
λ_0≠0.




For any system of the form (<ref>), the Hamiltonian H itself is a
constant of motion, as may be verified easily from the equations. The
conservation of the angular momentum is a consequence of the invariance of H
under simultaneous rotations of (a,b) and (p_a,p_b); see Proposition
2.30 and Conclusion 2.31 in <cit.>. This result can also be verified by
direct computation from (<ref>).

Finally, note from (<ref>) that if λ_0≠0, then
log|λ(t)| remains finite as long as the solution
to (<ref>) exists, so that λ(t) cannot pass through the
origin. We then compute that

    d/dttan(λ(t))=d/dtb/a=ḃ
    a-bȧ/a^2=ε p_εba-bε
    p_εa/a^2=0.

(If a=0, we instead compute the time-derivative of (λ),
which also equals zero.)



The Hamiltonian H in (<ref>) in invariant under the
one-parameter group of symplectic linear transformations given by

    (a,b,ε,p_a,p_b,p_ε)↦(e^σ/2
    a,e^σ/2b,e^σε,e^-σ/2p_a,e^-σ/2
    p_b,e^-σp_ε),

with σ varying over ℝ. Thus, the generator of this family of
transformations, namely,

    Ψ:=ε p_ε+1/2(ap_a+bp_b)

is a constant of motion for the system (<ref>). The constant Ψ
may be computed in terms of ε_0 and λ_0 as

    Ψ=p_0(a_0(a_0-1)+b_0^2+ε_0)

where p_0 is as in (<ref>).



The claimed invariance of H is easily checked from the formula
(<ref>). One can easily check that Ψ is the generator of
this family. That is to say, if we replace H by Ψ in (<ref>),
the solution is given by the map in (<ref>). Thus, by a simple
general result, Ψ will be a constant of motion; see Conclusion 2.31 in
<cit.>. Of course, one can also check by direct computation that the
function in (<ref>) is constant along solutions to
(<ref>). The expression (<ref>) then follows easily from the
initial conditions in (<ref>).



For all t, we have

    ε(t)p_ε(t)^2=ε_0p_0^2e^-Ct,

where C=2Ψ-1 and Ψ is as in (<ref>). The constant
C in (<ref>) may be computed in terms of ε_0 and
λ_0 as

    C=p_0(|λ_0| ^2-1+ε_0
    )=|λ_0| ^2-1+ε_0/|λ_0-1| ^2+ε_0.





We compute that

    ε̇     =∂ H/∂ p_ε=H/p_ε-ε p_ε(a^2+b^2-ε
    )
    ṗ_ε     =-∂ H/∂ε=-H/ε-ε p_ε^2

and then that

    d/dt(ε p_ε^2)       =ε̇p_ε^2+2ε p_εṗ_ε
         =p_εH-ε p_ε^3(a^2+b^2
    -ε)-2Hp_ε-2ε^2p_ε^3

This result simplifies to

    d/dt(ε p_ε^2)       =ε p_ε^2[  1-2(  ε p_ε+1/2(ap_a
    +bp_b))  ] 
         =-ε p_ε^2(2Ψ-1).

The unique solution to this equation is (<ref>). The expression
(<ref>) is obtained by evaluating Ψ at t=0, using the initial
conditions (<ref>), and simplifying.


We now make an important application of preceding results.


Suppose a solution to (<ref>) exists with
ε(t)>0 for 0≤ t<t_∗, but that lim_t→
t^∗ε(t)=0. Then

    lim_t→ t_∗log|λ(t)|
    =Ct_∗/2,

where C=2Ψ-1 is as in Proposition <ref>. Furthermore, we have

    lim_t→ t_∗(ap_a+bp_b)=lim_t→ t_∗
    2log|λ(t)|/t+1.




Equation (<ref>) is a key step in the derivation of our main result; see
Section <ref>. We will write (<ref>) in a more explicit
way in Proposition <ref>, after the time t_∗ has been
determined. We note also from Proposition <ref> that since
ε(t) approaches zero as t approaches t_∗, then
p_ε(t) must be blowing up, so that ε
(t)p_ε(t)^2 can remain positive in this limit.


Using the constant of motion Ψ in (<ref>), we can
rewrite the Hamiltonian H as

    H=-ε p_ε(1+(a^2+b^2)p_ε-2Ψ
    +ε p_ε).

Now, by assumption, the variable ε approaches zero as t
approaches t_∗. Furthermore, by Proposition <ref>,
ε p_ε^2 remains finite in this limit, so that
ε p_ε=√(ε)√(ε
p_ε^2) tends to zero. Thus, in the t→ t_∗
limit, the ε p_ε terms in (<ref>) vanish
while ε p_ε^2 remains finite, leaving us with

    H=-lim_t→ t∗ε p_ε^2(a^2+b^2).

Since H is a constant of motion, we may write this result as

    lim_t→ t_∗(a^2+b^2)=-lim_t→ t_∗
    H_0/ε p_ε^2=lim_t→ t_∗
    ε_0p_0^2/ε p_ε^2=e^Ct_∗

where we have used Lemma <ref> in the second equality and Proposition
<ref> in the third. The formula (<ref>) follows.

Meanwhile, as t approaches t_∗, the ε p_ε
term in the formula (<ref>) for Ψ vanishes and we find,
using (<ref>), that

    lim_t→ t^∗(ap_a+bp_b)=2Ψ=C+1=lim_t→
    t_∗2log|λ(t)|/t+1,

as claimed in (<ref>), where we have used (<ref>) in the last equality.




 §.§ Solving the equations


We now solve the system (<ref>) subject to the initial conditions
(<ref>) and (<ref>). The formula in
Proposition <ref> for ε(t)p_ε(t)^2 will be
a key tool. Although we are mainly interested in the case ε_0>0,
we will need in Section <ref> to allow ε_0 to be
slightly negative.

We begin by with the following elementary lemma.


Consider a number a^2∈ℝ and let a be
either of the two square roots of a^2. Then the solution to the equation

    ẏ=y^2-a^2

subject to the initial condition y(0)=y_0>0 is

    y(t)=y_0cosh(at)-asinh(at)/cosh(at)-y_0sinh(at)/a



If a^2≥ y_0^2, the solution exists for all t>0. If a^2
<y_0^2, then y(t) is a strictly increasing function of t until the
first positive time t_∗ at which the solution blows up. This time is
given by

    t_∗     =1/atanh^-1(  a/y_0)
    
         =1/2alog(  1+a/y_0/1-a/y_0)  .

Here, we use the principal branch of the inverse hyperbolic tangent, with
branch cuts (-∞,-1] and [1,∞) on the real axes, which
corresponds to using the principal branch of the logarithm. When a=0, we
interpret the right-hand side of (<ref>) or (<ref>) as
having its limiting value as a approaches zero, namely 1/y_0.


In passing from (<ref>) to (<ref>), we have used the
standard formula for the inverse hyperbolic tangent,

    tanh^-1(x)=1/2log(  1+x/1-x)  .

In (<ref>), we interpret sinh(at)/a as having the value
t when a=0. If a^2<0, so that a is pure imaginary, one can rewrite
the solution in terms of ordinary trigonometric functions, using the
identities cosh(iα)=cosα and sinh(iα)=isinα. For
each fixed t, the solution is an even analytic function of a and therefore
an analytic function of a^2.


If a is nonzero and real, we may integrate (<ref>) to
obtain

    t       =1/2a∫_0^t(  1/y(τ)-a-1/y(τ
    )+a)  ẏ(τ) dτ
         =1/2a.  log(  y(τ)-a/y(τ)+a)
    | _τ=0^t
         =1/2alog(  y(t)-a/y(t)+ay_0+a/y_0
    -a)  .

It is then straightforward to solve for y(t) and simplify to obtain
(<ref>). Similar computations give the result when a is
zero (recalling that we interpret sinh(at)/a as equaling t when a=0)
and when a is nonzero and pure imaginary. Alternatively, one may check by
direct computation that the function on the right-hand side of
(<ref>) satisfies the equation (<ref>) for
all a∈ℂ.

Now, if a^2≥ y_0^2>0, the denominator in (<ref>)
is easily seen to be nonzero for all t and there is no singularity. If
a^2 is positive but less than y_0^2, the denominator remains
positive until it becomes zero when tanh(at)=a/y_0. If a^2 is
negative, so that a=iα for some nonzero α∈ℝ, we write
the solution using ordinary trigonometric functions as

    y(t)=y_0cos(α t)+α/y_0sin(α t)/cos(α
    t)-y_0/αsin(α t).

The denominator in (<ref>) becomes zero at α t=tan^-1
(α/y_0)<π/2. Finally, if a^2=0, the solution is y(t)=y_0
/(1-y_0t), which blows up at t=1/y_0.

It is then not hard to check that for all cases with a^2<y_0^2, the
blow-up time can be computed as t_∗=1/atanh^-1(a/y_0),
where we use the principal branch of the inverse hyperbolic tangent, with
branch cuts (-∞,-1] and [1,∞) on the real axis. (At a=0 we
have a removable singularity with a value of 1/y_0.) This recipe
corresponds to using the principal branch of the logarithm in the last
expression in (<ref>).


We now apply Lemma <ref> to compute the p_ε-component of the solution to (<ref>). We use the following
notations, some of which have been introduced previously:

    p_0     =1/|λ_0-1| ^2+ε_0
    
    δ     =|λ_0| ^2+1+ε_0
    /|λ_0|
    
    C       =2Ψ-1=p_0(|λ_0| ^2-1+ε
    _0)
    
    y_0     =p_0+C/2=1/2p_0|λ_0|δ
    
    a^2     =C^2/4+ε_0p_0^2.

We now make the following standing assumptions:

    λ_0   ≠0
    
    p_0     >0
    δ     >0.



We note that under these assumptions, y_0 is positive. Furthermore, we may
compute that

    a=1/2p_0|λ_0|√(δ^2-4).

from which we obtain

    a^2/y_0^2=δ^2-4/δ^2<1,

so that a^2<y_0^2. Now, the assumptions p_0>0 and δ>0 can
be written as ε_0>-|λ_0-1| ^2 and
ε_0>-(1+|λ_0| ^2). Thus, for
λ_0≠0, the assumptions (<ref>) are always
satisfied if ε_0>0. Furthermore, except when λ_0=1,
some negative values of ε_0 are allowed.


Under the assumptions (<ref>), the
p_ε-component of the solution to (<ref>) subject to the
initial conditions (<ref>) and (<ref>)
is given by

    p_ε(t)=p_0cosh(at)+2|λ
    _0| -δ/√(δ^2-4)sinh(at)/cosh(at)-δ/√(δ^2-4)sinh(at)e^-Ct

for as long as the solution to the system (<ref>) exists. Here we
write a as in (<ref>) and we use the same choice of √(δ^2-4) in the computation of a as in the two times √(δ^2-4) appears explicitly in (<ref>). If δ=2, we
interpret sinh(at)/√(δ^2-4) as equaling 1/2
p_0|λ_0| t.

If ε_0≥0, the numerator in the fraction on the right-hand
side of (<ref>) is positive for all t. Hence when ε
_0≥0, we see that p_ε(t) is positive for as long as the
solution exists and 1/p_ε(t) extends to a real-analytic function
of t defined for all t∈ℝ.

The first time t_∗(λ_0,ε_0) at which the expression
on the right-hand side of (<ref>) blows up is

    t_∗(λ_0,ε_0)       =2(δ-2cosθ_0
    )/√(δ^2-4)tanh^-1(  √(δ^2-4)/δ) 
         =δ-2cosθ_0/√(δ^2-4)log(  δ+√(δ^2-4)/δ-√(δ^2-4))  ,

where θ_0=λ_0 and √(δ^2-4) is either of the
two square roots of δ^2-4. The principal branch of the inverse
hyperbolic tangent should be used in (<ref>), with branch cuts
(-∞,-1] and [1,∞) on the real axis, which corresponds to using
the principal branch of the logarithm in (<ref>). When δ=2, we
interpret t_∗(λ_0,ε_0) as having its limiting value
as δ approaches 2, namely δ-2cosθ_0.


Note that the expression

    1/atanh^-1(  a/b)

is an even function of a with b fixed, with a removable singularity at
a=0. This expression is therefore an analytic function of a^2 near the
origin. In particular, the value of t_∗(λ_0,ε_0)
does not depend on the choice of square root of δ^2-4.

We assume at first that
ε_0≠0. We recall from Proposition <ref> that
ε(t)p_ε(t)^2 is equal to ε_0p_0
^2e^-Ct, which is never zero, since we assume ε_0 is
nonzero and p_0 is positive. Thus, as long as the solution to the system
(<ref>) exists, both ε(t) and p_ε(t) must be
nonzero—and must have the same signs they had at t=0. Using (<ref>)
and the fact that H is a constant of motion, we obtain

    ṗ_ε(t)=ε_0p_0^2/ε
    (t)-ε_0p_0^2e^-Ct

But ε_0p_0^2/ε(t)=p_ε(t)^2e^Ct and
we obtain

    ṗ_ε(t)=p_ε(t)^2e^Ct-ε_0p_0
    ^2e^-Ct.

Then if y(t)=e^Ctp_ε(t)+C/2, we find that y satisfies
(<ref>). Thus, we obtain p_ε
(t)=(y(t)-C/2)e^-Ct, where y(t) is as in (<ref>), which
simplifies to the claimed formula for p_ε. The same formula
holds for ε_0=0, by the continuous dependence of the solutions
on initial conditions. (It is also possible to solve the system (<ref>
) with ε_0=0 by postulating that ε(t) is identically
zero and working out the equations for the other variables.)

In this paragraph only, we assume ε_0≥0. Then a^2≥0,
with a=0 occurring only if ε_0=0 and |λ
_0| =1, so that δ=2. In that case, the numerator on the
right-hand side of (<ref>) is identically equal to 1. If a^2>0,
then the numerator will always be positive provided that

    (  2|λ_0| -δ/√(δ^2-4)
    )  ^2≤1,

which is equivalent to

    (δ^2-4)-(2|λ_0| -δ)^2≥0.

But a computation shows that

    (δ^2-4)-(2|λ_0| -δ)^2=4ε
    _0,

and we are assuming ε_0≥0. Now, since the numerator in
(<ref>) is always positive, we conclude that p_ε
remains positive until it blows up.

For any value of ε_0, the blow-up time for the function on the
right-hand side of (<ref>) is computed by plugging the expression
(<ref>) for a/y_0 into the formula (<ref>), giving

    t_∗(λ_0,ε_0)       =1/y_01/a/y_0
    tanh^-1(  a/y_0) 
         =2/p_0|λ_0|δδ/√(δ^2-4)tanh^-1(  √(δ^2-4)/δ)  .

After computing that

    1/p_0|λ_0|=|λ
    _0-1| ^2+ε_0/|λ_0|=δ-2cosθ_0,

we obtain the claimed formula (<ref>) for t_∗(λ
_0,ε_0).



If ε_0<0, then numerator on the
right-hand side of (<ref>) can become zero. The time σ at
which this happens is computed using (<ref>) and
(<ref>) as

    σ=2/p_0|λ_0|√(δ^2-4)
    tanh^-1(  -(  1+4ε_0/(2|λ
    _0| -δ)^2)  ^1/2)  .

By considering separately the cases |λ_0|≠1
and |λ_0| =1, we can verify that σ tends
to infinity, locally uniformly in λ_0, as ε_0 tends to
zero from below. Thus, for small negative values of ε_0, the
function on the right-hand side of (<ref>) will remain positive
until the time t_∗(λ_0,ε_0) at which it blows up.


We now show that the whole system (<ref>) has a solution up to the
time at which the function on the right-hand side of (<ref>) blows up.


Assume that ε_0 and λ_0 satisfy
the assumptions (<ref>). Assume further that if
ε_0<0, then |ε_0| is
sufficiently small that p_ε remains positive until it blows up,
as in Remark <ref>. Then the solution to the system
(<ref>) exists up to the time t_∗(λ_0,ε_0)
in Proposition <ref>.

For any ε_0, we have

    lim_t→ t_∗(λ_0,ε_0)ε(t)=0.

If ε_0=0, the solution has ε(t)≡0 and
λ(t)≡λ_0.



Let T be the maximum time such that the solution to (<ref>) exists
on [0,T). We now compute formulas for the solution on this interval. Recall
from Proposition <ref> that if ε_0≥0, then
p_ε(t) remains positive for as long as the solution exists; by
Remark <ref>, the same assertion holds if ε_0
is small and negative.

Now, since ε p_ε^2=ε_0p_0^2e^-Ct,
we see that

    ε(t)=1/p_ε(t)^2ε_0p_0^2
    e^-Ct.

Since p_ε(t) remains positive until it blows up, ε
(t) remains bounded until time t_∗(λ_0,ε_0), at
which time ε(t) approaches zero, as claimed in (<ref>
). We recall from Proposition <ref> that the argument
of λ(t) remains constant. Then as in shown in (<ref>
), we have

    log|λ(t)| =log|λ_0|
    +∫_0^tε(s)p_ε(s) ds.

Finally,

    dp_a/dt=-∂ H/∂ a=-2aε p_ε^2+ε p_εp_a

which is a first-order, linear equation for p_a, which can be solved using
an integrating factor. A similar calculation applies to p_b.

Suppose now that the existence time T of the whole system were smaller than
the time t_∗(λ_0,ε_0) at which the right-hand side
of (<ref>) blows up. Then from the formulas (<ref>),
(<ref>), and (<ref>), we see that all functions involved
would remain bounded up to time T. But then by a standard result, T could
not actually be the maximal time. The solution to the system (<ref>)
must therefore exist all the way up to time t_∗(λ_0
,ε_0).

Finally, we note that when ε_0=0, (<ref>) gives
ε(t)≡0 and (<ref>) gives |λ(t)|≡|λ_0| . Since also
the argument of λ(t) is constant, we see that λ(t)≡λ_0.




 §.§ More about the lifetime of the solution


In light of Propositions <ref> and <ref>, the
lifetime of the solution to the system (<ref>) is t_∗
(λ_0,ε_0), as computed in (<ref>) or
(<ref>). In this subsection, we (1) analyze the behavior of
log|λ(t)| as t approaches t_∗
(λ_0,ε_0), (2) analyze the behavior of t_∗
(λ_0,ε_0) as ε_0 approaches zero, and (3)
show that t_∗(λ_0,ε_0) is an increasing function of
ε_0 with λ_0 fixed.


Assume that ε_0 and λ_0 satisfy the
assumptions (<ref>). Then

    lim_t→ t_∗(λ_0,ε_0)log|λ(t)| =δ-2/|λ_0|/2√(δ^2-4)log(  δ+√(δ^2-4)/δ-√(δ^2-4))  ,

where δ is as in (<ref>).


Notice that there is a strong similarity between the formula (<ref>)
for t_∗(λ_0,ε_0) and the expression on the
right-hand side of (<ref>).


By (<ref>) in Theorem <ref>, we have lim
_t→ t_∗(λ_0,ε_0)log|λ(t)| =Ct_∗(λ_0,ε_0)/2, where by
(<ref>),

    C=(|λ_0| ^2-1+ε_0)/|λ_0|/(|λ_0-1| ^2
    +ε_0)/|λ_0|=δ
    -2/|λ_0|/δ-2cosθ_0.

From this result and the second expression (<ref>) for t_∗(λ_0,ε_0), (<ref>) follows easily.



If t_∗(λ_0,ε_0) is defined by
(<ref>), then for all nonzero λ_0 we have

    t_∗(λ_0,0)=T(λ_0),

where the function T is defined in (<ref>). Furthermore, when
ε_0=0, we have

    lim_t→ t_∗(λ_0,ε_0)log|λ(t)| =log|λ_0| .




Recall that the formula for t_∗(λ_0,ε_0) is defined
under the standing assumptions in (<ref>). Note that for
all λ_0≠0, the value ε_0=0 satisfies these assumptions.

Since log(x)/(x-1)→1 as x→1, we see that t_∗(λ_0,0) is a continuous function of λ_0∈ℂ^∗. Comparing the formula for t_∗(λ_0,0) to Theorem
<ref>, we have the following consequence.


For λ_0∈Σ_t, we have t_∗
(λ_0,0)<t, while for λ_0∈∂Σ_t, we have
t_∗(λ_0,0)=t, and for λ_0∉Σ_t,
we have t_∗(λ_0,0)>t.


In the limit as ε
_0→0, we have

    δ=|λ_0| ^2+1/|λ
    _0|,

and

    δ^2-4=(  |λ_0| ^2
    -1/|λ_0|)  ^2,

so that

    √(δ^2-4)=±|λ_0| ^2
    -1/|λ_0|.



In the case |λ_0| =1, the limiting value of
δ is 2. We then make use of the elementary limit

    lim_δ→2^+1/√(δ^2-4)log(
    δ+√(δ^2-4)/δ-√(δ^2-4))  =1.

Thus, using (<ref>), we obtain in this case,

    lim_ε_0→0t_∗(λ_0,ε
    _0)=2-2cosθ_0=|λ_0-1| ^2
    ,  |λ_0| =1,

which agrees with the value of T(λ_0) when |λ
_0| =1.

In the case |λ_0|≠1, we note that the
quantity (1/b)log((a+b)/(a-b)) is an even function of b with a fixed.
We may therefore choose the plus sign on the right-hand side of
(<ref>), regardless of the sign of |λ
_0| ^2-1. We then obtain, using (<ref>),

    lim_ε_0→0t_∗(λ_0,ε_0)     
    =(|λ_0| ^2+1)/| λ
    _0| -2cosθ_0/(|λ_0|
    ^2-1)/|λ_0|log(  2|λ_0| ^2/|λ_0|/2/|λ_0|) 
         =|λ_0| ^2+1-2|λ
    _0|cosθ_0/|λ_0| ^2-1
    log(|λ_0| ^2)
         =T(λ_0).



A similar calculation, beginning from (<ref>), establishes
(<ref>).



If we began with (<ref>) instead of (<ref>), we would obtain by
similar reasoning

    t_∗(λ_0,0)=2|λ_0-1| ^2
    /|λ_0| ^2-1tanh^-1(  |λ_0| ^2-1/|λ_0| ^2
    +1)  .

Using (<ref>), this expression is easily seen to agree with
T(λ_0) but is more transparent in its behavior at |λ_0| =1.



For each λ_0, the function t_∗(λ_0,ε_0) is a strictly increasing function of
ε_0 for ε_0≥0, and

    lim_ε_0→+∞t_∗(λ_0,ε
    _0)=+∞.





We note that the quantity δ in (<ref>) is an increasing
function of ε_0 with λ_0 fixed, with δ tending
to infinity as ε_0 tends to infinity. We note also that if
ε_0≥0, then

    δ≥|λ_0| +1/|λ
    _0|≥2.

It therefore suffices to show that for each angle θ_0, the function

    g_θ_0(δ):=δ-2cosθ_0/√(δ^2-4)
    log(  δ+√(δ^2-4)/δ-√(δ^2-4)
    )  ,

is strictly increasing, non-negative, continuous function of δ for
δ≥2 that tends to +∞ as δ tends to infinity. Here when
δ=2, we interpret g_θ_0(δ) as having the value
2-2cosθ_0, in accordance with the limit (<ref>).

Throughout the proof, we use the notation

    γ=√(δ^2-4).

We note that

    lim_δ→∞δ-2cosθ_0/γ=1.

Meanwhile, for large δ, we have

    δ-γ     =δ(  1-√(1-4/δ^2)) 
         =δ(  1-(  1-1/24/δ^2+O(
    1/δ^3)  )  ) 
         =2/δ+O(  1/δ^3)  ,

whereas

    δ+γ=2δ+O(  1/δ)  .

Thus, g_θ_0(δ) grows like log(δ^2) as δ→∞.

Our definition of g_θ_0(δ) for δ=2, together with
(<ref>), shows that g_θ_0 is non-negative and continuous
there. To show that g_θ_0 is an increasing function of δ, we
show that ∂ g_θ_0/∂δ is positive for δ>2.
The derivative is computed, after simplification, as

    ∂ g_θ_0/∂δ=2/γ^3(
    (δ-2cosθ_0)γ+(δcosθ_0-2)log(
    δ+γ/δ-γ)  )  .

Since this expression depends linearly on cosθ_0 with δ
fixed, if it is positive when cosθ_0=1 and also when cosθ
_0=-1, it will be positive always. Thus, it suffices to verify the
positivity of the functions

    (δ-2)(  γ+log(  δ+γ/δ-γ)  )

and

    (δ+2)(  γ-log(  δ+γ/δ-γ)  )  .



Now, (<ref>) is clearly positive for all δ>2. Meanwhile, a
computation shows that

    d/dδ(  γ-log(  δ+γ/δ
    -γ)  )  =δ-2/γ>0

and

    lim_δ→2^+(  γ-log(  δ+γ/δ-γ)  )  =0,

from which we conclude that (<ref>) is also positive for all
δ>2.




 §.§ Surjectivity


In Section <ref>, we will compute s_t(λ):=lim
_ε→0^+S(t,λ,ε) for λ in
Σ_t. We will do so by evaluating S (and its derivatives) along
curves of the form (t,λ(t),ε(t)) and then the taking the
limit as we approach the time t_∗ when ε(t) becomes zero.
For this method to be successful, we need the following result, whose proof
appears on p. surjectivityProof.


Fix t>0. Then for all λ∈Σ_t, there
exists a unique λ_0∈ℂ and ε_0>0 such that
the solution to (<ref>) with these initial conditions exists on
[0,t) with lim_u→ t^-ε(u)=0 and lim
_u→ t^-λ(u)=λ. For all λ∈Σ_t, the
corresponding λ_0 also belongs to Σ_t.

Define functions Λ_0^t:Σ_t→Σ_t and
E_0^t:Σ_t→(0,∞) by letting Λ_0
^t(λ) and E_0^t(λ) be the corresponding values of
λ_0 and ε_0, respectively. Then Λ_0^t and
E_0^t extend to continuous maps of Σ_t into
Σ_t and [0,∞), respectively, with the continuous
extensions satisfying Λ_t(λ)=λ and E_0^t
(λ)=0 for λ∈∂Σ_t.


We first recall that we have shown (Proposition <ref>) that
the lifetime of the path to be a strictly increasing function of
ε_0≥0 with λ_0 fixed. If λ_0 is outside
Σ_t, then by Theorem <ref> and Proposition
<ref>, the lifetime is at least t, even at ε_0=0.
(That is to say, T(λ_0)=t_∗(λ_0,0)≥ t for
λ_0 outside Σ_t.) Thus, for λ_0 outside
Σ_t, the lifetime cannot equal t for ε_0>0. On the
other hand, if λ_0∈Σ_t, then t_∗(λ_0,0)<t
and Proposition <ref> tells us that there is a unique
ε_0>0 with t_∗(λ_0,ε_0)=t.


Fix t>0. Define maps

    ε_0^t     :Σ_t→0,∞)
    λ_t     :Σ_t→ℂ∖{0}

as follows. For λ_0∈Σ_t, we let ε_0
^t(λ_0) denote the unique positive value of ε_0 for
which t_∗(λ_0,ε_0)=t. Then we set

    λ_t(λ_0)=lim_u→ t^-λ(u),

where λ(·) is computed with initial conditions λ
(0)=λ_0 and ε(0)=ε_0^t(λ_0). Then
both ε_0^t and λ_t extend continuously from
Σ_t to Σ_t, with the extended maps satisfying
ε_0^t(λ_0)=0 and λ_t(λ_0)=λ
_0 for λ_0∈∂Σ_t. The extended map λ_t
is a homeomorphism of Σ_t to itself.


We note that the desired function Λ_0^t in Theorem
<ref> is the inverse function to λ_t and that
E_0^t(λ)=ε_0^t(λ_t^-1(λ)).

Recall from Proposition <ref> that the argument of
λ(t) is constant. By the formula (<ref>) in Theorem
<ref> together with the expression (<ref>) for the
constant C, we can write

    λ_t(λ_0)=λ_0/|λ_0|e^Ct/2=λ_0/|λ_0|exp(
    t/2|λ_0| ^2-1+ε_0
    ^t(λ_0)/|λ_0-1| ^2+ε
    _0^t(λ_0))  ,

where we have used that t_∗(λ_0,ε_0^t(λ
_0))=t. As noted in the proof of Proposition <ref>, this formula
can also be written as

    λ_t(λ_0)=λ_0/|λ_0|exp(  δ-2/|λ_0|/2√(δ^2-4)log(  δ+√(δ^2-4)/δ
    -√(δ^2-4))  )  ,

where δ=(|λ_0| ^2+1+ε_0
^t(λ_0))/|λ_0| .


We start by trying to compute the function ε_0^t, which we
will do by finding the correct value of δ and then solving for
ε_0^t. Recall that the lifetime t_∗(λ
_0,ε_0) is computed as g_θ_0(δ), where δ
is as in (<ref>) and g_θ_0 is as in (<ref>).
As we have computed in (<ref>), we have

    g_θ_0(  r_0^2+1/r_0)  =T(r_0e^iθ
    _0).

Assume, then, that the ray with angle θ_0 intersects Σ_t and
let r_t(θ_0) be the outer (for definiteness) radius at which this
ray intersects the boundary of Σ_t. Then Theorem
<ref> tells us that T(r_t(θ_0)e^iθ_0)=t,
and we conclude that

    g_θ_0(  r_t(θ_0)^2+1/r_t(θ_0
    ))  =t.



Consider, then, some λ_0∈Σ_t with (λ_0
)=θ_0. By the formula (<ref>), to find ε_0 with
t_∗(λ_0,ε_0)=t, we first find δ so that
g_θ_0(δ)=t. (Note that the value of δ depends only on
the argument of λ_0.) We then adjust ε_0 so that
(|λ_0| ^2+ε_0+1)/|λ_0| =δ. Since the correct value of δ is given
in (<ref>), this means that we should choose ε_0 so
that

    |λ_0| ^2+ε_0+1/|λ_0|=r_t(θ_0)^2+1/r_t(θ_0).

We can solve this relation for ε_0 to obtain

    ε_0^t(λ_0)=|λ_0|(
    r_t(λ_0)^2+1/r_t(λ_0)-|λ_0| ^2+1/|λ_0|)  .

Now, we have shown that r_t(θ) is continuous for the full range of
angles θ occurring in Σ_t. Since 0 is not in
Σ_t, we can then see that the formula (<ref>)
is well defined and continuous on all of Σ_t. For
λ_0∈∂Σ_t, we have that |λ
_0| equals r_t(λ_0) or 1/r_t(λ
_0), so that ε_0^t(λ_0) equals zero.

Now, the point 0 is always outside Σ_t, while the point
1 is always in Σ_t and therefore not on the boundary of Σ
_t. Thus, since ε_0^t is continuous on Σ_t and zero precisely on the boundary, we see from (<ref>)
that λ_t is continuous on Σ_t. Furthermore, on
∂Σ_t, we compute λ_t(λ_0) by putting
ε_0^t(λ_0)=0 in (<ref>). Suppose now
that λ_0 is in ∂Σ_t. Then ε_0
^t(λ_0)=0 and, by Theorem <ref>, the function
T(λ_0) in (<ref>) has the value t, so that

    t/2|λ_0| ^2-1/|λ_0-1| ^2=log(|λ_0| ).

Thus, from (<ref>), we see that λ_t(λ
_0)=λ_0.

Consider an angle θ_0 for which the ray Ray(θ_0)
with angle θ_0 intersects Σ_t and let δ be
chosen so that g_θ_0(δ)=t, noting again that the value of
δ depends only on θ_0=λ_0. We now observe from
(<ref>) that |λ_t(λ_0)|
is a strictly increasing function of |λ_0| with
δ fixed. Thus, λ_t is a strictly increasing function of the
interval Ray(θ_0)∩Σ_t into
Ray(θ_0) that fixes the endpoints. Thus, actually,
λ_t maps this interval bijectively into itself. Since this holds for
all θ_0, we conclude that λ_t maps Σ_t
bijectively into itself. The continuity of the inverse then holds because
λ_t is continuous and Σ_t is compact.


We have
noted before the statement of Lemma <ref> that if the desired
pair (λ_0,ε_0) exists, λ_0 must be in
Σ_t. The lemma then tells us that a unique pair (λ
_0,ε_0) exists with λ_0∈Σ_t. We compute
Λ_0^t(λ) as λ_t^-1(λ) and E_0
^t(λ) as ε_0^t(λ_t^-1(λ)), both of
which extend continuously to Σ_t. For λ∈∂Σ_t, we have λ_t^-1(λ)=λ and
ε_0^t(λ_t^-1(λ))=ε_0^t
(λ)=0.




§ LETTING Ε TEND TO ZERO




 §.§ Outline


Our goal is to compute the Laplacian with respect to λ of the function
s_t(λ):=lim_ε→0^+S(t,λ,ε),
using the Hamilton–Jacobi method of Theorem <ref>. We want the curve
ε(·) occurring in (<ref>) and (<ref>) to
approach zero at time t; a simple way we might try to accomplish this is to
let the initial condition ε_0 approach zero. Suppose, then, that
ε_0 is very small. Using various formulas from Section
<ref>, we then find that for as long as the solution to
the system (<ref>) exists, the whole curve ε(·) will
be small and the whole curve λ(·) will be approximately constant.
Thus, by taking ε_0≈0 and λ_0≈λ, we
obtain a curve with ε(t)≈0 and λ(t)≈λ.
We may then hope to compute s_t(λ) by letting λ_0 and
λ(t) approach λ and ε_0 approach zero in the
Hamilton–Jacobi formula (<ref>), with the result that

    s_t(λ)=log(|λ-1| ^2).



It is essential to note, however, that this approach is only valid if the
solution to system (<ref>) exists up to time t. Corollary
<ref> tells us that for ε_0≈0, the solution
will exist beyond time t provided λ is outside Σ_t. Thus, we expect that for λ outside Σ_t,
the function s_t will be given by (<ref>) and therefore that
Δ s_t will be zero. (The function log(|λ
-1| ^2) is harmonic except at the point λ=1, which is
always inside Σ_t.)

To analyze s_t(λ) for λ inside Σ_t, we first make
use of the surjectivity result in Theorem <ref>. The theorem
says that for each t>0 and λ∈Σ_t, there exist
ε_0>0 and λ_0∈Σ_t such that ε(u)
approaches 0 and λ(u) approaches λ as u approaches t. We
then use the formula (<ref>) in Theorem <ref>. In light of
the second Hamilton–Jacobi formula (<ref>), we can write
(<ref>) as

    lim_u→ t(  a∂ S/∂ a+b∂
    S/∂ b)  (u,λ(u),ε(u))       =lim_u→
    t2log|λ(t)|/t+1
         =2log|λ|/t+1.

Once we have established enough regularity in the function S(t,λ
,ε) near ε=0, we will be able to identify the left-hand
side of (<ref>) with the corresponding derivative of s_t,
giving the following explicit formula for one of the derivatives of s_t:

    (  a∂ s_t/∂ a+b∂ s_t/∂
    b)  (λ)=2log|λ|/t+1.



We now compute in logarithmic polar coordinates, with ρ=log|λ| and θ=λ. We may recognize the left-hand
side of (<ref>) as the derivative of s_t with respect to
ρ, giving

    ∂ s_t/∂ρ=2ρ/t+1

for points inside Σ_t. Remarkably, ∂ s_t/∂ρ is
independent of θ! Thus,

    ∂/∂ρ∂ s_t/∂θ
    =∂/∂θ∂ s_t/∂ρ=0,

meaning that ∂ s_t/∂θ is independent of ρ.

Now, we will show in Section <ref> that the first derivatives of
s_t have the same value as we approach a point λ∈∂Σ_t from the inside as when we approach λ from the outside. We
can therefore give a complete description of the function ∂
s_t/∂θ on Σ_t as follows. It is the unique function on
Σ_t that is independent of ρ (or, equivalently, independent of
r=|λ|) and whose boundary values agree

    ∂/∂θlog(|λ-1|
    ^2)=2b/|λ-1| ^2=2rsinθ/r^2+1-2rcosθ.

Since the points on the outer boundary of Σ_t have the polar form
(r_t(θ),θ), we conclude that

    ∂ s_t/∂θ=2r_t(θ)sinθ/r_t(θ)^2+1-2r_t(θ)cosθ.

From this result, the expression (<ref>), and the formula for
the Laplacian in logarithmic polar coordinates, we obtain

    Δ s_t(λ)       =1/|λ| ^2(
    ∂^2s_t/∂ρ^2+∂^2s_t/∂θ^2) 
         =1/|λ| ^2(  2/t
    +∂/∂θ2r_t(θ)sinθ/r_t
    (θ)^2+1-2r_t(θ)cosθ)

for points inside Σ_t, accounting for the formula in Theorem
<ref>.

We now briefly discuss what is needed to make the preceding arguments
rigorous. If λ is outside Σ_t and ε is
small and positive, we need to know that we can find a λ_0 close to
λ and a small, positive ε_0 such that with these initial
conditions, ε(t)=ε and λ(t)=λ. To show
this, we apply the inverse function theorem to the map U_t(λ
_0,ε_0):=(λ(t),ε(t)) in a neighborhood of the
point (λ_0,ε_0)=(λ,0).

For λ inside Σ_t, we need to know first that S(t,λ
,ε) is continuous—in all three variables—up to ε=0.
After all, s_t(λ) is defined letting ε tend to zero in
the expression S(t,λ,ε), with t and
λ fixed. But the Hamilton–Jacobi formula (<ref>)
gives a formula for S(u,λ(u),ε(u)), in which the first two
variables in S are not remaining constant. Furthermore, we want to apply
also the Hamilton–Jacobi formula (<ref>) for the derivatives of
S, which means we need also continuity of the derivatives of S with
respect to λ up to ε=0. Using another inverse function
theorem argument, we will show that after making the change of variable
z=√(ε), the function S will extend smoothly up to
ε=z=0, from which the needed regularity will follow.

We use the following notation throughout the section.


We will let

    ε(t;λ_0,ε_0)

denote the ε-component of the solution to (<ref>) with
λ(0)=λ_0 and ε(0)=ε_0 (and with
initial values of the momenta given by (<ref>)), and
similarly for the other components of the solution.




 §.§ Outside Σ_t


The goal of this subsection is to prove the following result.


Fix a pair (t,λ) with λ outside
Σ_t. Then

    s_t(λ):=lim_ε→0^+S(t,λ,ε
    )=log(|λ-1| ^2).

Thus,

    Δ s_t(λ)=0

whenever λ is outside Σ_t.


As we have discussed in Section <ref>, the idea is that for
λ outside Σ_t and ε small and
positive, we should try to find a λ_0 close to λ and a
small, positive ε_0 such that ε(u) and λ(u)
will approach 0 and λ, respectively, as u approaches t. To that
end, we define, for each t>0, a map U_t from an open subset of
ℝ×ℂ into ℝ×ℂ by

    U_t(λ_0,ε_0)=(λ(t;λ_0,ε
    _0),ε(t;λ_0,ε_0)).

We wish to evaluate the derivative of this map at the point (λ
_0,ε_0)=(λ,0). For this idea to make sense,
λ(t;λ_0,ε_0) and ε(t;λ
_0,ε_0) must be defined in a neighborhood of (λ,0); it
is for this reason that we have allowed ε_0 to be negative in
Section <ref>.

The domain of U_t consists of pairs (λ_0,ε_0) such
that (1) the assumptions (<ref>) are satisfied; (2) the
function p_ε(·) remains positive until it blows up, as in
Remark <ref>; and (3) we have t_∗(λ
_0,ε_0)>t. We note that these conditions allow ε
_0 to be slightly negative and that all the results of Section
<ref> hold under these conditions. We also note that by
Proposition <ref>, if ε_0=0, then ε
(t)≡0 and λ(t)≡λ_0; thus,

    U_t(λ_0,0)=(λ_0,0).



We now fix a pair (t,λ) with λ outside of Σ_t (so that λ≠1). By Corollary <ref>, we then have
t_∗(λ,0)>t.


The Jacobian of U_t at (λ,0) has the form

    U_t^'(λ,0)=(
    [          I_2×2 ∂λ/∂ε_0(t;λ,0);              0 ∂ε/∂ε_0(t;λ,0) ])

with ∂ε/∂ε_0(t;λ,0)>0. In
particular, the inverse function theorem applies at (λ,0).



The claimed form of the second column of U_t^'(λ,0) follows
immediately from (<ref>). We then compute from (<ref>) that

    ∂ε(t;λ_0,ε_0)/∂ε_0(0,λ_0)       =.  1/p_ε(t)^2
    p_0^2e^-Ct+ε_0∂/∂ε_0
    [  1/p_ε(t)^2p_0^2e^-Ct]  |
    _ε_0=0
         =1/p_ε(t)^2p_0^2e^-Ct,

which is positive.


We note that the inverse of the matrix in
(<ref>) will have a positive entry in the bottom right corner, meaning
that U_t^-1 has the property that ∂ε_0
/∂ε>0. It follows that the ε_0-component of
U_t^-1(λ,ε) will be positive for ε small and
positive. In that case, the solution to the system (<ref>) will have
ε(u)>0 up to the blow-up time. The blow-up time, in turn, exceeds
t for all points in the domain of U_t.

We may, therefore, apply the Hamilton–Jacobi formula (<ref>), which
we write as follows. We let HJ denote the right-hand side of the
Hamilton–Jacobi formula (<ref>):

    HJ(t,λ_0,ε_0)       =log(|λ
    _0-1| ^2+ε_0)-ε_0t/(|λ_0-1| ^2+ε_0)^2
         +log|λ(t;λ_0,ε_0)|
    -log|λ_0|

and we then have

    S(t,λ(t;λ_0,ε_0),ε(t;λ
    _0,ε_0))=HJ(t,λ_0,ε_0).

If ε is small and positive, we therefore obtain

    S(t,λ,ε)=HJ(t,U_t^-1(λ,ε)),

where we note that by definition λ(t;U_t^-1(λ,ε
))=λ.

Now, in the limit ε→0^+ with λ fixed, the
inverse function theorem tells us that U_t^-1(ε,λ
)→(0,λ). Thus, the limit (<ref>) may be computed
by putting λ(t;λ_0,ε_0)=λ in
(<ref>) and letting ε_0 tend to zero and λ_0
tend to λ. This process gives (<ref>).

Finally, when λ_0=0, we can use continuous dependence of the
solutions on the initial conditions. The formula for p_ε(t) in
Proposition <ref> has a limit as |λ
_0| tends to zero, so that δ tends to +∞. From
(<ref>), we find that a^2=y_0^2, so that from
(<ref>), y(t)≡ y_0. We then obtain

    p_ε(t)=e^-Ctp_0,

which remains nonsingular for all t. We can then continue to use the formula
(<ref>) for ε(t). Furthermore, by exponentiating
(<ref>) and letting |λ_0| tend to
zero, we find that λ(t)≡0. We then continue to use the remaining
formulas in the proof of Proposition <ref> and find that the
solution to the system exists for all time.

When λ_0=0, we apply the Hamilton–Jacobi formula in the form
(<ref>), which is to say that we replace the last two terms in
(<ref>) by ∫_0^tε(s)p_ε(s) ds. We then
compute as in (<ref>) that the derivative of ε
(t;0,ε_0) with respect to ε_0 is positive at
ε_0=0. Thus, by the inverse function theorem, for small positive
ε, we can find a small positive ε_0 that gives
ε(t;0,ε_0)=ε. We then apply (<ref>
) with λ_0=0 and λ(t)=0, and let ε tend to zero,
which means that ε_0 also tends to zero. As ε_0
tends to zero, the function

    ε(s)p_ε(s)=ε(s)p_ε(s)^2
    /p_ε(s)=ε_0p_0^2e^-Cs/p_ε(s)

tends to zero uniformly and we obtain (<ref>).




 §.§ Inside Σ_t


In this subsection, we establish the needed regularity of S(t,λ
,ε) as ε tends to zero, for λ in Σ_t.
This result, whose proof is on p. stildeProof, together with Theorem
<ref>, will allow us to understand the structure of s_t
and its derivatives on Σ_t.


Define

    S̃(t,λ,z)=S(t,λ,z^2),   z>0.

Fix a pair (σ,μ) with μ in Σ_σ. Then S̃(t,λ,z), initially defined for z>0, extends to a real-analytic
function in a neighborhood of (σ,μ,0) inside ℝ
×ℂ×ℝ.


We emphasize that the analytically extended S̃ does not
satisfy the identity S̃(t,λ,z)=S(t,λ,z^2). Indeed,
since √(ε(t))p_ε(t) is always bounded away from
zero (Proposition <ref>), the second Hamilton–Jacobi formula
(<ref>) tells us that ∂S̃/∂ z(t,λ
,z)=2√(ε)∂ S/∂ε(t,λ,z^2) has a
nonzero limit as z tends to zero, ruling out a smooth extension that is even
in z.


Fix a pair (σ,μ) with μ in
Σ_σ. Then the functions

    S(t,λ,ε),  ∂ S/∂ a(t,λ
    ,ε),  ∂ S/∂ b(t,λ,ε
    ),  √(ε)∂ S/∂ε(t,λ
    ,ε)

all have extensions that are continuous in all three variables to the set of
(t,λ,ε) with λ∈Σ_t and ε≥0.
Furthermore, for each t>0, the function s_t is infinitely differentiable
on Σ_t, and its derivatives with respect to a and b agree with
the ε→0^+ limit of ∂ S/∂ a and
∂ S/∂ b. If we let t_∗ be short for t_∗
(λ_0,ε_0), then for all λ_0 and ε
_0>0, we have

    s_t(t_∗,λ(t_∗;λ_0,ε_0))     
    =log(|λ_0-1| ^2+ε_0)-ε_0t_∗/(|λ_0-1| ^2
    +ε_0)^2
         +log|λ(t_∗;λ_0,ε_0)|
    -log|λ_0|

and

    ∂ s_t/∂ a(t_∗,λ(t_∗;λ
    _0,ε_0))       =lim_t→ t_∗p_a(t)
    ∂ s_t/∂ b(t_∗,λ(t_∗;λ
    _0,ε_0))       =lim_t→ t_∗p_b(t).





We note that the four functions in (<ref>) may be computed as

    S̃(t,λ,√(ε)),  ∂S̃/∂
    a(t,λ,√(ε)),  ∂S̃/∂
    b(t,λ,√(ε)),  1/2∂S̃
    /∂ z(t,λ,√(ε)),

respectively, and that S(t,λ,0)=S̃(t,λ,0). The first
claim then follows from Theorem <ref>. Now that the
continuity of S and its derivatives has been established, we may let t
approach t_∗(λ_0,ε_0) in the Hamilton–Jacobi
formulas (<ref>) and (<ref>) to obtain the second claim.



Let us write λ∈Σ_t in logarithmic polar
coordinates, with ρ=log|λ| and θ
=λ. Then for each pair (t,λ) with λ∈Σ_t,
we have

    ∂ s_t/∂ρ(t,λ)=2ρ/t+1.

Furthermore, ∂ s_t/∂θ is independent of ρ; that
is,

    ∂ s_t/∂θ=m_t(θ),

for some smooth function m_t. Thus,

    ∂^2s_t/∂ρ^2+∂^2s_t/∂θ^2=2/t+∂/∂θm_t(θ)

for some smooth function m_t, and

    (  ∂^2/∂ a^2+∂^2/∂
    b^2)  s_t(λ)=1/|λ| ^2
    (  2/t+∂/∂θm_t(θ))  .




In Section <ref>, we will obtain a formula for the function
m_t(θ) appearing in Corollary <ref>.


The derivative ∂/∂ρ may be computed in ordinary polar
coordinates as r∂/∂ r or in rectangular coordinates as
a∂/∂ a+b∂/∂ b. It then follows from the
Hamilton–Jacobi formula (<ref>) that

    (  a∂ S/∂ a+b∂ S/∂ b)
    (t,λ(t),ε(t))=a(t)p_a(t)+b(t)p_b(t).

Now, for each pair (t,λ) with λ∈Σ_t, Theorem
<ref> tells us that we can find (λ_0,ε
_0) so that

    lim_u→ tε(u)=0;  lim_u→ t
    λ(u)=λ.

In light of (<ref>), the formula (<ref>) then follows from the
formula (<ref>) in Theorem <ref>.

Now, ∂ s_t/∂ρ is manifestly independent of θ.
Since, by Corollary <ref>, s_t is an analytic, hence
C^2, function on Σ_t, we conclude that

    ∂/∂ρ∂ s_t/∂θ
    =∂/∂θ∂ s_t/∂ρ=0,

showing that ∂ s_t/∂θ is independent of ρ. The
formula (<ref>) then follows by differentiating (<ref>) with
respect to ρ. Finally, if we use the standard formula for the Laplacian
in polar coordinates,

    Δ=1/r^2(  (  r∂/∂ r)
    ^2+∂^2/∂θ^2)  =1/r^2(
    ∂^2/∂ρ^2+∂^2/∂θ^2
    )  .

we obtain (<ref>) from (<ref>).


We now begin preparations for the proof of Theorem <ref>.
Recall from Proposition <ref> that ε(t)p_ε(t)^2=ε_0p_0^2e^-Ct, where C=2Ψ-1 is a constant
computed from ε_0 and λ_0 as in (<ref>).
Recall also from Proposition <ref> that for ε
_0>0, the function 1/p_ε(t) extends to real analytic function
of t defined for all t∈ℝ. We then define, for ε
_0>0,

    z(t;λ_0,ε_0)=√(ε_0)p_0e^-Ct/21/p_ε(t;λ_0,ε_0)

for all t∈ℝ. For t<t_∗(λ_0,ε_0), the
function z(t;λ_0,ε_0) is positive and satisfies

    z(t;λ_0,ε_0)^2=ε(t;λ_0,ε
    _0),

while for t=t_∗(λ_0,ε_0), we have z(t;λ
_0,ε_0)=0 and for t>t_∗(λ_0,ε_0),
the function z(t;λ_0,ε_0) is negative.

Furthermore, using (<ref>) and Point <ref> of
Proposition <ref>, we see that

    λ(t;λ_0,ε_0)=λ_0e^∫_0^t
    ε(s)p_ε(s) ds,

where by Proposition <ref>, we have

    ε(s)p_ε(s)=ε(s)p_ε(s)^2
    /p_ε(s)=ε_0p_0^2e^-Cs/p_ε(s).

Since 1/p_ε(s) extends to an analytic function of s∈ℝ, we see that λ(t) extends to an analytic function of
t∈ℝ. We may therefore define a map

    V(t,λ_0,ε_0):=(t,λ(t;λ_0,ε
    _0),z(t;λ_0,ε_0)),

for all t∈ℝ, λ_0∈ℂ, and ε_0>0.


Suppose (t,λ_0,ε_0) has the property that λ
_0≠0 and t_∗(λ_0,ε_0)=t, so that
z(t;λ_0,ε_0)=0. Then the Jacobian matrix of V at
(t,λ_0,ε_0) is invertible.



We make some convenient changes of variables. First, we replace (t,λ
_0,ε_0) with (t,λ_0,δ), where δ is as in
(<ref>). This change has a smooth inverse, since we can recover
ε_0 from δ as

    ε_0=|λ_0|δ-|λ_0| ^2-1.

Then we write λ_0 in terms of its polar coordinates, (r_0
,θ_0). Finally, we write λ(t;λ_0,ε_0) in
logarithmic polar coordinates,

    ρ(t;λ_0,ε_0):=log|λ_0(t;λ
    _0,ε_0)| ;  θ(t;λ_0,ε
    _0):=(λ(t;ε_0,λ_0)),

where by Point <ref> of Proposition
<ref>, θ(t;λ_0,ε_0
)=θ_0.

Thus, to prove the proposition, it suffices to verify that the Jacobian matrix
of the map

    W(t,θ_0,r_0,δ):=(t,θ_0,ρ(t;λ_0,ε
    _0),z(t;λ_0,ε_0))

is invertible. We observe that this Jacobian has the form

    W^'=(
    [ I_2×2     0;     ∗     K ])  ,

where

    K=(
    [  ∂ρ/∂ r_0     ∂ρ/∂δ; ∂ z/∂ r_0    ∂ z/∂δ ])  .

Now, by Proposition <ref>, the lifetime is independent of
r_0 with δ and θ_0 fixed. Thus, if we start at a point
with t_∗(λ_0,ε_0)=t and vary r_0, the lifetime
will remain equal to t and z(t;ε_0,λ_0) will remain
equal to 0. Thus, at the point in question, ∂ z/∂ r_0=0.
Meanwhile, Proposition <ref> gives a formula for the value of
ρ(t;λ_0,ε_0) at t=t_∗(λ_0
,ε_0), from which we can easily see that ∂ρ/∂
r_0>0. It therefore remains only to verify that ∂ z/∂δ
is nonzero.

Now, z(t_∗(λ_0,ε_0);λ_0,ε_0)=0.
If we differentiate this relation with respect to ε_0 with
λ_0 fixed, we find that

    ∂ z/∂ε_0=-∂ z/∂ t
    ∂ t_∗(λ_0,ε_0)/∂ε_0
    .

The derivative ∂ t_∗/∂ε_0 may be computed as

    ∂ t_∗(λ_0,ε_0)/∂ε_0
    =∂ g_θ_0(δ)/∂δ∂δ/∂ε_0,

where g_θ_0 is as in (<ref>). But the proof of Proposition
<ref> shows that ∂ g_θ/∂δ>0 for
all δ>2, while from the formula (<ref>) for δ, we
see that ∂δ/∂ε_0>0. (Note also that
δ>2 whenever ε_0>0.) Thus, ∂ t_∗(λ
_0,ε_0)/∂ε_0>0.

Meanwhile, from (<ref>) and (<ref>), we have

    z(t)=√(ε_0)e^Ct/2/cosh(at)+2|λ_0| -δ/√(δ^2-4)sinh(at)(
    cosh(at)-δ/√(δ^2-4)sinh(at))  .

If we differentiate with respect to t and evaluate at the time t_∗
when the last factor is zero, the product rule gives

    .  ∂ z(t;λ_0,ε_0)/∂
    t| _t=t_∗(λ_0,ε_0)
         =0+√(ε_0)e^Ct/2/cosh(at)+2|λ_0| -δ/√(δ^2-4)sinh(at)a(
    sinh(at)-δ/√(δ^2-4)cosh(at))  ,

which is negative because δ/√(δ^2-4)>1 and the denominator
is positive (Proposition <ref>). Thus, from (<ref>), we
conclude that ∂ z/∂ε_0>0.


We are now ready for the proof of the main result of this section.

By
(<ref>), we have

    S̃(t;λ(t;λ_0,ε_0),z(t;λ_0
    ,ε_0))       =S(t;λ(t;λ_0,ε_0
    ),ε(t;λ_0,ε_0))
         =HJ(t,λ_0,ε_0),

where HJ is as in (<ref>), whenever t_∗(λ
_0,ε_0)>σ. Fix a point (σ,μ) with μ∈Σ_σ. Then by Theorem <ref>, we can find a pair
(λ_0,ε_0) with t_∗(λ_0,ε_0
)=t—so that z(t;λ_0,ε_0)=0—and λ
(t;λ_0,ε_0)=λ. We now construct a local inverse
V^-1 to V around the point V(t,λ_0,ε_0
)=(t,λ,0).

For any triple (t,λ,z) in the domain of V^-1, we write
V^-1(t, λ,z) as (t,λ_0,ε_0). We note that if
z>0 then t_∗(λ_0,ε_0) must be greater than t,
because if we had t_∗(λ_0,ε_0)≤ t, then
z(t;λ_0,ε_0)=z would be zero or negative. Thus, we may
apply (<ref>) at (t,λ_0,ε_0)=V^-1
(t,λ,z) to obtain

    S̃(t,λ,z)=HJ(V^-1(t,λ,z)),

whenever (t,λ,z) is in the domain of V^-1 and z>0.

Recall now that λ(t;λ_0,ε_0) extends to an
analytic function of t∈ℝ. Thus, the function HJ in
(<ref>) extends to a smooth function of t∈ℝ, λ
_0∈ℂ∖{0}, and ε_0>0, defined even if
t_∗(λ_0,ε_0)<t. Therefore, the right-hand side of
(<ref>) provides the claimed smooth extension of S̃.




 §.§ Near the boundary of Σ_t


We start by considering what is happening right on the boundary of Σ
_t.


Neither the method of Section <ref> nor the method of Section
<ref> allows us to compute the value of s_t(λ) for
λ in the boundary of Σ_t. Although we expect that this value
will be log(|λ-1| ^2), the question is
irrelevant to the computation of the Brown measure. After all, we are supposed
to consider Δ s_t computed in the distribution sense, that is,
the distribution whose value on a test function ψ is

    ∫_ℂs_t(λ)Δψ(λ) d^2λ.

The value of (<ref>) is unaffected by the value of s_t(λ) for
λ in ∂Σ_t, which is a set of measure zero in
ℂ.


It is nevertheless essential to understand the behavior of s_t(λ) as
λ approaches the boundary of Σ_t.


We say that a function f:ℂ→ℝ is analytic
up to the boundary from inside Σ_t if the following conditions hold.
First, f is real analytic on Σ_t. Second, for each λ∈∂Σ_t, we can find an open set U containing λ and a
real analytic function g on U such that g agrees with f on
U∩Σ_t. We may similarly define what it means for f to be
analytic up to the boundary from outside Σ_t.



For each t>0, the function s_t is analytic
up to the boundary from inside Σ_t and analytic up to the boundary
from outside Σ_t.


Note that the proposition is not claiming that s_t is an analytic function
on all of ℂ. Indeed, our main results tell us that 1/4πΔ s_t(λ) is identically zero for λ outside Σ_t
but approaches a typically nonzero value as λ approaches a boundary
point from the inside. As we approach from the inside a boundary point with
polar coordinates (r,θ), the limiting value of 1/4πΔ
s_t(λ) is w_t(θ)/r^2. This quantity certainly cannot
always be zero, or the Brown measure of b_t would be identically zero.
Actually, we will see in Section <ref> that w_t(θ)
is strictly positive except when t=4 and θ= π.


We have shown that s_t(λ)=log(|λ-1| ^2)
for λ in (Σ_t)^c. Since 1∈Σ_t, we see
that s_t is analytic from the outside of Σ_t.

To address the analyticity from the inside, first note that by applying
(<ref>) with z=0, we have

    s_t(λ)=S(t,λ,0)=S̃(t,λ,0)=HJ
    (V^-1(t,λ,0)),

where HJ is as in (<ref>). But if ε_0
^t:Σ_t→ℝ and λ_t:Σ_t
→ℂ are as in Lemma <ref>, then we can see
that

    V^-1(t,λ,0)=(t,λ_t^-1(λ),ε_0^t
    (λ_t^-1(λ)))

and we conclude that

    s_t(λ)=HJ(t,λ_t^-1(λ),ε_0
    ^t(λ_t^-1(λ))).



We now claim that the function ε_0^t(λ_0), initially
defined for λ_0∈Σ_t, extends to an analytic
function in a neighborhood of Σ_t. For t≥4, we can
simply use the formula (<ref>) for all nonzero λ_0. For
t<4, however, the formula (<ref>) becomes undefined in a
neighborhood of a point where ∂Σ_t intersects the unit circle.

Nevertheless, we can make a general argument as follows. To compute
ε_0^t(λ_0), we solve the equation t_∗
(λ_0,ε_0)=t for ε_0 as a function of
λ_0. To do this, we first solve the equation g_θ_0
(δ)=t for δ_θ_0,t and then solve for ε_0
in terms of δ as ε_0=|λ_0|δ-|λ_0| ^2-1. Now, we know from the proof
of Proposition <ref> that g_θ_0(δ)=t has a
solution when |θ_0|≤θ_max(t)=cos
^-1(1-t/2), with the solution being δ=2 when θ_0=±θ_max(t). We can also verify that ∂ g_θ_0
/∂δ>0 for all δ≥2. This was verified for δ>2 in
the proof of Proposition <ref>. To see that the result
holds even when δ=2, it suffices to verify that the expressions in
(<ref>) and (<ref>) have positive limits as
δ→2^+. We omit this verification and simply note that the
limits have the values 1 and 1/3, respectively. It then follows from the
implicit function theorem that (1) the solution δ_θ_0,t
continues to exist (with δ<2) for |θ_0|
slightly larger than θ_max(t), and (2) the solution δ
_θ_0,t depends analytically on θ_0. Then, the expression

    ε_0^t(λ_0)=|λ_0|δ_θ_0,t-|λ_0| ^2-1

makes sense and is analytic for all nonzero λ_0 with |λ_0| <θ_max(t)+α_t, for some positive
quantity α_t. We note that in this expression, ε_0
^t(λ_0) can be negative—for example if |λ
_0| =1 and λ_0>θ_max(t).

We now consider the function λ_t, defined as

    λ_t(λ_0)=λ(t;λ_0,ε_0^t(λ
    _0)),

and we recall that λ_t(λ_0)=λ_0 for λ_0
∈∂Σ_t. Although λ_t was initially defined for
λ_0 in Σ_t, it has an analytic extension to a
neighborhood of Σ_t, namely the set of λ_0 in
the domain of the extended function ε_0^t for which the pair
(λ_0,ε_0) satisfy the assumptions in
(<ref>). We now claim that the derivative of λ
_t(λ_0) is invertible at each point in its domain. We use polar
coordinates in both domain and range. Since (λ_t(λ
_0))=λ_0, the derivative will have the form

    λ_t^'(λ_0)=(
    [ ∂|λ_t|/∂ r    ∂λ_t/∂θ;          0          1 ])  ,

and it therefore suffices to check that ∂|λ
_t| /∂ r is nonzero. To see this, we use the formula
(<ref>), where δ=δ_θ_0,t as in the
previous paragraph. Since δ is independent of |λ
_0| with t and λ_0 fixed, we can easily verify
from (<ref>) that ∂|λ_t|
/∂ r>0.

Now, we have already established that s_t is analytic in the interior of
Σ_t. Consider, then, a point λ in ∂Σ_t, so
that λ_t(λ)=λ. Since λ_t^'(λ) is
invertible, it has a analytic local inverse λ_t^-1 defined near
λ. Then the formula (<ref>) gives an analytic extension
of s_t to a neighborhood of λ.



Fix a point μ on the boundary of Σ_t.
Then the functions

    s_t(λ),  ∂ s_t/∂ a(λ),  ∂ s_t/∂ b(λ)

all approach the same value when λ approaches μ from inside
Σ_t as when λ approaches μ from outside Σ_t.



We begin by considering s_t itself. The limit as λ approaches
μ from the inside may be computed by using (<ref>). By Lemma
<ref>, as λ approaches μ from the inside,
λ_t^-1(λ) approaches λ_t^-1(μ)=μ, and
ε_0^t(λ_t^-1(λ)) approaches 0. Thus, the
limiting value of s_t from the inside is

    HJ(t,μ,0)=log(|μ-1| ^2),

where HJ is given by (<ref>) and were we have used that
λ(t;μ,0)=μ. (See the last part of Proposition <ref>
.) Since s_t(λ)=log(|λ-1| ^2) outside
Σ_t, the limit of s_t from the outside agrees with the
limit from the inside.

Next we consider the derivatives, which we compute in logarithmic polar
coordinates ρ=log|λ| and θ=λ. By (<ref>), we have

    ∂ s_t/∂ρ(λ)=(  a∂ s_t
    /∂ a+b∂ s_t/∂ b)  (λ)=log(|λ| ^2)/t+1

for λ∈Σ_t. Letting λ approach μ from the inside
gives the value log(|μ| ^2)/t+1. Since μ is on
the boundary of Σ_t, Theorem <ref> says that
T(μ)=t, so that

    log|μ| ^2/t+1       =|μ| ^2-1/|μ-1| ^2+1
         =2(|μ| ^2-Reμ)/|μ-1| ^2.

Taking the corresponding derivative of the “outside” function log(|λ
-1| ^2) and letting λ tend to μ from the outside
gives the same result.

Finally, we recall from Proposition <ref> that
ap_b-bp_a is a constant of motion. Thus, by the second Hamilton–Jacobi
formula (<ref>) and the initial conditions
(<ref>), we have

    a∂ s_t/∂ b(u,λ(u),ε(u))-b∂ s_t/∂ a(u,λ(u),ε(u))       =a_0
    p_b,0-b_0p_a,0
         =(2a_0b_0-2b_0(a_0-1))p_0
         =2b_0/|λ_0-1| ^2+ε_0.

If we choose ε_0 and λ_0 so that t_∗(λ
_0,ε_0)=t we can use the regularity result in Corollary
<ref> to let u tend to t. This gives

    a∂ s_t/∂ b(λ)-b∂ s_t/∂
    a(λ)=2b_0/|λ_0-1| ^2
    +ε_0,

where now λ_0=λ_t^-1(λ) and ε
_0=ε_0^t(λ_t^-1(λ)). As λ approaches
μ, Theorem <ref> says that the value of λ_0
approaches μ and ε_0 approaches 0, so we get

    lim_λ→μ^inside(  a∂ s_t
    /∂ b(λ)-b∂ s_t/∂ a(λ))
    =2Imμ/|μ-1| ^2.

Taking the corresponding derivative of log(|λ-1| ^2) and letting λ tend to μ from the
outside gives the same result.




 §.§ Proof of the main result


In this subsection, we prove our first main result, Theorem <ref>.
Proposition <ref> will then be proved in Section
<ref>, while Propositions <ref> and
<ref> will be proved in Section <ref>.


For each fixed t, the restriction to Σ_t of
the function

    ∂ s_t/∂θ(t,λ)

is the unique function that on Σ_t that (1) extends continuously to
the boundary, (2) agrees with the θ-derivative of log(|λ-1| ^2) on the boundary, and (3) is independent of
r=|λ| . Thus, the function m_t in Corollary
<ref> is given by

    m_t(θ)=2r_t(θ)sinθ/r_t(θ)^2+1-2r_t
    (θ)cosθ,

where r_t(θ) is the outer radius of the domain Σ_t (Figure
<ref>).



We have already established in Corollary <ref> that ∂
s_t/∂θ is independent of ρ (or equivalently, of r) in
Σ_t. Then Propositions <ref> and
<ref> tell us that ∂ s_t/∂θ is
continuous up to the boundary and agrees there with the angular derivative of
log(|λ-1| ^2). Thus, to compute ∂
s_t/∂θ at a point in Σ_t, we travel along a radial
segment (in either direction) until we hit the boundary at radius
r_t(θ) or 1/r_t(θ). We then evaluate the angular derivative
of log(|λ-1| ^2), as in (<ref>),
giving the claimed expression for ∂ s_t/∂θ=m_t
(θ).



For each t>0, the distributional Laplacian of
s_t(λ) with respect to λ may be computed as follows. Take
the pointwise Laplacian of s_t outside Σ_t (giving
zero), take the pointwise Laplacian of s_t inside Σ_t (giving the
expression (<ref>) in Corollary <ref>) and ignore
the boundary of Σ_t.



Since, by Proposition <ref>, s_t is analytic up to
the boundary of Σ_t from the inside, Green's second identity says
that

    ∫_Σ_ts_t(λ)Δψ(λ) d^2λ   
    =∫_Σ_t(Δ s_t(λ))ψ(λ) d^2λ
         +∫_∂Σ_t(s_t(λ)∇ψ(λ)-ψ
    (λ)∇ s_t(λ))·n̂ dS,

for any test function ψ, where in the last integral, the limiting value
of ∇ s_t from the inside should be used. This identity holds because
the boundary of Σ_t is smooth for t≠4 and piecewise smooth when
t=4 (Point <ref> of Theorem <ref>).
We also have similar formula for the integral over the complement of
Σ_t, provided that ψ is compactly supported, but with
the direction of the unit normal reversed. Proposition <ref>
then tells us that the boundary terms in the two integrals cancel, giving

    ∫_ℂs_t(λ)Δχ(λ) d^2λ=∫
    _(Σ_t)^c(Δ s_t(λ))χ(λ
    ) d^2λ+∫_Σ_t(Δ s_t(λ))χ(λ
    ) d^2λ,

where the integral over (Σ_t)^c is actually zero, since
Δ s_t(λ)=0 there. The formula (<ref>) says
that the distributional Laplacian of s_t may be computed by taking the
ordinary, pointwise Laplacian in Σ_t and in Σ_t
and ignoring the boundary of Σ_t.


We now have all the ingredients for a proof of Theorem <ref>.

Proposition <ref>
tells us that we can compute the distributional Laplacian of s_t
separately inside Σ_t and outside Σ_t, ignoring
the boundary. Theorem <ref> tells us that the Laplacian outside
Σ_t is zero. Corollary <ref> gives us the
form of Δ s_t inside Σ_t, while Proposition <ref>
identifies the function m_t appearing in Corollary <ref>.
The claimed formula for the Brown measure therefore holds.




§ FURTHER PROPERTIES OF THE BROWN MEASURE




 §.§ The formula for ω


In this subsection, we derive the formula for w_t given in Proposition
<ref> in terms of the density ω. Throughout, we will write
the function T in (<ref>) in polar coordinates as

    T(r,θ)=(r^2+1-2rcosθ)log(r^2)/r^2-1.

We start with a simple rewriting of the expression for w_t in Theorem
<ref>.


The density w_t(θ) in Theorem <ref> may also be written as

    w_t(θ)=1/2π t(  1+∂/∂θ[h(r_t(θ))sinθ])  ,

where

    h(r)=rlog(r^2)/r^2-1.





We start by noting that the point with polar coordinates (r_t
(θ),θ) is on the boundary of Σ_t. Thus, by Theorem
<ref>, we have T(r_t(θ),θ)=t, from which we
obtain

    1/r_t(θ)^2+1-2r_t(θ)cosθ=1/tlog(r_t(θ))/r_t(θ)^2-1.

Thus, we may write

    2r_t(θ)sinθ/r_t(θ)^2+1-2r_t(θ)cosθ=2/th(r_t(θ))sinθ,

from which the claimed formula follows easily from the expression in Theorem
<ref>.


We now formulate the main result of this subsection, whose proof is on p.
densityWithoutProof.


Consider the function ω(r,θ)
defined in (<ref>). Although the right-hand side of
(<ref>) is indeterminate at r=1, the function ω has a
smooth extension to all r>0 and all θ. The function w_t(θ)
in Theorem <ref> can then be expressed as

    w_t(θ)=1/2π tω(r_t(θ),θ).



The function ω has the following properties.



  * We have ω(1/r,θ)=ω
(r,θ) for all r>0 and all θ.


  * When r=1, we have

    ω(1,θ)=31+cosθ/2+cosθ.

In particular, ω(1,0)=2 and ω(1,π)=0.


  * The density ω(r,θ) is strictly
positive except when r=1 and θ=±π. Furthermore, ω
(r,θ)≤2 with equality precisely when r=1 and θ=0.


  * We have

    lim_r→0ω(r,θ)=1,

where the limit is uniform in θ.



We now derive consequences for w_t. For t≤4, the density
w_t(θ) is only defined for -θ_max(t)<θ<θ_max(t), where θ_max(t)=cos^-1(1-t/2), while for t>4, the density
w_t(θ) is defined for all θ. (Recall Theorem
<ref>.)

If t>4, then w_t(θ) is strictly positive for all
θ. If t<4, then w_t(θ) is strictly positive for
-θ_max(t)<θ<θ_max(t) and the limit as θ
approaches ±θ_max(t) of w_t(θ) is strictly positive.
Finally, if t=4, then w_t(θ) is strictly positive for -π
<θ<π, but lim_θ→±πw_t(θ)=0.



The only time ω(r,θ) equals zero is when r=1 and θ=±π. When t>4, the function r_t(θ) is continuous and and greater
than 1 for all θ, so that w_t(θ) is strictly positive in
this case. When t≤4, we know from Proposition <ref> that
r_t(θ) is greater than 1 for |θ|
<θ_max(t) and approaches 1 when θ approaches ±θ_max(t). Thus, w_t(θ)=ω(r_t(θ),θ) is strictly positive
for |θ| <θ_max(t). When t<4, we have
θ_max(t)=cos^-1(1-t/2)<π and the limiting value of
w_t(θ)—namely ω(1,θ_max)—will be positive.
Finally, when t=4, we have θ_max(t)=π and the limiting value of
w_t(θ) is ω(1,π)=0.
















The density w_t(θ) has the
property that

    w_t(θ)∼1/π t

for small t. More precisely, for all sufficiently small t and all
θ∈(-θ_max(t),θ_max(t)), the quantity π
tw_t(θ) is close to 1. Furthermore,

    w_t(θ)∼1/2π t

for large t. More precisely, for all sufficiently large t and all
θ, the quantity 2π tw_t(θ) is close to 1.


See Figures <ref> and <ref>. The small- and
large-t behavior of the region Σ_t can also be determined using the
behavior of the function T(λ) near λ=1 (small t) and near
λ=0 (large t), together with the invariance of the region under
λ↦1/λ. For small t, the region resembles a disk of
radius √(t) around 1, while for large t, the region resembles an
annulus with inner radius e^-t/2 and outer radius e^t/2. In
particular, the expected behavior of the Brown measure for small t can be
observed: it resembles the uniform probability measure on a disk of radius
√(t) centered at 1.














When t is small, the entire boundary of Σ_t will be close to
λ=1, since this is the only point where T(λ)=0. Furthermore,
when t is small, θ_max(t)=cos^-1(1-t/2) is close to zero. When
t is small, therefore, the quantity

    π tw_t(θ)=1/2ω(r_t(θ),θ)

will be close to ω(1,0)/2=1 for all θ∈(-θ_max
(t),θ_max(t)), by Point <ref> of Theorem
<ref>.

When t is large (in particular, greater than 4), the inner boundary of the
domain will be close to λ=0, since this is the only point in the unit
disk where T(λ) is large. Thus, for large t, the inner radius
1/r_t(θ) of the domain will be uniformly small, and therefore

    2π tw_t(θ)=ω(r_t(θ),θ)=ω(1/r_t(θ
    ),θ)

will be uniformly close to 1, by Point <ref> of Theorem
<ref>.


We
note that the function T in (<ref>) can be written as

    T(r,θ)=(  r+1/r-2cosθ)  h(r),

so that

    ∂ T/∂ r     =(  1-1/r^2)
    h(r)+(  r+1/r-2cosθ)  h^'(r);
    ∂ T/∂θ     =2sinθ h(r).

Applying implicit differentiation to the identity T(r_t(θ),θ)=t
then gives

    dr_t(θ)/dθ=-∂ T/∂θ/∂
    T/∂ r.



By the chain rule and (<ref>), d/dθ[h(r_t
(θ))sinθ]=q(r_t(θ),θ), where

    q(r,θ)       =h(r)cosθ-h^'(r)sinθ∂
    T/∂θ/∂ T/∂ r
         =h(r)cosθ- 2h^'(r)sin^2θ h(r)/(
    1-1/r^2)  h(r)+(  r+1/r-2cosθ)
    h^'(r).

After computing that

    h^'(r)=2/r^2-1-r^2+1/r(r^2-1)h(r),

it is a straightforward but tedious exercise to simplify (<ref>) and
obtain the claimed formula (<ref>).

Since h(1/r)=h(r), we may readily verify Point (<ref>);
both numerator and denominator in the fraction on the right-hand side of
(<ref>) change by a factor of 1/r^2 when r is replaced by
1/r.

To understand the behavior of ω at r=1, we need to understand the
function h better. We may easily calculate that h has a removable
singularity at r=1 with h(1)=1, h^'(1)=0, and h^''(1)=-1/3. We also claim that h satisfies 0<h(r)≤1, with h(r)=1 only
at r=1. To verify the claim, we first compute that lim_r→
0h(r)=0 and that

    h^'(r)=2(r^2-1)+(r^2+1)log(1/r^2)/(r^2-1)^2.

Using the Taylor expansion of logarithm, we may then compute that

    h^'(r)=1/(r^2-1)^2∑_k=3^∞(  2/k-1/k+1)  (1-r^2)^k>0

for 0<r<1. Thus, h(r) increases from 0 to 1 on [0,1].

We now write h in the form

    h(r)=1-c(r)(r-1)^2

for some analytic function c(r), with c(1)=1/6. The minus sign in
(<ref>) is convenient because h has a strict global maximum at 1,
which means c(r) is strictly positive everywhere.

Now, since h(1)=1, the fraction on the right-hand side of
(<ref>) is of 0/0 form when r=1. To rectify this situation,
we observe that α and β may be written as

    α(r)=(r-1)^2[1+2rc(r)];  β(r)=(r-1)^2[1-(r^2+1)c(r)].

Thus, we can take a factor of (r-1)^2 out of numerator and denominator to
obtain

    ω(r,θ)=1+h(r)α̃(r)cosθ+β̃
    (r)/β̃(r)cosθ+α̃(r),

where α̃(r)=1+2rc(r) and β̃(r)=1-(r^2+1)c(r).
This expression is no longer of 0/0 form at r=1. Indeed, since h(1)=1
and c(1)=1/6, we may easily verify the claimed formula for ω
(1,θ) in Point <ref> of the theorem. We will shortly
verify that the denominator in the fraction on the right-hand side of
(<ref>) is positive for all r>0 and all θ, from which
the claimed smooth extension of ω follows.

To verify the claimed positivity of ω, we first observe that
β̃(r)z+α̃(r) is positive when z=1 (with a value of
2-(r-1)^2c(r)=1+h(r)) and also positive when z=-1 (with a value of
(r+1)^2c(r)), and hence positive for all -1≤ z≤1. Thus, the
denominator in the fraction on the right-hand side of (<ref>) is
never zero. We then compute that

    d/dzα̃(r)z+β̃(r)/β̃(r)z+α̃(r)=α̃(r)^2-β̃(r)^2
    /(β̃(r)z+α̃(r))^2=(r+1)^2h(r)/(β̃(r)z+α̃(r))^2>0

for all r and θ. Thus, (α̃(r)z+β̃
(r))/(β̃(r)z+α̃(r)) increases from -1 to 1 as z
increases from -1 to 1. Since h(r) is positive, we conclude that

    1-h(r)≤ω(r,θ)≤1+h(r)

for all r and θ, with equality when cosθ=-1 in the first case
and when cosθ=1 in the second case. Since h(r)≤1 with equality
only at r=1, we see that ω(r,θ) is positive except when r=1
and cosθ=-1. Similarly, ω(r,θ)≤2 with equality only if
r=1 and cosθ=1.

Finally, from the definition (<ref>) and the fact that lim
_r→0h(r)=0, we find that lim_r→0c(r)=1. Thus, as
r→0, we have α̃(r)→1 and β̃(r)→0. In this limit, the fraction on the right-hand side of
(<ref>) converges uniformly to cosθ, while h(r) tends
to zero, giving Point <ref>.




 §.§ The connection to free unitary Brownian
motion


Recall from Theorem <ref> that the spectral measure ν_t of
the free unitary Brownian motion u_t was computed by Biane. In this
subsection, we prove Proposition <ref>, which connects the
Brown measure of b_t to Biane's measure ν_t. The support of ν
_t is a proper subset of the unit circle for t<4 and the entire unit
circle for t≥4. For t<4, the support of ν_t consists of points
with angles ϕ satisfying |ϕ|≤ϕ_max(t),
where

    ϕ_max(t)=1/2√(t(4-t))+cos^-1(1-t/2).



Recall the definition in (<ref>) of the function f_t. Then f_t
maps the boundary of Σ_t into the unit circle. (This is true by the
definition (<ref>) for points in ∂Σ_t outside the unit
circle and follows by continuity for points in ∂Σ_t in the
unit circle.) Indeed, let the outer boundary of Σ_t, denoted
∂Σ_t^out, be the portion of ∂Σ_t
outside the open unit disk. Then f_t is a homeomorphism of ∂Σ_t^out to the support of ν_t:

    f_t:∂Σ_t^out↔supp(ν_t).

In particular, for t<4, let us define

    θ_max(t)=cos^-1(1-t/2),

so that the two points in ∂Σ_t∩ S^1 have angles
±θ_max(t) (Theorem <ref>). Then

    f_t(e^iθ_max(t))=e^iϕ_max(t),

as may be verified by direct computation from the definition of f_t. (Use
the formula (<ref>) below with r=1 and cosθ=1-t/2.)

We now describe the map (<ref>) more concretely. We denote by
λ_t(θ) the point at angle θ in ∂Σ
_t^out:

    λ_t(θ)=r_t(θ)e^iθ,

where for t<4, we require |θ|≤θ_max(t). Then the map in (<ref>) can be thought of as a map of
θ to ϕ determined by the relation

    f_t(  λ_t(θ))  =e^iϕ.



We now observe a close relationship between the density w_t(θ) in
Theorem <ref> and the map in (<ref>).


Let ϕ and θ be related as in
(<ref>), where if t<4, we require |ϕ|≤ϕ_max(t) and |θ|≤θ_max(t).
Then the density w_t in Theorem <ref> may be computed as

    w_t(θ)=1/2π tdϕ/dθ.

We may also write this formula as a logarithmic derivative of f_t along
the outer boundary of Σ_t:

    w_t(θ)=1/2π t1/id/dθf_t
    (λ_t(θ))/f_t(λ_t(θ)).





We compute that

    Im(  1+λ/1-λ)  =2Imλ/|λ-1| ^2=2rsinθ/r^2+1-2rcosθ.

Thus, using the definition (<ref>) of f_t, we find that

    (f_t(λ))=λ+ e^t/21+λ/1-λ=θ+trsinθ/r^2+1-2rcosθ.

Evaluating this expression at the point λ_t(θ) gives

    ϕ     =(f_t(λ_t(θ)))
         =θ+tr_t(θ)sinθ/r_t(θ)^2+1-2r_t
    (θ)cosθ.

(Strictly speaking, ϕ and θ are only defined “mod
2π,” but for any local continuous version of θ,
the last expression in (<ref>) gives a local continuous version of
ϕ.) Thus,

    dϕ=(  1+td/dθr_t(θ)sinθ/r_t
    (θ)^2+1-2r_t(θ)cosθ)   dθ

and the formula (<ref>) follows easily by recalling the definition
(<ref>) of w_t. The expression (<ref>) is then obtained
by noting that ϕ=1/ilog f_t(λ_t(θ)).



Biane's measure ν_t may be computed as

    dν_t(ϕ)=r_t(θ)^2-1/r_t(θ)^2+1-2r_t
    (θ)cosθdϕ/2π

or as

    dν_t(ϕ)=log(r_t(θ))/π tdϕ.

Here, as usual, r_t(θ) is the outer radius of the domain Σ_t
and θ is viewed as a function of ϕ by inverting the relationship
(<ref>). When t<4, the formula should be used only for |ϕ|≤ϕ_max(t).



We make use of the expression for ν_t in Theorem <ref>. If
ϕ is in the interior of the support of ν_t, then χ_t
(e^iϕ) is in the open unit disk, so that the density of ν_t is
nonzero at this point. Now, since χ_t is an inverse function to f_t
we see that χ_t(e^iϕ) is (for ϕ in the interior of the
support of ν_t) the unique point λ with |λ| <1 for which f_t(λ)=e^iϕ. Thus,

    dν_t(ϕ)=1-1/r_t(θ)^2/1+1/r_t(θ)^2-2cosθ/r_t(θ)dϕ/2π,

which reduces to (<ref>). Finally, since T(λ)=t on
∂Σ_t (Theorem <ref>), we have

    (r_t(θ)^2+1-2r_t(θ)cosθ)log(r_t(θ)^2
    )/r_t(θ)^2-1=t,

which allows us to obtain (<ref>) from (<ref>).


We are now ready for the proof of Proposition <ref>.

The distribution of
λ with respect to the Brown measure of b_t is given in
(<ref>) as 2log(r_t(θ))w_t(θ) dθ, which we write
using Proposition <ref> and Proposition
<ref> as

    2log(r_t(θ))w_t(θ) dθ     =2log(r_t(θ))1/2π tdϕ/dθ dθ
         =log(r_t(θ))dϕ/π t
         =dν_t(ϕ),

as claimed.


The value Φ_t(λ)
is computed by first taking the argument of λ to obtain θ and
then applying the map in (<ref>) to obtain ϕ. Thus, the first
result is just a restatement of Proposition <ref>. For the
uniqueness claim, suppose a measure μ on Σ_t has the form

    dμ(λ)=1/r^2g(θ) r dr dθ.

Then the distribution of the argument θ of λ will be, by
integrating out the radial variable, 2log(r_t(θ))g(θ) dθ.
The distribution of ϕ will then be

    2log(r_t(θ)g(θ)dθ/dϕ dϕ=2log(r_t
    (θ)g(θ)1/2π tw_t(θ) dϕ.

The only way this can reduce to Biane's measure as computed in (<ref>)
is if g coincides with w_t.




 §.§ Acknowledgments


The authors thank Ching-Wei Ho and Maciej Nowak for useful discussions.

99                                                                                               


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