# Expected signature of stopped Brownian motion on $d$-dimensional
$C^{2,\alpha}$-domains has finite radius of convergence everywhere: $2\leq
d\leq 8$

Siran Li  Siran Li: New York University – Shanghai, Office 1146, 1555 Century
Avenue, Pudong District, Shanghai, China (200122)  Siran Li: NYU-ECNU
Institute of Mathematical Sciences, Room 340, Geography Building, 3663 North
Zhongshan Road, Shanghai, China (200062<EMAIL_ADDRESS>and  Hao Ni
Hao Ni: Department of Mathematics University College, Gower Street, WC1E 6BT,
London, UK<EMAIL_ADDRESS>
###### Abstract.

A fundamental question in rough path theory is whether the expected signature
of a geometric rough path completely determines the law of signature. One
sufficient condition is that the expected signature has infinite radius of
convergence, which is satisfied by various stochastic processes on a fixed
time interval, including the Brownian motion. In contrast, for the Brownian
motion stopped upon the first exit time from a bounded domain $\Omega$, it is
only known that the radius of convergence for the expected signature on
sufficiently regular $\Omega$ is strictly positive everywhere, and that the
radius of convergence is finite at some point when $\Omega$ is the
$2$-dimensional unit disc ([1]).

In this paper, we prove that on any bounded $C^{2,\alpha}$-domain
$\Omega\subset\mathbb{R}^{d}$ with $2\leq d\leq 8$, the expected signature of
the stopped Brownian motion has finite radius of convergence everywhere. A key
ingredient of our proof is the introduction of a “domain-averaging hyperbolic
development” (see Definition 4.1), which allows us to symmetrize the PDE
system for the hyperbolic development of expected signature by averaging over
rotated domains.

###### Key words and phrases:

Rough path; expected signature; stopped Brownian motion; first exit time

###### 2020 Mathematics Subject Classification:

Primary: 60L20; Secondary: 35R45

##  1\. Introduction

The theory of rough paths is a generalisation of the classical control theory.
It sets forth a framework that makes sense of solutions to differential
equations driven by irregular signals; for example, the Brownian motion.

A fundamental concept in rough path theory is the _signature_ of a path.
Playing a similar role to that of the moment generating function of a random
variable, the _expected signature_ is of both theoretical and practical
significance. For example, computation for the expected signature of the
Brownian motion on $[0,1]$ leads to the notion of _cubature on Wiener space_ ,
which is a high order numerical method for high dimensional SDEs (stochastic
differential equations) and semi-elliptic PDEs (partial differential
equations); see [2]. Recently, in [3] a metric for laws of stochastic
processes has been proposed based on the normalised expected signature. It can
be used for nonparametric two-sample hypothesis tests for laws of stochastic
processes, and potentially has much wider applications in machine learning.

One central question concerning the expected signature, known as the _moment
problem_ , asks if the expected signature uniquely determines the law of
random signature. A sufficient condition for the affirmative answer has
recently been identified for the moment problem by Chevyrev–Lyons ([4],
Proposition 6.1): the radius of convergence of expected signature is infinite.

The infiniteness of radius of convergence for the expected signature has been
verified for several popular stochastic processes on a fixed time horizon,
_e.g._ , fractional Brownian motions with Hurst parameter in
$\left]\frac{1}{4},1\right]$ (see [4, 5]). It is nonetheless considerably
challenging to check whether this is satisfied by processes _up to a random
time_. Even for the simplest case of the Brownian motion up to the first exit
time from a bounded domain $\Omega\subset\mathbb{R}^{d}$ (in brief, “ _stopped
Brownian motion_ ” in the sequel), it has remained open ever since it was
proposed in [6].

As a first step towards the aforementioned problem, it was established in [6]
that, under suitable regularity assumptions for the domain (see Proposition
3.1 and Remark 3.2), the expected signature of the stopped Brownian motion has
a geometric upper bound for its decay rate. This is achieved by deriving a
system of nested PDEs (_i.e._ , partial differential equations graded by a
parameter $n\in\mathbb{N}$) satisfied by the expected signature and applying
the standard boundary regularity theory for elliptic PDEs. Despite its
insufficiency to resolve the moment problem, the geometric upper bound
warrants the positivity of the radius of convergence.

Recently, Boedihardjo–Diehl–Mezzarobba–Ni [1] showed that the stopped Brownian
motion on the $2$-dimensional unit disc has finite radius of convergence. The
proof in [1] relies crucially on the technique of _hyperbolic development_ ,
which plays an essential role in Hambly–Lyons’ proof of the unique
determination of paths of bounded variation by the signature; see [7]. Indeed,
by taking a morphism (which appears in [7]) from the tensor algebra space to
$\mathfrak{gl}(3;\mathbb{R})$ and exploiting the partial symmetries of the
resulting matrix-valued PDE systems, a nested system of three ODEs has been
obtained in [1]. These ODEs can be solved explicitly, from which one deduces
the finiteness of radius of convergence. Nonetheless, [1] relies heavily on
the rotational invariance of the domain, thus making it difficult to be
extended to general domains. In addition, even for the case of the unit disc,
it remains unknown if the stopped Brownian motion starting at a point apart
from the centre has finite radius of convergence.

The main result of our paper gives a complete solution to the finiteness
problem of the expected signature of stopped Brownian motions on bounded
domains under mild regularity assumptions, for all dimensions up to $8$:

######  Theorem 1.1.

Let $\Omega$ be bounded $C^{2,\alpha}$-domain in $\mathbb{R}^{d}$; $2\leq
d\leq 8$. The expected signature $\Phi$ of a Brownian motion stopped upon the
first exit time from $\Omega$ has finite radius of convergence at every point
in $\Omega$.

The strategy of our proof is outlined as follows. First, by averaging the
hyperbolic development of expected signature over the rotated images of the
domain, we construct a quantity
$\overline{\mathcal{H}_{\lambda,\epsilon}}(z)$, the “ _domain-averaging
development_ ”, that has the following features:

  * •

$\overline{\mathcal{H}_{\lambda,\epsilon}}(z)$ is rotationally invariant (with
respect the domain rotations);

  * •

$\overline{\mathcal{H}_{\lambda,\epsilon}}(z)$ satisfies the same PDE as that
for $\mathcal{H}_{\lambda,\Omega}$; and

  * •

$\overline{\mathcal{H}_{\lambda,\epsilon}}(z)$ inherits the finiteness of
radius of convergence from the non-averaged hyperbolic development
$\mathcal{H}_{\lambda,\Omega}$ of the expected signature. Thus, the operation
of domain averaging preserves the lower bound for radius of convergence of the
hyperbolic development.

Working locally near $z\in\Omega$, we arrive at the same PDE for
$\overline{\mathcal{H}_{\lambda,\epsilon}}(z)$ as for its non-averaged
analogue on a small ball around $z$. Loosely speaking, “half” of the boundary
conditions will gone missing for the PDE for
$\overline{\mathcal{H}_{\lambda,\epsilon}}(z)$; nevertheless, geometric
properties of the hyperbolic development enable us to establish uniform lower
bounds for the remaining component of
$\overline{\mathcal{H}_{\lambda,\epsilon}}(z)$.

Organisation. The remaining parts of the paper is organised as follows. §2
summarises background materials on the rough path theory, expected signature,
and hyperbolic development. In §3 we discuss the regularity theory and
probabilistic consequences of the PDE associated to the expected signature.
The crucial technique in this work (a symmetrization argument) shall be
introduced in §4. Our main result, Theorem 1.1, is proved in §5 for $d=2$ and
in §6 for $3\leq d\leq 8$. Some technical computations are given in the
appendix.

We denote $E:=\mathbb{R}^{d}$ throughout this paper.

##  2\. Preliminaries

###  2.1. Signature

We first introduce the tensor algebra space over $E$:

######  Definition 2.1.

A formal $E$-tensor series is a sequence of tensors $\left(a_{n}\in E^{\otimes
n}\right)_{n\in\mathbb{N}}$ which we write as
$a=\left(a_{0},a_{1},\ldots\right)$. There are two binary operations on
$E$-tensor series, addition $+$ and product $\otimes$ — let
$\mathbf{a}=(a_{0},a_{1},\ldots)$ and $\mathbf{b}=(b_{0},b_{1},\ldots)$ be
$E$-tensor series; then
$\mathbf{a}+\mathbf{b}:=(a_{0}+b_{0},a_{1}+b_{1},\ldots)$ and
$\mathbf{a}\otimes\mathbf{b}:=(c_{0},c_{1},\ldots)$, where
$c_{n}:=\sum_{k=0}^{n}a_{k}\otimes b_{n-k}$ for each $n$.

In addition, we write $\mathbf{1}:=(1,0,\ldots)$, $\mathbf{0}:=(0,0,\ldots)$,
and $\lambda\mathbf{a}:=(\lambda a_{0},\lambda a_{1},\ldots)$ for
$\lambda\in\mathbb{R}$. Also, $T\left(\left(E\right)\right)$ denotes the
vectorspace of formal $E$-tensor series.

The space $T\left(\left(E\right)\right)$ equipped with $+$ and $\otimes$ is an
associative unital algebra over $\mathbb{R}$. An element
$\mathbf{a}=(a_{0},a_{1},\ldots)\in T\left(\left(E\right)\right)$ is
invertible if and only if $a_{0}\neq 0$. In this case, its inverse is

$\mathbf{a}^{-1}=\frac{1}{a_{0}}\sum_{n\geq
0}\left(\mathbf{1}-\frac{\mathbf{a}}{a_{0}}\right)^{n}.$

It is well defined because, at any given degree, only finitely many terms of
the summation are non-zero. In particular, the subset $\left\\{\mathbf{a}\in
T\left(\left(E\right)\right):\,a_{0}=1\right\\}$ forms a group.

######  Definition 2.2.

Let $n\geq 1$ be an integer. Let
$B_{n}:=\\{\mathbf{a}=(a_{0},a_{1},\ldots):\,a_{0}=\ldots=a_{n}=0\\}.$ The
truncated tensor algebra $T^{(n)}(E)$ of order $n$ over $E$ is defined as the
quotient algebra

$T^{(n)}(E)=T\left(\left(E\right)\right)/B_{n}.$

The canonical epimorphism $T\left(\left(E\right)\right)\rightarrow T^{(n)}(E)$
is denoted by $\mathbf{Proj}_{n}$.

We collect below a few basic facts about geometric rough paths. See [8, 9, 10]
for details.

For $p\geq 1$ and $J\subset\mathbb{R}$ an open interval, one can enhance a
path $X:J\rightarrow E$ of finite $1$-variation to a function defined on the
simplex $\left\\{(s,t):\,s\leq t,s,t\in J\right\\}$ with values in
$T^{(\lfloor p\rfloor)}(E)$ via its iterated integrals; _i.e._ , $S^{\lfloor
p\rfloor}(X)=\left(1,X^{1}_{J},\cdots X^{\lfloor p\rfloor}_{J}\right)$ where

$\displaystyle X^{n}_{s,t}=\underset{s\leq u_{1}\leq\cdots\leq u_{n}\leq
t}{\int\cdots\int}\,{\rm d}X_{u_{1}}\otimes\,{\rm
d}X_{u_{2}}\otimes\cdots\otimes\,{\rm d}X_{u_{n}},$

for $n=1,2,\cdots,\lfloor p\rfloor$. We call the enhancement of $X$ in
$T^{(\lfloor p\rfloor)}(E)$ a _smooth rough path_. The signature of $X$ is the
collection $S(X)=\left(1,X^{1}_{J},\cdots,X^{n}_{J},\cdots\right)$ of all the
iterated integrals of $X$.

######  Definition 2.3 ($p$-variation distance).

Let $p\geq 1$. For smooth rough paths $\mathbf{X},\mathbf{Y}:J\to T^{(\lfloor
p\rfloor)}(E)$, the $p$-variation metric between $\mathbf{X}$ and $\mathbf{Y}$
is given by

$\displaystyle d(\mathbf{X},\mathbf{Y})=\max_{k=1,\cdots\lfloor
p\rfloor}\left(\sup_{\mathcal{D}}\sum_{i=1}^{n-1}\left|\mathbf{X}_{u_{i},u_{i+1}}^{k}-\mathbf{Y}_{u_{i},u_{i+1}}^{k}\right|^{\frac{p}{k}}\right)^{\frac{k}{p}}+\sup_{t\in
J}\left|X^{1}_{t}-Y^{1}_{t}\right|,$

with the supremum taken over all finite divisions
$\mathcal{D}=(u_{1},u_{2},\cdots,u_{n})$ of $J$.

######  Definition 2.4 (Geometric rough path).

$\mathbf{X}:J\rightarrow T^{(\lfloor p\rfloor)}(E)$ is said to be a geometric
$p$-rough path if there exists a sequence of smooth rough path
$(\mathbf{X}^{n})_{n\in\mathbb{N}}$ such that $\mathbf{X}$ is the limit of
$\mathbf{X}^{n}$ in the $p$-variation metric. The space of geometric $p$-rough
paths is denoted as $G\Omega_{p}(J,E)$.

###  2.2. Stopped Brownian motion and its expected signature

Denote by $(e_{1},\ldots,e_{d})$ the canonical basis for $E=\mathbb{R}^{d}$.
Let $\left(B_{t}\right)_{t\geq 0}$ be a standard Brownian motion on $E$ under
a probability space $(\mathbf{\Omega},\mathcal{F},P^{z})$ with its canonical
filtration $\mathcal{F}=\left(\mathcal{F}_{t}\right)_{t\geq 0}$, where
$P^{z}(B_{0}=z)=1$ for $z\in E$.

######  Definition 2.5.

Let $\Omega$ be a domain (_i.e._ , a connected open set) in $E$. Then

$\tau_{\Omega}=\inf\\{t\geq 0:B_{t}\in E\setminus\Omega\\}$

is the first exit time of Brownian motion from $\Omega$.

######  Definition 2.6.

Let $J$ be a compact time interval. Let $B:J\rightarrow E$ be an $E$-valued
Brownian motion path. The signature of $B$, denoted by $S(B_{J})$, is the
element $(1,B^{1},...,B^{n},...)$ of $T\left(\left(E\right)\right)$ defined
for each $n\geq 1$ as follows:

$B^{n}_{J}=\underset{\underset{u_{1},\ldots,u_{n}\in
J}{u_{1}<\ldots<u_{n}}}{\int\cdots\int}\,{\rm
d}B_{u_{1}}\otimes\cdots\otimes\,{\rm d}B_{u_{n}},$

where the integral is taken in the Stratonovich sense. The truncated signature
of $B$ of order $n$ is denoted by $S^{n}(B_{J})$, _i.e._ ,
$S^{n}(B_{J})=\mathbf{Proj}_{n}(S(B_{J}))$.

One also writes

$\mathbf{B}_{J}=\sum_{n=0}^{\infty}\sum_{i_{1},\ldots i_{n}\in\\{1,\ldots
d\\}}\left(\underset{\underset{u_{1},\ldots,u_{n}\in
J}{u_{1}<\ldots<u_{n}}}{\int\cdots\int}\,{\rm
d}B_{u_{1}}^{(i_{1})}\otimes\ldots\otimes\,{\rm
d}B_{u_{n}}^{(i_{n})}\right)e_{i_{1}}\otimes\ldots\otimes e_{i_{n}}$

where

$\underset{\underset{u_{1},\ldots,u_{n}\in
J}{u_{1}<\ldots<u_{n}}}{\int\cdots\int}\,{\rm
d}B_{u_{1}}^{(i_{1})}\otimes\ldots\otimes\,{\rm
d}B_{u_{n}}^{(i_{n})}=:\mathbf{Proj}^{I}(S(B_{J}))$

is the co-ordinate signature of Brownian motion indexed by
$I:=(i_{1},\ldots,i_{n})$. The (Stratonovich) signature is defined for _a.e._
Brownian path $B$ and for all pairs of times $(s,t)$ where $s\leq t$.

######  Lemma 2.7 (See [11]).

For $T>0$, the Stratonovich signature of Brownian motion
$t\mapsto(S(B_{0,t}))_{t\in[0,T]}$ is a geometric $p$-rough path almost surely
for $p>2$.

We are interested in the random signature
$S\left(B_{[0,\tau_{\Omega}]}\right)$ of the Brownian path up to the first
exit time $\tau_{\Omega}$. It is shown in [6] that
$\pi^{I}\left(S\left(B_{\left[0,\tau_{\Omega}\right]}\right)\right)$ has
finite expectation with respect to the Wiener measure for every index $I$.
Thus, the tensor-valued function
$S\left(B_{\left[0,\tau_{\Omega}\right]}\right)$ is integrable.

######  Definition 2.8.

We denote by $\Phi_{\Omega}(z)$ the expected signature of Brownian motion
starting at $z$ and stopped upon the first exit time $\tau_{\Omega}$ from a
domain $\Omega$. That is,

$\Phi_{\Omega}(z)=\mathbb{E}^{z}\left[S\left(B_{\left[0,\tau_{\Omega}\right]}\right)\right].$

One of the main results in [6] states that $\Phi$ satisfies a system of nested
PDEs:

######  Theorem 2.9.

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain. Then $\Phi$ satisfies

$\Delta\Phi(z)=-\left(\sum_{i=1}^{d}e_{i}\otimes
e_{i}\right)\otimes\Phi(z)-2\sum_{i=1}^{d}\left(e_{i}\otimes\frac{\partial\Phi}{\partial
z^{i}}(z)\right)\quad\text{for each }z\in\Omega,$ (1)

with the boundary condition that for every $z\in\Omega$,

$\lim_{t\nearrow\tau_{\Omega}}\Phi(B_{t})=\mathbf{1}\qquad\text{a.s. in
}P^{z},$ (2)

and the initial conditions

$\displaystyle\mathbf{Proj}_{0}(\Phi_{\Omega}(z))$ $\displaystyle=$
$\displaystyle 1\qquad\text{for each }z\in\overline{\Omega},$ (3)
$\displaystyle\mathbf{Proj}_{1}(\Phi_{\Omega}(z))$ $\displaystyle=$
$\displaystyle 0\qquad\text{for each }z\in\overline{\Omega}.$ (4)

###  2.3. Hyperbolic development

Let $\mathscr{A}$ be a normed algebra. Any linear operator $M:E\to\mathscr{A}$
can be extended to $T((E))$ _by naturality_. Indeed, one first defines
$\hat{M}$ on $E^{\otimes k}$ for each $k$ via

$\displaystyle\hat{M}\left(e_{i_{1}}\otimes e_{i_{2}}\otimes\cdots\otimes
e_{i_{k}}\right):=M(e_{i_{1}})M(e_{i_{2}})\cdots M(e_{i_{k}}),$

and then extends to $T((E))$ by linearity. The right-hand side of the above
identity is understood as a product in the algebra $\mathscr{A}$. By an abuse
of notations, we always write $\hat{M}\equiv M$.

For any $\lambda>0$, the action of $\lambda M$ on the signature of a path
$\gamma$ of bounded $1$-variation is given by $M(\lambda\gamma)$. Similarly,
for $\lambda>0$, the action of $\lambda M$ on $\Phi_{\Omega}$ is given by

$\mathcal{M}_{\lambda,\Omega}(z):=(\lambda M)\Phi_{\Omega}(z)\quad\text{ for
each }z\in\Omega.$ (5)

This can be recast into

$\mathcal{M}_{\lambda,\Omega}(z)=\sum_{n=0}^{\infty}\lambda^{n}M\left[\mathbf{Proj}_{n}(\Phi_{\Omega}(z))\right]$
(6)

at least for small enough $|\lambda|$. See Lyons–Ni [12], Theorem 3.6.

The hyperbolic development (_cf._ Boedihardjo–Diehl–Mezzarobba–Ni [1];
Hambly–Lyons [7]) refers to following particular choice of $M$, denoted as
$H:T((E))\to\mathfrak{gl}(d+1;\mathbb{R})$ —

$\displaystyle H(c)=cI_{d+1}\qquad\text{ for }c\in\mathbb{R},$ $\displaystyle
H(x,y):=\begin{bmatrix}0&0&\cdots&0&z^{1}\\\ 0&0&\cdots&0&z^{2}\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&\cdots&0&z^{d}\\\
z^{1}&z^{2}&\cdots&z^{d}&0\end{bmatrix}\qquad\text{ for
}z=\left(z^{1},\ldots,z^{d}\right)\in E,$

and $H$ is extended to $T((E))$ by naturality as above.

Lyons–Xu [13] characterises the space of the hyperbolic development of bounded
1-variation $\mathbb{R}^{d}$ path and provides an explicit expression for the
hyperbolic development in terms of the signature. Specifically, for
$\gamma:[0,1]\rightarrow\mathbb{R}^{d}$ of bounded $1$-variation and
$h_{\lambda}(\gamma):=H\left(S\left(\lambda\gamma\right)\right)(0,\cdots,0,1)^{\top}$,
it is shown that $h_{\lambda}(\gamma)$ is confined within the $d$-dimensional
hyperboloid

$\displaystyle\mathbb{H}_{d}:=\left\\{x\in\mathbb{R}^{d+1}:\,\sum_{j=1}^{d+1}\left(x^{j}\right)^{2}-\left(x^{d+1}\right)^{2}=-1,\,x^{d+1}>0\right\\}.$

Denote the set of “squared words” by

$\displaystyle\mathcal{E}_{2n}=\Big{\\{}w=(i_{1},i_{1},i_{2},i_{2},\cdots,i_{n},i_{n}):\,i_{1},i_{2},\cdots,i_{n}\in\\{1,2,\cdots,d\\}\Big{\\}}$

and, for $k\in\\{1,2,\cdots,d\\}$, put

$\displaystyle\mathcal{E}_{2n}^{(k)}=\Big{\\{}w=(i_{1},i_{1},i_{2},i_{2},\cdots,i_{n},i_{n},k):\,i_{1},i_{2},\cdots,i_{n}\in\\{1,2,\cdots,d\\}\Big{\\}}.$

######  Lemma 2.10.

Let $\gamma:[0,T]\rightarrow\mathbb{R}^{d}$ be a path of bounded
$1$-variation, and set

$h_{\lambda}(\gamma)\equiv\left(h_{\lambda}^{(1)}(\gamma),\cdots,h_{\lambda}^{(d+1)}(\gamma)\right)^{\top}:=H\left(S\left(\lambda\gamma\right)\right).$

We may express

$\displaystyle h_{\lambda}^{(k)}(\lambda\gamma)=\sum_{n\geq
0}\sum_{w\in\mathcal{E}_{2n}^{(k)}}\lambda^{2n+1}\mathbf{Proj}_{w}(S(\gamma)),$
$\displaystyle h_{\lambda}^{(d+1)}(\lambda\gamma)=\sum_{n\geq
0}\sum_{w\in\mathcal{E}_{2n}}\lambda^{2n}\mathbf{Proj}_{w}(S(\gamma)).$

As in Chevyrev–Lyons [4], the radius of convergence of the expected signature
of a rough path contains crucial information about the law of the path.

######  Definition 2.11.

The radius of convergence of a tensor $\varphi=(\varphi_{n})\in T((E))$ is the
radius of convergence of the series
$\lambda\mapsto\sum_{n=0}^{\infty}\lambda^{n}\||\varphi_{n}\||$. Throughout
this paper, $\||\bullet\||$ is taken to be the projective norm on $T((E))$.

The following characterisation for the finiteness of radius of convergence of
the expected signature can be found in [4].

######  Proposition 2.12.

$\varphi\in T((E))$ has infinite radius of convergence if and only if it lies
in the closure of $T((E))$ with respect to the coarsest topology for which the
following holds: for any normed algebra $\mathscr{A}$ and any morphism
$H\in{\rm Hom}(E;\mathscr{A})$, the natural extension
$\hat{H}:T((E))\to\mathscr{A}$ of $H$ (not relabelled in the sequel) is
continuous in this topology.

###  2.4. Further notations

A domain $\Omega\subset\mathbb{R}^{d}$ is a connected open set. The symbols
$z^{\dagger}$, ${\rm Re}(z)$, and ${\rm Im}(z)$ denote respectively complex
conjugate, real part, and imaginary part of $z\in\mathbb{C}$, $A^{\top}$ is
the transpose of matrix $A$, and ${\bf id}$ is the identity map (the domain
being clear from the context). We write $U\Subset V$ if the closure of $U$ is
in $V$. Our notations in §5 below are largely identical to those in [1];
_e.g._ , the usage of symbols $\alpha$, $\zeta$, $M$, etc. For essentially
analogous but slightly different symbols we shall distinguish by an overhead
bar; for instance, we write $\bar{A}_{\lambda}$, $\bar{B}_{\lambda}$, and
$\bar{C}_{\lambda}\equiv\overline{\mathcal{F}_{\lambda,\epsilon}}$ in contrast
to $A_{\lambda}$, $B_{\lambda}$, and $C_{\lambda}={F}_{\lambda,\Omega}$ in
[1], respectively. Finally, $\mathfrak{gl}(D;\mathbb{R})=\\{D\times D\text{
real matrices}\\}$ is equipped with the Hilbert–Schmidt norm as a normed
algebra.

A bounded domain $\Omega\Subset\mathbb{R}^{d}$ is said to be of regularity $X$
($=C^{k}$, $C^{2,\alpha}$, etc.) if its boundary are locally graphs of
functions of regularity $X$. More precisely, there exist open sets
$\mathcal{O}_{1},\ldots,\mathcal{O}_{N}\subset\mathbb{R}^{d}$ such that for
each $j\in\\{1,2,\ldots,N\\}$, it holds that

  * •

$\partial\Omega\subset\bigcup_{j=1}^{N}\mathcal{O}_{j}$;

  * •

$\mathcal{O}_{j}\cap\partial\Omega\neq\emptyset$;

  * •

there are an open set $V_{j}\subset\mathbb{R}^{d-1}$ and a function
$\psi_{j}:V_{j}\to\mathbb{R}$ of regularity $X$ such that
$\mathcal{O}_{j}\cap\partial\Omega=\left\\{(y,\psi_{j}(y)):\,y\in
V_{j}\right\\}$ and $\mathcal{O}_{j}\cap\Omega=\left\\{(y,z)\in
V_{j}\times\mathbb{R}:\,z<\psi_{j}(y)\right\\}$.

##  3\. PDE for the expected signature

Here we investigate the PDE (1) for the expected signature. Our results in
this section are valid for $E=\mathbb{R}^{d}$ for any $d\in\mathbb{Z}_{\geq
2}$. Also note that our regularity assumption on the domain is weaker than
that in Lyons–Ni [6]. See Remark 3.2 below.

######  Proposition 3.1.

Let $\Omega\subset E$ be a bounded $C^{2,\alpha}$-domain; $\alpha>0$. Then for
each $n=2,3,4,\ldots$, the $n^{\text{th}}$ term of $\Phi_{\Omega}$ satisfies
the PDE

$\displaystyle\Delta\left(\mathbf{Proj}_{n}(\Phi_{\Omega}(z))\right)=-2\sum_{i=1}^{d}e_{i}\otimes\frac{\partial\mathbf{Proj}_{n-1}(\Phi_{\Omega}(z))}{\partial
z^{i}}-\left(\sum_{i=1}^{d}e_{i}\otimes
e_{i}\right)\otimes\mathbf{Proj}_{n-2}(\Phi_{\Omega}(z))$ (7)

with the initial conditions (3), (4) and the boundary condition

$\Phi_{\Omega}={\bf 1}\qquad\text{ on }\partial\Omega.$ (8)

The above PDE has a unique classical solution in the sense that

$\mathbf{Proj}_{n}(\Phi_{\Omega})\in C^{\infty}\left(\Omega;E^{\otimes
n}\right)\cap C^{2,\alpha}\left(\overline{\Omega};E^{\otimes
n}\right)\qquad\text{ for each }n=0,1,2,\ldots.$

In addition, the radius of convergence of $\Phi$ is strictly positive.

###### Proof.

The proof is standard and shall only be sketched here. See [6], §3 for
details.

The boundary condition (8) follows from Eq. (2) and a continuity argument.

Given the initial conditions (3) and (4), one may apply the standard Schauder
theory ([14, 15]) inductively to show that Eq. (7) has a unique classical
solution

$\mathbf{Proj}_{n}(\Phi_{\Omega})\in C^{\infty}\left(\Omega;E^{\otimes
n}\right)\cap C^{2,\alpha}\left(\overline{\Omega};E^{\otimes n}\right).$

As a remark, although the Schauder theory is not applicable for systems of
elliptic PDEs in general, we can decouple the PDE (7) into scalar equations,
thus there is no danger of utilising the Schauder theory here. More precisely,
let $\\{e_{1},\ldots,e_{d}\\}$ be the canonical basis for $E$; we can find
scalarfields $f_{i_{1}\cdots i_{n}}:\Omega\to\mathbb{R}$ such that

$\Phi_{n}(z)=\sum_{i_{1},\ldots i_{n}\in\\{1,2,\ldots,d\\}}\,f_{i_{1}\cdots
i_{n}}(z)\,e_{i_{1}}\otimes\ldots\otimes e_{i_{n}}.$

Thus Eq. (7) is equivalent to finitely many scalar Poisson equations of
$f_{i_{1}\cdots i_{n}}$ for each $n$.

In addition, Schauder estimates together with a simple induction yield the
geometric bound:

$\|\mathbf{Proj}_{n}(\Phi_{\Omega})\|_{C^{0}(\Omega)}\leq C^{n},$ (9)

with $C$ depending only on $\Omega$ and $d$, which implies that the radius of
convergence of $\Phi$ is strictly positive. See [6], Theorem 3.6 or the
appendix in [1]. ∎

######  Remark 3.2.

In [6], §3, a variant of Proposition 3.1 is established for strong solutions
in Sobolev spaces $W^{m,2}(\Omega)$ for each $m\in\mathbb{N}$. Then the
geometric bound (9) can be deduced from the Sobolev–Morrey embedding, provided
that $m>\lfloor n/2\rfloor+1$. But this requires, as in [6], that the domain
$\Omega$ is $C^{m}$. Proposition 3.1 indicates that such regularity assumption
can be relaxed to $C^{2,\alpha}$.

In fact, thanks to the Kellogg theorem [16] one may further relax the
regularity of $\Omega$ to $C^{2,{\rm Dini}}$, i.e., $\Omega$ admits
$C^{2}$-local charts with parametrisation maps being merely Dini continuous.
In this case we have the classical solution in
$C^{\infty}\left(\Omega;E^{\otimes n}\right)\cap
C^{2}\left(\overline{\Omega};E^{\otimes n}\right)$, while the geometric bound
(9) remains valid.

Recall the multi-index notation:

$\displaystyle\left|I\right|=i_{1}+\ldots+i_{d},\qquad
D^{I}u=\frac{\partial^{i_{1}+\ldots i_{d}}u}{\partial z_{1}^{i_{1}}\partial
z_{2}^{i_{2}}\cdots\partial z_{d}^{i_{d}}},\qquad
I=\left(i_{1},\ldots,i_{d}\right)\in\mathbb{N}^{d}.$

As an immediate consequence of Proposition 3.1 and Remark 3.2, we have

######  Corollary 3.3.

Let $\Omega\subset E$ be a bounded domain of class $C^{2,\alpha}$ (or even
$C^{2,{\rm Dini}}$). Let $M\in{\rm Hom}\big{(}\Omega;T((E))\big{)}$ be an
algebra homomorphism with the projective norm on $T((E))$. Then there exists a
constant $L>0$ depending only on $\Omega$ and the operator norm of $M$ such
that, for all $\lambda\in[0,L[$, the series
$z\mapsto\sum_{n=0}^{\infty}\lambda^{n}M\left[\mathbf{Proj}_{n}\left(\Phi_{\Omega}(z)\right)\right]$
defines a $C^{2}$-function on $\Omega$. In addition, for any multi-index $I$
with $|I|\leq 2$, it holds that

$D^{I}\left(\sum_{n=0}^{\infty}\lambda^{n}M\left[\mathbf{Proj}_{n}\left(\Phi_{\Omega}(z)\right)\right]\right)=\sum_{n=0}^{\infty}\lambda^{n}D^{I}\Big{\\{}M\left[\mathbf{Proj}_{n}\left(\Phi_{\Omega}(z)\right)\right]\Big{\\}}.$

Theorem 2.9 shows that for a bounded domain, the expected signature of the
stopped Brownian motion satisfies the characteristic PDE (3.4). Conversely,
the classical solution ($C^{2}$ in the interior and $C^{1}$ up to the
boundary) for the characteristic PDE must coincide with the expected signature
of the stopped Brownian motion.

######  Theorem 3.4.

Let $\Omega$ be a bounded domain in $E$. Suppose that
$\xi:\overline{\Omega}\rightarrow T((E))$ is a classical solution to the
following PDE, in the regularity class $C^{2}\big{(}\Omega;T((E))\big{)}\cap
C^{1}\big{(}\overline{\Omega};T((E))\big{)}$:

$\begin{cases}\Delta\mathbf{Proj}_{n}(\xi)=-2\sum_{i=1}^{d}e_{i}\otimes\frac{\partial\mathbf{Proj}_{n-1}(\xi)}{\partial
z^{i}}-\sum_{i=1}^{d}e_{i}\otimes
e_{i}\otimes\mathbf{Proj}_{n-2}(\xi)\qquad\text{in }\Omega;\\\ \xi={\bf
1}\qquad\text{ on }\partial\Omega;\\\ \mathbf{Proj}_{0}(\xi)\equiv 1\text{ and
}\mathbf{Proj}_{1}(\xi)\equiv 0\qquad\text{in }\Omega.\end{cases}$

Then $\xi(z)$ is the expected signature of Brownian motion starting at $z$ up
to the first exit time $\tau_{\Omega}$; _i.e._ , $\xi=\Phi_{\Omega}$ on
$\overline{\Omega}$.

Let us outline the main ideas. Following the martingale approach in [12] for
the diffusion process up to a deterministic time, if we can construct a
martingale ${N}_{t}$ based on $\xi$ such that:

  * (a)

$\lim_{t\rightarrow\infty}{N}_{t}=S(B_{0,\tau_{\Omega}})$ where the limit are
taken in both the almost surely and $L^{1}$-sense (note that
$S(B_{0,\tau_{\Omega}})$ is proven to be $L^{1}$-integrable); and that

  * (b)

${N}_{0}=\xi(z)$,

then we have
$\mathbb{E}^{z}\left[\underset{t\rightarrow\infty}{\lim}{N}_{t}\right]=\mathbb{E}^{z}[{N}_{0}]$
and hence $\mathbb{E}^{z}[S(B_{0,\tau_{\Omega}})]=\xi(z)$ by Doob’s martingale
convergence theorem.

The next question is how to construct the martingale ${N}_{t}$. For this
purpose, note that
$\hat{N}_{t}:=\mathbb{E}^{z}[S(B_{0,\tau_{\Omega}})|\mathcal{F}_{t\wedge\tau_{\Omega}}]$
is a martingale (by Tower’s property) and satisfies the above condition. Thus,
in light of Chen’s identity, $\hat{N}_{t}$ can be rewritten as
$S(B_{0,t\wedge\tau_{\Omega}})\otimes\Phi(B_{t\wedge\tau_{\Omega}})$. This
motivates us to consider
$\hat{N}_{t}:=S(B_{0,t\wedge\tau_{\Omega}})\otimes\xi(B_{t\wedge\tau_{\Omega}})$.
Once we prove that $\hat{N}_{t}$ is a martingale, we may conclude that the PDE
solution $\xi$ coincides with the expected signature $\Phi_{\Omega}$.

###### Proof.

Since $\xi_{0}=\mathbf{Proj}_{0}(\Phi_{\Omega})\equiv 1$ and
$\xi_{1}=\mathbf{Proj}_{1}(\Phi_{\Omega})\equiv 0$, we only need to prove that
for any $n\geq 2$, one has $\xi_{n}=\mathbf{Proj}_{n}(\Phi_{\Omega})$. To this
end, set $N_{t}:=S(B_{0,t})\otimes\xi(B_{t})$. First of all, let us show that
$N_{t}$ is a martingale, or equivalently, that $\mathbf{Proj}_{n}(N_{t})$ is a
martingale for every $n\geq 2$.

Indeed, note that

$\displaystyle{\rm d}\pi_{n}(N_{t})={\rm
d}\mathbf{Proj}_{n}\Big{(}S(B_{0,t})\otimes\xi(B_{t})\Big{)}={\rm
d}\left\\{\sum_{i=0}^{n}\mathbf{Proj}_{i}\big{(}S(B_{0,t})\big{)}\otimes\xi_{n-i}(B_{t})\right\\}$
(10)

and, by the definition of the signature of Brownian motion, it holds that

$\displaystyle{\rm d}\mathbf{Proj}_{n}(S(B_{0,t}))$
$\displaystyle=\mathbf{Proj}_{n-1}(S(B_{0,t}))\otimes\,{\rm d}B_{t}$
$\displaystyle=\mathbf{Proj}_{n-1}(S(B_{0,t}))\otimes\cdot\,{\rm
d}B_{t}+\frac{1}{2}\mathbf{Proj}_{n-2}(S(B_{0,t}))\otimes\sum_{j=1}^{d}e_{j}\otimes
e_{j}\,{\rm d}t,$

where $\cdot{\rm d}B_{t}$ is understood in the Itô sense and $n\geq 2$.

On the other hand, we infer from the Itô formulae that

$\displaystyle{\rm
d}\xi_{n}(B_{t})=\sum_{i=1}^{d}\partial_{z_{i}}\xi_{n}(B_{t})\,\cdot{\rm
d}B^{i}_{t}+\frac{1}{2}\Delta\xi_{n}(B_{t})\,{\rm d}t.$

Thus, the SDE of $\pi_{n}({N}_{t})$ in Eq. (10) can be rewritten as

$\displaystyle{\rm
d}\Big{(}\mathbf{Proj}_{i}\big{(}S(B_{0,t})\big{)}\otimes\xi_{n-i}(B_{t})\Big{)}$
$\displaystyle\quad={\rm
d}\mathbf{Proj}_{i}\big{(}S(B_{0,t})\big{)}\otimes\xi_{n-i}(B_{t})+\mathbf{Proj}_{i}\big{(}S(B_{0,t})\big{)}\otimes{\rm
d}\xi_{n-i}(B_{t})+{\rm
d}\Big{\langle}\mathbf{Proj}_{i}\big{(}S(B_{0,t})\big{)},\xi_{n-i}(B_{t})\Big{\rangle}$
$\displaystyle\quad=\mathbf{1}_{i\geq
2}\frac{1}{2}\mathbf{Proj}_{i-2}\big{(}S(B_{0,t})\big{)}\otimes\sum_{j=1}^{d}e_{j}\otimes
e_{j}\,{\rm d}t\otimes\xi_{n-i}(B_{t})$
$\displaystyle\qquad+\mathbf{Proj}_{i}\big{(}S(B_{t})\big{)}\otimes\frac{1}{2}\Delta\xi_{n-i}(B_{t})\,{\rm
d}t+\sum_{j=1}^{d}\mathbf{1}_{i\geq
1}\mathbf{Proj}_{i-1}\big{(}S(B_{t})\big{)}\otimes
e_{j}\otimes\partial_{z_{j}}\xi_{n-i}(B_{t})+Q^{n,i}_{t}\cdot{\rm d}B_{t},$

where $i\leq n$ and

$\displaystyle Q^{n,i}_{t}$
$\displaystyle=\mathbf{Proj}_{i}(S(B_{0,t}))\otimes\sum_{j=1}^{d}\partial_{z_{j}}\xi_{n-i}(B_{t})\,\cdot{\rm
d}B_{t}+\mathbf{1}_{i\geq 1}\mathbf{Proj}_{i-1}(S(B_{0,t}))\otimes\cdot{\rm
d}B_{t}\otimes\xi_{n-i}(B_{t}).$

Summing the above equation over $i\in\\{0,\ldots,n\\}$, we get

$\displaystyle{\rm
d}\mathbf{Proj}_{n}(N_{t})=\sum_{i=0}^{n-2}\mathbf{Proj}_{i}\big{(}S(B_{t})\big{)}\otimes\mathcal{R}_{n-i-2}(B_{t})\,{\rm
d}t+\sum_{i=0}^{n}Q^{n,i}_{t}\cdot{\rm d}B_{t},$

where

$\displaystyle\mathcal{R}_{i}(z)=\frac{1}{2}\sum_{j=1}^{d}e_{j}\otimes
e_{j}\otimes\xi_{i}(z)+\sum_{j=1}^{d}e_{j}\otimes\partial_{z_{j}}\xi_{i+1}(z)+\frac{1}{2}\Delta\xi_{i+2}(z).$

By the fact that $\xi$ is a solution to the given PDE, the drift term equals
zero. Therefore, $N_{t}$ is a local martingale. It then implies that
$Y_{t}:=N_{t\wedge\tau_{\Omega}}$ is a local martingale.

It remains to show that $Y_{t}$ is a martingale. Indeed, for any $n$ and
$i\in\\{0,1,\cdots,n\\}$, and for all $t\in\mathbb{R}^{+}$, we have
$\mathbb{E}\left[\left|Q^{n,i}_{t\wedge\tau_{\Omega}}\right|^{2}\right]<\infty$.
This is because for every $i\in\\{0,1,2,\cdots\\}$, it holds that

$\displaystyle\mathbb{E}^{z}\left[|\mathbf{Proj}_{i}(S(B_{0,t\wedge\tau_{\Omega}}))|^{2}\right]<\infty,$
$\displaystyle\mathbb{E}^{z}\left[\Big{|}\xi_{i}(B_{t\wedge\tau_{\Omega}})\Big{|}^{2}+\sum_{j=1}^{d}\Big{|}\partial_{z_{j}}\xi_{i}(B_{t\wedge\tau_{\Omega}})\Big{|}^{2}\right]<\infty.$

Then $\mathbb{E}^{z}[|Y_{t}|]<\infty$ by Hölder’s inequality, and hence
$Y_{t}$ is a martingale. Moreover, $\mathbb{E}^{z}[|Y_{t}|]$ is uniformly
bounded, thus $\mathbb{E}^{z}[Y_{0}]=\mathbb{E}^{z}[Y_{t}]$ for all $t>0$. In
addition, $Y_{\infty}=S(B_{0,\tau_{\Omega}})\in L^{1}$, so

$\displaystyle\mathbb{E}^{z}[Y_{0}]=\lim_{t\rightarrow\infty}\mathbb{E}^{z}[Y_{t}]=\mathbb{E}^{z}\left[\lim_{t\rightarrow\infty}Y_{t}\right].$

We conclude by noting that $\mathbb{E}^{z}[Y_{0}]=\xi(z)$ and
$\mathbb{E}^{z}[\lim_{t\rightarrow\infty}Y_{t}]=\mathbb{E}^{z}[S(B)_{0,\tau_{\Omega}}]$.
∎

##  4\. Symmetrization of PDEs and Hyperbolic development

To overcome the difficulty that general domains are not rotationally
invariant, we integrate the development $\mathcal{M}_{\lambda,\Omega}$ (see
Eq. (5)) over all the rotated domains centred at $x\in\Omega$. The resulting
object is referred to as the “domain-averaging development” of expected
signature in the sequel.

In this section, we first explore general properties of domain-averaging
developments for any $M\in{\rm
Hom}\big{(}T((E)),\mathfrak{gl}(d+1;\mathbb{R})\big{)}$, and then specialise
to the hyperbolic development $H=M$.

###  4.1. Domain-averaging development of the expected signature

We start with defining several operations on domains. The translation of
domain $\Omega$ by $h$ is $\Omega+h:=\\{z+h:\,z\in\Omega\\}$. For $R\in
SO(d)$, write $\Omega_{R}:=R(\Omega)$, the domain obtained by rotating
$\Omega$ with respect to the centre $x=0$ (see Figure 1 for a 2-dimensional
illustration).

Fix an arbitrary $x\in\Omega$. As $\Omega$ is open, there is
$\epsilon\in]0,1[$ such that $\overline{\mathbf{B}(x,\epsilon)}\subset\Omega$.
We first note that the expected signature is invariant under translations,
namely that $\Phi_{\Omega}(x):=\Phi_{\Omega+h}(x+h)$ whenever $x\in\Omega$.
Without loss of generality we may assume $x=0\in\Omega$, since otherwise one
simply takes $\tilde{\Omega}=\Omega-x$.

Recall that

$\mathcal{M}_{\lambda,\Omega}(z):=\lambda
M(\Phi_{\Omega}(z))=\sum_{n=0}^{\infty}\lambda^{n}M\mathbf{Proj}_{n}[\Phi_{\Omega}(z)]$
(11)

for sufficiently small $\lambda$ ([12], Theorem 3.6). Any
$z\in\mathbf{B}(0,\epsilon)$ lies in $\bigcap_{S\in SO(d)}\Omega_{S}$, so
$\mathcal{M}_{\lambda,\Omega_{R}}(z)$ is well defined for each $R\in SO(d)$.

Figure 1. Rotation of the 2-dimensional domain $\Omega$ (whose boundary is the
solid blue curve) by degree $\alpha=\pi/4$ with respect to the centre marked
in orange. The boundary of the rotated domain $R_{\alpha}(\Omega)$ is the
dashed blue curve, where $R_{\alpha}\in SO(2)$ is the rotation matrix by
$\alpha$.

######  Definition 4.1 (Domain-averaging development of expected signature).

Let $\Omega\Subset\mathbb{R}^{d}$ be a domain containing
$\overline{\mathbf{B}(0,\epsilon)}$. Set

$\overline{\mathcal{M}_{\lambda,\epsilon,\Omega}}(z)=\int_{SO(d)}\mathcal{M}_{\lambda,\Omega_{R}}(z)\,{\rm
d}\chi(R)\in\mathfrak{gl}(d+1;\mathbb{R})\qquad\text{ for each
}z\in\overline{\mathbf{B}(0,\epsilon)},$ (12)

where $\chi$ is the normalised Haar measure on $SO(d)$, and integration of
matrices in $\mathfrak{gl}(d+1;\mathbb{R})$ is understood in the entry-wise
sense. For ease of notations, we write
$\overline{\mathcal{M}_{\lambda,\epsilon}}:=\overline{\mathcal{M}_{\lambda,\epsilon,\Omega}}$.

######  Lemma 4.2 (Rotational “symmetry” of $\mathcal{M}_{\lambda,\Omega}$).

For any $z\in\mathbf{B}(0,\epsilon)\subset\Omega$ and $R\in SO(d)$,

$\displaystyle\left(R\oplus{\bf
id}\right)\mathcal{M}_{\lambda,\Omega}(z)\left(R^{\top}\oplus{\bf
id}\right)=\mathcal{M}_{\lambda,\Omega_{R}}(Rz).$ (13)

###### Proof.

This extends the proof of Corollary 4 for the unit disc in [1], where ${\bf
id}$ is the identity map on $\mathbb{R}^{1}$ and $R^{\top}$ the transpose of
$R$. Since the Brownian motion $B^{Rz}$ starting at $Rz$ has the same
distribution as the rotated Brownian motion $R(B^{z})$ (with $B^{z}$ starting
from $z$), for any $z\in\bigcap_{R\in SO(d)}R(\Omega)$ it holds that

$\displaystyle\mathbb{E}^{z}\left[S\left(RB\right)_{\tau_{\Omega}}\right]=\mathbb{E}^{Rz}\left[S(B)_{\tau_{R(\Omega)}}\right].$

Applying $\lambda M$ to both sides of the above equation, we have

$\displaystyle\lambda
M\left(\mathbb{E}^{z}\left[S\left(RB\right)_{\tau_{\Omega}}\right]\right)=\lambda
M\left(\mathbb{E}^{R(z)}\left[S(B)_{\tau_{R(\Omega)}}\right]\right).$

By Lemma 3 in [1] (which clearly holds in arbitrary dimension), it equals

$\displaystyle\lambda
M\left(S\left(RB\right)_{\tau_{\Omega}}\right)=\left(R\oplus{\bf
id}\right)\left\\{\lambda
M\left[S\left(B\right)_{\tau_{\Omega_{R}}}\right]\right\\}\left(R^{\top}\oplus{\bf
id}\right).$

As $R$ is deterministic, it is furthermore equal to

$\displaystyle\left(R\oplus{\bf id}\right)\Big{\\{}(\lambda
M)\mathbb{E}^{z}\left[S(B)_{\tau_{\Omega}}\right]\Big{\\}}\left(R^{\top}\oplus{\bf
id}\right)=(\lambda M)\mathbb{E}^{Rz}\left[S(B)_{\tau_{\Omega_{R}}}\right].$

This concludes the proof. ∎

######  Lemma 4.3.

Write $r=|z|$ for $z\in\mathbf{B}(0,\epsilon)\subset\Omega$. Then for any
$R\in SO(d)$,

$(R\oplus{\bf
id})\overline{{\mathcal{M}}_{\lambda,\epsilon}}(r)\left(R^{\top}\oplus{\bf
id}\right)=\overline{\mathcal{M}_{\lambda,\epsilon}}(z).$ (14)

###### Proof.

Let $z=R(r,\underbrace{0,\cdots,0}_{d-1\text{
zeros}})^{\top}\in\mathbf{B}(0,\epsilon)\subset\bigcap_{O\in
SO(d)}\Omega_{O}\subset\Omega$. We deduce from Eq. (13) that

$(R\oplus{\bf id})\mathcal{M}_{\lambda,\Omega_{S}}(r)\left(R^{\top}\oplus{\bf
id}\right)=\mathcal{M}_{\lambda,\Omega_{R\circ
S}}\left(R(r,0,\cdots,0)^{\top}\right)=\mathcal{M}_{\lambda,\Omega_{R\circ
S}}(z)$

for any $S\in SO(d)$. We can now conclude by integrating both sides of this
equation against the Haar measure $\mu$ on $SO(d)$. ∎

As in [1] let us consider
$F_{\lambda,\Omega}:=M_{\lambda,\Omega}(z)\cdot[\underbrace{0,\cdots,0}_{d-1\text{
zeros}},1]^{\top}$, and similarly

$\overline{\mathcal{F}_{\lambda,\epsilon}}(z):=\overline{\mathcal{M}_{\lambda,\epsilon}}(z)\cdot[\underbrace{0,\cdots,0}_{d-1\text{
zeros}},1]^{\top}.$ (15)

The following can be deduced directly from Lemma 4.3.

######  Lemma 4.4 (Separation of variables).

For any $z\in\mathbf{B}(0,\epsilon)\subset\Omega$ and $R\in SO(d)$, it holds
that

$\overline{\mathcal{F}_{\lambda,\epsilon}}(z)=(R\oplus{\bf
id})\overline{{\mathcal{F}}_{\lambda,\epsilon}}\left((r,\underbrace{0,\cdots,0}_{d-1\text{
zeros}})^{\top}\right).$ (16)

In addition, $\overline{\mathcal{F}_{\lambda,\epsilon}}$ and
$\mathcal{F}_{\lambda,\Omega}$ are related in the following manner:

######  Lemma 4.5.

For $\mathbf{B}(0,\epsilon)\subset\Omega$, let
$\overline{C_{\lambda,\epsilon}}(r)=[\underbrace{0,\cdots,0}_{d-1\text{
zeros}},1]\overline{\mathcal{F}_{\lambda,\epsilon}}(r)$ and
$C_{\lambda}(z)=[\underbrace{0,\cdots,0}_{d-1\text{
zeros}},1]\mathcal{F}_{\lambda,\Omega}(z)$. Then
$C_{\lambda}(0)=\overline{C_{\lambda,\epsilon}}(0)$.

###### Proof.

By Lemma 4.3 we have, for each $R\in SO(d)$, that

$\displaystyle(R\oplus{\bf
id})\mathcal{M}_{\lambda,\Omega}(0)\left(R^{\top}\oplus{\bf
id}\right)=\mathcal{M}_{\lambda,\Omega_{R}}(0).$

The $(d+1,d+1)^{\text{th}}$ entry of both sides of the above equation is
$C_{\lambda,\Omega}(0)=C_{\lambda,\Omega_{R}}(0)$. So, integrating both sides
over the angular variable gives us
$C_{\lambda,\Omega}(0)=\overline{C_{\lambda,\epsilon}}(0).$ ∎

######  Lemma 4.6 (Preservation of finite radius of convergence).

For $\mathbf{B}(0,\epsilon)\subset\Omega$, if
$\overline{\mathcal{M}_{\lambda,\epsilon}}(0)$ has finite radius of
convergence, then so does $\mathcal{M}_{\lambda,\Omega}(0)$.

###### Proof.

By Lemma 4.3 we have $(R\oplus{\bf
id})\mathcal{M}_{\lambda,\Omega}(0)\left(R^{\top}\oplus{\bf
id}\right)=\mathcal{M}_{\lambda,\Omega_{R}}(0)$, which is equivalent to

$\displaystyle\mathcal{M}_{\lambda,\Omega}(0)=(R^{\top}\oplus{\bf
id})\mathcal{M}_{\lambda,\Omega_{R}}(0)\left(R\oplus{\bf id}\right).$

The assertion follows immediately from the definition of radius of
convergence. ∎

###  4.2. Symmetrization of PDEs

By Corollary 3.3 (compare also with [12]), the map $z\mapsto\Phi_{\Omega}(z)$
is twice differentiable on $\Omega$. This leads to

######  Theorem 4.7 (PDE for $\overline{\mathcal{F}_{\lambda,\epsilon}}$).

Assume $\mathbf{B}(0,\epsilon)\subset\Omega$. There exists a $\lambda^{*}>0$
such that for any $\lambda\in\mathbb{C}$ with $|\lambda|<\lambda^{*}$,
$\mathcal{F}_{\lambda,\Omega}$ is twice continuously differentiable on
$\mathbf{B}(0,\epsilon)$. Moreover, it satisfies

$\Delta\overline{\mathcal{F}_{\lambda,\epsilon}}(z)=-2\lambda\sum_{i=1}^{d}Me_{i}\frac{\partial\overline{\mathcal{F}_{\lambda,\epsilon}}}{\partial
z_{i}}(z)-\lambda^{2}\left(\sum_{i=1}^{d}\left(Me_{i}\right)^{2}\right)\overline{\mathcal{F}_{\lambda,\epsilon}}(z)\qquad\text{for
any $z\in\mathbf{B}(0,\epsilon)$}.$ (17)

###### Proof.

Consider

$\displaystyle\lambda^{*}=\sup\left\\{|\lambda|:\,\sum_{n=0}^{\infty}\lambda^{n}\left|\left|\mathbf{Proj}_{n}(\Phi_{\Omega}(z))\right|\right|<\infty\text{
for all }z\in\mathbf{B}(0,\epsilon)\right\\}.$ (18)

As remarked above, $z\mapsto\overline{\mathcal{F}_{\lambda,\epsilon}(z)}$ is
$C^{2}$ on $\mathbf{B}(0,\epsilon)$. The definition of
$\overline{\mathcal{F}_{\lambda,\epsilon}}$ and Eq. (12) give us

$\overline{\mathcal{F}_{\lambda,\epsilon}}(z)=\int_{SO(d)}\mathcal{F}_{\lambda,\Omega_{R}}(z)\,{\rm
d}\chi(R).$ (19)

We shall use the PDE for $\mathcal{F}_{\lambda,\Omega_{R}}$ to derive that for
$\overline{\mathcal{F}_{\lambda,\epsilon}}$. Indeed, for any $R\in SO(d)$ we
have

$\Delta\mathcal{F}_{\lambda,\Omega_{R}}(z)=-2\lambda\sum_{i=1}^{d}Me_{i}\frac{\partial\mathcal{F}_{\lambda,\Omega_{R}}}{\partial
z_{i}}(z)-\lambda^{2}\left(\sum_{i=1}^{d}\left(Me_{i}\right)^{2}\right)\mathcal{F}_{\lambda,\Omega_{R}}(z).$

As $\mathcal{F}_{\lambda,\Omega_{R}}$ is $C^{2}$, we can differentiate under
integral signs to get

$\displaystyle\int_{SO(d)}\Delta\mathcal{F}_{\lambda,\Omega_{R}}(z)\,{\rm
d}\chi(R)$
$\displaystyle\qquad=-2\lambda\sum_{i=1}^{d}Me_{i}\int_{SO(d)}\frac{\partial\mathcal{F}_{\lambda,\Omega_{R}}}{\partial
z_{i}}(z)\,{\rm
d}\chi(R)-\lambda^{2}\left(\sum_{i=1}^{d}\left(Me_{i}\right)^{2}\right)\overline{\mathcal{F}_{\lambda,\epsilon}(z)}.$
(20)

∎

###  4.3. Hyperbolic development

Recall that the hyperbolic development (Hambly–Lyons [7];
Boedihardjo–Diehl–Mezzarobba–Ni [1]) is the morphism
$H:T((E))\to\mathfrak{gl}(d+1;\mathbb{R})$ —

$\displaystyle H(x,y):=\begin{bmatrix}0&0&\cdots&0&z^{1}\\\
0&0&\cdots&0&z^{2}\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&\cdots&0&z^{d}\\\ z^{1}&z^{2}&\cdots&z^{d}&0\end{bmatrix}\qquad\text{ for
}z=(z^{1},\ldots,z^{d})\in E,$ $\displaystyle H(v_{1}\otimes\cdots\otimes
v_{n}):=H(v_{1})\cdot\ldots\cdot H(v_{n})\quad\text{ for any $n\in\mathbb{N}$
and $v_{1},\ldots,v_{n}\in E$}.$

Let $\overline{\mathcal{H}_{\lambda,\epsilon}}$ denote the last column of
domain-averaging development of expected stopped Brownian motion
$\overline{\mathcal{F}_{\lambda,\epsilon}}$ corresponding to the hyperbolic
development $H$.

######  Theorem 4.8 (ODE system of
$\overline{\mathcal{H}_{\lambda,\epsilon}}$).

Write, in the case $d=2$,

$\displaystyle\overline{\mathcal{H}_{\lambda,\epsilon}}(r)=:\left[\bar{A}_{\lambda}(r),\bar{B}_{\lambda}(r),\bar{C}_{\lambda}(r)\right]^{\top}.$

When $|\lambda|<\lambda^{\star}$ in Eq. (18), the following ODE system holds:

$\displaystyle
r^{2}\bar{A}_{\lambda}^{{}^{\prime\prime}}+r\bar{A}_{\lambda}^{{}^{\prime}}(r)-\bar{A}_{\lambda}(r)+\lambda^{2}r^{2}\bar{A}_{\lambda}(r)+2\lambda
r^{2}\bar{C}_{\lambda}^{{}^{\prime}}(r)=0,$ (21) $\displaystyle
r^{2}\bar{B}_{\lambda}^{{}^{\prime\prime}}(r)+r\bar{B}_{\lambda}^{\prime}(r)+(\lambda^{2}r^{2}-1)\bar{B}_{\lambda}(r)=0,$
(22)
$\displaystyle\bar{C}^{{}^{\prime}}_{\lambda}(r)+r\bar{C}^{{}^{\prime\prime}}_{\lambda}(r)+2\lambda^{2}r\bar{C}_{\lambda}(r)+2\lambda
r\bar{A}^{{}^{\prime}}_{\lambda}(r)+2\lambda\bar{A}_{\lambda}(r)=0.$ (23)

To ensure that $\overline{\mathcal{H}_{\lambda,\epsilon}}$ is $C^{2}$ at the
origin $x=0$, one needs to impose the conditions

$\bar{A}_{\lambda}(0)=0,\quad\bar{B}_{\lambda}(0)=0,\quad\bar{C}^{{}^{\prime}}_{\lambda}(0)=0.$
(24)

###### Proof.

As in [1], a straightforward computation using Eq. (14) and Lemma 4.3 yields
that

$\overline{\mathcal{H}_{\lambda,\epsilon}}\left(re^{i\theta}\right)=(R_{\theta}\oplus{\bf
id})\overline{\mathcal{H}_{\lambda,\epsilon}}(r),$ (25)

where $R_{\theta}=\begin{bmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{bmatrix}\in SO(2)$. The PDE for
$\overline{\mathcal{H}_{\lambda,\epsilon}}$ is the same as that for
$\mathcal{H}_{\lambda,\mathbf{B}(0,1)}$. Adapting almost verbatim the
arguments in [1], Lemma 6, we conclude that the ODE for
$\overline{\mathcal{H}_{\lambda,\epsilon}}$ is the same as that for
$\mathcal{H}_{\lambda,\mathbf{B}(0,1)}$ given in Eqs. (21), (22), and (23). ∎

In contrast to the case that $\Omega$ is the unit disc, roughly speaking, the
PDE for $\overline{\mathcal{H}_{\lambda,\epsilon}}$ misses “half of the
boundary conditions”, namely the conditions on
$\partial\mathbf{B}(0,\epsilon)$. To address this problem, we make use of
geometric properties of the hyperbolic development.

######  Lemma 4.9.

Let $\mathbf{B}(0,\epsilon)\subset\Omega$ and let $\lambda^{\ast}$ be as in
Eq. (18). For every $\lambda\in[0,\lambda^{*}[$ and $r\in[0,\epsilon]$, set
$\overline{\mathcal{H}_{\lambda,\epsilon}}(r):=\left[h_{\lambda}^{(1)}(r),\cdots,h_{\lambda}^{(d+1)}(r)\right]^{\top}$.
Then $h^{(d+1)}_{\lambda}(r)\geq 1$ for every $r\in[0,\epsilon]$.

###### Proof.

As the hyperbolic development is a normed algebra morphism, for any
$\lambda\in[0,\lambda^{*}[$ with $\lambda^{*}$ defined in Eq. (17),

$\displaystyle||(\lambda H)(\Phi_{\Omega}(z))||$
$\displaystyle=\left\|\sum_{n\geq
0}\lambda^{n}H(\mathbf{Proj}_{n}(\Phi_{\Omega}(z))\right\|$
$\displaystyle\leq\sum_{n\leq
0}\lambda^{n}\|H\|^{n}\|\mathbf{Proj}(\Phi_{\Omega}(z))\|$
$\displaystyle\leq\sum_{n\leq
0}\lambda^{n}\|\mathbf{Proj}(\Phi_{\Omega}(z))\|<\infty,$

where the norm of $H$ is $1$. By the dominated convergence theorem, we can
interchange the expectation and $\lambda H$ to obtain that

$\displaystyle\lambda H(\Phi_{\Omega}(z))=\mathbb{E}^{z}\left[\lambda
H\left(S(B)_{0,\tau_{\Omega}}\right)\right].$

By Lemma 2.7, the Stratonovich signature of the Brownian motion
$\left(S(B)_{0,t}\right)_{t\in[0,T]}$ is a geometric rough path for any finite
$T>0$. That is, there exists a sequence of bounded 1-variation paths
$\left(B^{m}_{[0,T]}\right)_{m=1}^{\infty}$ whose signatures converge almost
surely to $\left(S(B)_{0,t}\right)_{t\in[0,T]}$ in the $p$-variation distance
for any $p>2$. For instance, we can choose $B^{m}_{[0,T]}$ as the dyadic
piecewise linear approximation of Brownian motion with mesh size $2^{-m}$ up
to time $T$.

It then holds that

$\displaystyle\lim_{m\rightarrow\infty}\lambda
H\left(S\left(B^{m}_{[0,T]}\right)\right)\overset{a.s}{=}\lambda
H\left(S\left(B_{[0,T]}\right)\right).$

As $\mathbb{E}^{z}[\tau_{\Omega}]$ is almost surely finite, we can send $T$
and $m$ to $+\infty$ to get

$\displaystyle\lim_{T\rightarrow\infty}\lim_{m\rightarrow\infty}\lambda
H\left(S\left(B^{m}_{\left[0,T\wedge\tau_{\Omega}\right]}\right)\right)\overset{a.s}{=}\lambda
H\left(S\left(B_{\left[0,\tau_{\Omega}\right]}\right)\right).$

Note that for any $T>0$ and $m\in\mathbb{N}$, the boundness of $1$-variation
of $B^{m}_{[0,T]}$ in the almost sure sense ensures that $\lambda
H\left(S\left(B^{m}_{[0,T]}\right)\right)$ is finite almost surly. Lyons–Xu
[13] showed that any bounded 1-variation path $\gamma$ satisfies
$H\left(S(\gamma)\right)\cdot[0,\cdots,0,1]^{\top}\in\mathbb{H}_{d}$, with

$\displaystyle\mathbb{H}_{d}:=\left\\{\left(x^{(1)},\cdots,x^{(d)},x^{(d+1)}\right)\in\mathbb{R}^{d+1}:\sum_{i=1}^{d}\left(x^{(i)}\right)^{2}-\left(x^{(d+1)}\right)^{2}=-1,\,x^{(d+1)}>0\right\\}.$

That is, the last co-ordinate of the hyperbolic development of the signature
of $B^{m}_{[0,T\wedge\tau_{\Omega}]}\in\mathbb{N}$ is no less than $1$ almost
surely for all $m$ and $T>0$. Thus, the expectation of the limit of
$\left(\lambda H(S(B^{m})_{[0,T\wedge\tau_{\Omega}}])^{(d+1)}\right)_{m,T}$,
as $m,T\nearrow\infty$, is greater than or equal to $1$.

Therefore, we can conclude the desired result:

$\displaystyle h^{(d+1)}_{\lambda}(r)=\left(\mathbb{E}^{z}\left[\lambda
H\left(S\left(B_{\left[0,\tau_{\Omega}\right]}\right)\right)\right]\left(0,\cdots,0,1\right)^{\top}\right)^{(d+1)}\geq
1.$

∎

######  Corollary 4.10.

Let $\mathbf{B}(0,\epsilon)$, $\lambda^{\ast}$, and
$\overline{\mathcal{H}_{\lambda,\epsilon}}(r)$ be the same as in Lemma 4.9
above, where $r\in[0,\epsilon]$. Then
$h^{(d+1)}_{\lambda}(r)\geq\left|h^{(1)}_{\lambda}(r)\right|$ for every
$\lambda\in[0,\lambda^{*}[$ and $r\in[0,\epsilon]$.

###### Proof.

By [13] again,
$H\left(S(\gamma)\right)\cdot[0,\cdots,0,1]^{\top}:=\left(H^{(1)},\cdots,H^{(d+1)}\right)^{\top}\in\mathbb{H}_{d}$
and $H^{(d+1)}>0$ for any bounded 1-variation path $\gamma$. In particular,
$\left(H^{(d+1)}\right)^{2}=1+\sum_{i=1}^{d}\left(H^{(i)}\right)^{2}$. Thus
$H^{(d+1)}\geq\left|H^{(1)}\right|$. Let $B^{m}_{0,T\wedge\tau}$ denote the
dyadic approximation of Brownian motion as in the proof of Lemma 4.9, which
has bounded $1$-variation almost surely. Moreover, for
$\lambda\in[0,\lambda^{*}]$,

$\lim_{T\rightarrow\infty}\lim_{m\rightarrow\infty}\lambda
H(S(B^{m})_{0,T\wedge\tau}))\overset{a.s.}{=}\lambda H(S(B)_{0,\tau})).$

Since for any $m\in\mathbb{N}$ and $T>0$, it holds almost surely that

$\lambda H(S(B^{m})_{0,T\wedge\tau}))^{(d+1)}-\left|\lambda
H(S(B^{m})_{0,T\wedge\tau}))^{(1)}\right|\geq 0,$

we may infer by sending $m,T\nearrow\infty$ that
$h^{(d+1)}_{\lambda}(r)-\left|h^{(1)}_{\lambda}(r)\right|\geq 0.$ ∎

##  5\. Proof of Theorem 1.1: $d=2$

The domain-averaging hyperbolic development
$\overline{\mathcal{H}_{\lambda,\epsilon}}$ introduced in earlier sections
overcomes the issue of lack of rotational invariance on general domains.
Combining $\overline{\mathcal{H}_{\lambda,\epsilon}}$ with the techniques
developed in [1, 4] and utilising properties of the Bessel functions $J_{\nu}$
and $Y_{\nu}$ (see [17, 18]), we arrive at the following proof of Theorem 1.1
in the 2-dimensional case.

###### Proof of Theorem 1.1, $d=2$.

Assume for contradiction that the expected signature $\Phi\equiv\Phi_{\Omega}$
had infinite radius of convergence. We divide our arguments into six steps
below.

Step 1. Recall the PDE (7) for $\Phi$ (reproduced below, see Theorem 3.4):

$\begin{cases}\Delta\left(\mathbf{Proj}_{n}\Phi\right)=-2\sum_{i=1}^{2}e_{i}\otimes\frac{\partial(\mathbf{Proj}_{n-1}\Phi)}{\partial
z^{i}}-\sum_{i=1}^{2}e_{i}\otimes
e_{i}\otimes\mathbf{Proj}_{n-2}\Phi\quad\text{ in }\Omega;\\\ \Phi={\bf
1}\qquad\text{ on }\partial\Omega;\\\ \Phi_{0}\equiv 1\text{ and
}\Phi_{1}\equiv 0\quad\text{ in }\Omega.\end{cases}$ (26)

By standard elliptic PDE theory and induction, for each $n=0,1,2,\ldots$ there
exists a unique solution for Eq. (26) that is real-analytic in the interior.
Indeed, one may begin with the same arguments as for Proposition 3.1 to reduce
Eq. (26) to scalar elliptic PDEs, which renders applicable the standard
elliptic regularity theory. For any subdomain
$\widetilde{\Omega}\Subset\Omega$ and $n=2,3,4\ldots$, a simple induction and
the Lax–Milgram lemma ([15], §5.8) yield the existence of $\Phi_{n}\in
W^{1,2}\left(\widetilde{\Omega};E^{\otimes n}\right)$, such that the boundary
value is attained in the trace sense as usual, and that for any test function
$\psi\in C^{\infty}_{c}\left(\widetilde{\Omega}\right)$ we have

$\displaystyle\int_{\Omega}\nabla\psi\cdot\nabla\Phi_{n}\,{\rm
d}x=\int_{\Omega}\left\\{2\sum_{i=1}^{d}e_{i}\otimes\frac{\partial\Phi_{n-1}}{\partial
z^{i}}+\sum_{i=1}^{d}e_{i}\otimes e_{i}\otimes\Phi_{n-2}\right\\}\psi\,{\rm
d}x.$

Then, by the interior regularity Theorem 8.10 in [15] we can bootstrap the
regularity to $\Phi_{n}\in C^{\infty}\left(\widetilde{\Omega};E^{\otimes
n}\right)=\bigcap_{k=0}^{\infty}W^{k,2}\left(\widetilde{\Omega};E^{\otimes
n}\right)$. The real-analyticity of $\Phi_{n}$ follows since, by induction,
$\Delta\Phi_{n}$ equals a real-analytic function in the interior of $\Omega$;
_cf._ [19].

On the other hand, when $\Omega$ is a $C^{2,\alpha}$-domain, straightforward
adaptations of the arguments in Lyons–Ni [6], §§3.5.1–3.5.2 (by changing
Sobolev spaces to suitable Hölder spaces) together with the
$C^{2,\alpha}$-boundary regularity theory for elliptic PDE ([15], Theorem
6.19) yield the geometric decay bound for the expected signature:

$\displaystyle\|\mathbf{Proj}_{n}\Phi\|_{C^{2}(\Omega)}\leq(C_{0})^{n}.$

Here $C_{0}$ depends only on the geometry of $\Omega$ and the Hölder index
$\alpha$.

Step 2. Fix an arbitrary $x\in\Omega$. There is $\epsilon\in]0,1[$ such that
$\overline{\mathbf{B}(x,\epsilon)}\subset\Omega$. As
$\Phi_{\Omega}(z)=\Phi_{\Omega-x}(z-x)$ for any $z\in\Omega$, we can take
$x=0$ without loss of generality.

Recall from Theorem 4.8 that
$\overline{\mathcal{F}_{\lambda,\Omega}}(r)\equiv\left[\bar{A}_{\lambda}(r),\bar{B}_{\lambda}(r),\bar{C}_{\lambda}(r)\right]^{\top}$
satisfies

$\displaystyle
r^{2}\bar{A}_{\lambda}^{{}^{\prime\prime}}(r)+r\bar{A}_{\lambda}^{{}^{\prime}}(r)-\bar{A}_{\lambda}(r)+\lambda^{2}r^{2}\bar{A}_{\lambda}(r)+2\lambda
r^{2}\bar{C}_{\lambda}^{{}^{\prime}}(r)=0,$ (27) $\displaystyle
r^{2}\bar{B}_{\lambda}^{{}^{\prime\prime}}(r)+r\bar{B}_{\lambda}^{\prime}(r)+(\lambda^{2}r^{2}-1)\bar{B}_{\lambda}(r)=0,$
(28)
$\displaystyle\bar{C}^{{}^{\prime}}_{\lambda}(r)+r\bar{C}^{{}^{\prime\prime}}_{\lambda}(r)+2\lambda^{2}r\bar{C}_{\lambda}(r)+2\lambda
r\bar{A}^{{}^{\prime}}_{\lambda}(r)+2\lambda\bar{A}_{\lambda}(r)=0$ (29)

for $r\in[0,\epsilon[$, with the boundary conditions

$\displaystyle\bar{A}_{\lambda}(0)=0,\quad\bar{B}_{\lambda}(0)=0,\quad\text{
and }\quad\bar{C}^{{}^{\prime}}_{\lambda}(0)=0.$ (30)

Step 3. From now on, let us focus on Eqs. (27) and (29) for
$\bar{A}_{\lambda}$ and $\bar{C}_{\lambda}$; our blowup quantity will be
$\bar{C}_{\lambda}(0)$. Introduce the “missing boundary conditions”:

$\bar{A}_{\lambda}(\epsilon)=:{\mathfrak{b}}\quad\text{ and
}\quad\bar{C}_{\lambda}(\epsilon)=:\mathfrak{c}.$ (31)

Here ${\mathfrak{b}}$ and $\mathfrak{c}$ depend in general on $\lambda$ and
$\epsilon$.

An adaptation of Boedihardjo–Diehl–Mezzarobba–Ni [1], §6 allows us to
explicitly solve the system (27) – (31). Indeed, for $r\in[0,\epsilon[$ we
consider the ansatz

$\displaystyle\bar{A}_{\lambda}(r)=\alpha\mathscr{C}_{1}(\lambda\zeta
r)\quad\text{and}\quad\bar{C}_{\lambda}(r)=\mathscr{C}_{0}(\lambda\zeta r),$
(32)

where $\alpha,\zeta\in\mathbb{C}$ are complex parameters to be specified and
$\mathscr{C}_{\nu}$ are cylinder functions ($\nu=0,1$); _i.e._ , linear
combinations of $J_{\nu}$ and $Y_{\nu}$, the order-$\nu$ Bessel function of
the first and the second kind, respectively. The identities

$\displaystyle\mathscr{C}_{0}^{\prime}=\mathscr{C}_{1},$ $\displaystyle
s\mathscr{C}_{1}^{\prime}(s)+\mathscr{C}_{1}(s)=s\mathscr{C}_{0}(s)$

together with the defining equations for cylinder functions

$\displaystyle
s^{2}\mathscr{C}_{\nu}^{\prime\prime}(s)+s\mathscr{C}_{\nu}^{\prime}(s)+(s^{2}-\nu^{2})\mathscr{C}_{\nu}(s)=0$
(33)

show that for particular choices of $\zeta$ and $\alpha$, the ansatz (32)
indeed solves Eqs. (21) and (23). Here $\zeta$ is any one of the four complex
roots of $p(z)=z^{4}+z^{2}+2$, and $\alpha=\zeta+\zeta^{3}/2.$

Moreover, as $Y_{0}(s)\sim\mathcal{O}(\log s)$ and
$Y_{1}(s)\sim\mathcal{O}(s^{-1})$ for $s\searrow 0$ ([18], §10.7(i)), in order
for $\bar{A}_{\lambda}$ and $\bar{C}_{\lambda}$ to be bounded at the origin
$r=0$, the cylinder functions in the ansatz (32) cannot contain $Y_{0}$ and
$Y_{1}$. Then we are left with $J_{0}$ and $J_{1}$ only, and the corresponding
ansatz automatically satisfies the boundary condition (24) at $r=0$.

Note also that the $C^{2}$-solution for the PDE system (27) – (30) is unique.

To summarise, we have further reduced the ansatz (32) to

$\begin{cases}\bar{A}_{\lambda}(r)=u{\mathfrak{m}}(r)+v{\mathfrak{m}}^{\dagger}(r),\\\
\bar{C}_{\lambda}(r)=u{\mathfrak{n}}(r)+v{\mathfrak{n}}^{\dagger}(r),\end{cases}$
(34)

where $u,v$ are complex coefficients and ${\mathfrak{m}}$,
${\mathfrak{n}}:[0,\epsilon[\to\mathbb{C}$ are given by

$\begin{cases}{\mathfrak{m}}(r):=\alpha J_{1}(\lambda\zeta r),\\\
{\mathfrak{n}}(r):=J_{0}(\lambda\zeta r).\end{cases}$ (35)

######  Remark 5.1.

It should be emphasised that one needs $\lambda\in\mathbb{R}$ in the above
arguments, which guarantees that
${\mathfrak{m}}^{\dagger}(r)=\alpha^{\dagger}J_{1}\left(\lambda\zeta^{\dagger}r\right)$
and ${\mathfrak{n}}^{\dagger}=J_{0}\left(\lambda\zeta^{\dagger}r\right)$.

Step 4. The coefficients $u,v$ can be easily solved from the boundary
condition (31). Indeed, Eqs. (34) and (31) imply that
$\begin{bmatrix}{\mathfrak{m}}(\epsilon)&{\mathfrak{m}}(\epsilon)^{\dagger}\\\
{\mathfrak{n}}(\epsilon)&{\mathfrak{n}}(\epsilon)^{\dagger}\end{bmatrix}\begin{bmatrix}u\\\
v\end{bmatrix}=\begin{bmatrix}{\mathfrak{b}}\\\ \mathfrak{c}\end{bmatrix}$,
hence

$\displaystyle\begin{bmatrix}u\\\
v\end{bmatrix}=\frac{\begin{bmatrix}{\mathfrak{n}}(\epsilon)^{\dagger}&-{\mathfrak{m}}(\epsilon)^{\dagger}\\\
-{\mathfrak{n}}(\epsilon)&{\mathfrak{m}}(\epsilon)\end{bmatrix}\begin{bmatrix}{\mathfrak{b}}\\\
\mathfrak{c}\end{bmatrix}}{{\mathfrak{m}}(\epsilon){\mathfrak{n}}(\epsilon)^{\dagger}-{\mathfrak{n}}(\epsilon){\mathfrak{m}}(\epsilon)^{\dagger}}.$

Setting $r=0$ in Eq. (34), we get

$\displaystyle\bar{C}_{\lambda}(0)$
$\displaystyle={\mathfrak{n}}(0)u+{\mathfrak{n}}^{\dagger}(0)v$
$\displaystyle=\frac{\left[{\mathfrak{n}}^{\dagger}(\epsilon)-{\mathfrak{n}}(\epsilon)\right]\cdot{\mathfrak{b}}+\left[{\mathfrak{m}}(\epsilon)-{\mathfrak{m}}^{\dagger}(\epsilon)\right]\cdot\mathfrak{c}}{{\mathfrak{m}}(\epsilon){\mathfrak{n}}(\epsilon)^{\dagger}-{\mathfrak{n}}(\epsilon){\mathfrak{m}}(\epsilon)^{\dagger}}$
$\displaystyle=\frac{{\rm
Im}\left\\{-{\mathfrak{b}}J_{0}(\lambda\zeta\epsilon)+\mathfrak{c}\alpha
J_{1}(\lambda\zeta\epsilon)\right\\}}{{\rm
Im}\left\\{\alpha^{\dagger}\left[J_{1}(\lambda\zeta\epsilon)\right]^{\dagger}J_{0}(\lambda\zeta\epsilon)\right\\}}.$
(36)

Again, in the last line of Eq. (5) the boundary data $\mathfrak{c}$,
${\mathfrak{b}}$ depend on both $\lambda$ and $\epsilon$.

As computed in [1], §6, the denominator ${\rm
Im}\left\\{\alpha^{\dagger}\left[J_{1}(\lambda\zeta\epsilon)\right]^{\dagger}J_{0}(\lambda\zeta\epsilon)\right\\}$
has a root
$\lambda_{\star}\in\left]\frac{2.5}{\epsilon},\frac{3}{\epsilon}\right[$. On
the other hand, for such $\lambda_{\star}$ we have

$\displaystyle{\rm Im}\left\\{\mathfrak{c}\alpha
J_{1}(\lambda_{\star}\zeta\epsilon)\right\\}\leq-1.3\mathfrak{c}.$

See the proof of Lemma 8 in [1]. But Lemma 4.9 in this paper yields
$\mathfrak{c}\geq 1$, for the third component of the hyperbolic development
must be confined to the upper branch of the hyperbola. So the numerator in the
right-most term in Eq. (5) is non-zero. This implies that the radius of
convergence of $\overline{\mathcal{H}_{\lambda,\epsilon}}$ (the domain-
averaged hyperbolic development of the expected signature) is no larger than
$\lambda_{\star}$, _provided that ${\mathfrak{b}}=0$_.

Step 5. It remains to consider the case ${\mathfrak{b}}\neq 0$. Then the
numerator equals

$\displaystyle{\rm Im}\left\\{\mathfrak{c}\alpha
J_{1}(\lambda_{\star}\zeta\epsilon)-{\mathfrak{b}}J_{0}(\lambda_{\star}\zeta\epsilon)\right\\}={\mathfrak{b}}\,{\rm
Im}\left\\{\frac{\mathfrak{c}}{{\mathfrak{b}}}\alpha
J_{1}(\lambda_{\star}\zeta\epsilon)-J_{0}(\lambda_{\star}\zeta\epsilon)\right\\}$

when evaluated at $\lambda_{\star}$, since both
$\mathfrak{c},{\mathfrak{b}}\in\mathbb{R}$. Again, by the geometry of
hyperbolic developments (Lemma 4.9 and Corollary 4.10) we have
$|\mathfrak{e}|\equiv|\mathfrak{c}/{\mathfrak{b}}|\geq 1$.

To this end, we shall establish in Appendix, Lemma A.4 that

$\displaystyle\mathcal{N}(\mu;\mathfrak{e}):={\rm
Im}\left\\{\mathfrak{e}\alpha J_{1}(\mu\zeta)-J_{0}(\mu\zeta)\right\\}\neq
0\quad\text{ for all }\mu\in[2.5,3],\,|\mathfrak{e}|\geq 1.$ (37)

Step 6. As a consequence, there exists at least one $R_{0}\in SO(d)$ such that
$C_{\lambda,\Omega_{R_{0}}}(x)$, _i.e._ , the final component of the
hyperbolic development for the stopped Brownian motion on the rotated domain
$R_{0}$, blows up. Therefore, in view of Lemma 4.6 (in which
$\mathcal{M}=\mathcal{H}$), for any $x\in\Omega\Subset\mathbb{R}^{2}$ the
expected signature $\Phi_{\Omega}$ must have finite radius of convergence at
$x$. ∎

##  6\. Proof of Theorem 1.1: $3\leq d\leq 8$

In this final section, fix any $d=2,3,\ldots$ and write
$\mathbf{D}\equiv\mathbf{D}^{d}$ for the $d$-dimensional unit ball.

###  6.1. The unit ball

Let $\mathcal{H}_{\lambda,\mathbf{D}}(z)$ be the hyperbolic development of the
expected signature of a $d$-dimensional Brownian motion starting from $z$ and
stopped upon the first exit time from $\mathbf{D}$ scaled by $\lambda$. Note
that the domain-averaging hyperbolic development of the expected signature
coincides with that of the hyperbolic development; that is,
$\overline{\mathcal{H}_{\lambda,1,\mathbf{D}}}(z)=\mathcal{H}_{\lambda,\mathbf{D}}(z)$.

######  Lemma 6.1 (Sparseness of $\mathcal{H}_{\lambda,\mathbf{D}}$).

For each $\lambda>0$, the hyperbolic development
$\mathcal{H}_{\lambda,\mathbf{D}}\left(\mathbf{r}\right)=\left(\mathcal{H}^{(1)}_{\lambda,\mathbf{D}}(\mathbf{r}),\cdots,\mathcal{H}^{(d+1)}_{\lambda,\mathbf{D}}(\mathbf{r})\right)^{\top}$
evaluated at the point ${\bf r}=(r,0,\cdots,0)^{\top}\in\mathbb{R}^{d}$
satisfies $\mathcal{H}^{(i)}_{\lambda,\mathbf{D}}(\mathbf{r})=0$ for
$i\in\\{2,3,\cdots,d\\}$ and
$\mathcal{H}^{(d+1)}_{\lambda,\mathbf{D}}(\mathbf{r})\geq 1$ almost surely, as
long as it is well-defined for such $\lambda$.

###### Proof.

The second statement follows immediately from the construction of the
hyperbolic development, as $\mathcal{H}_{\lambda,\mathbf{D}}^{(d+1)}(\lambda
B)\geq 1$ almost surely.

To prove the first statement, we let $\\{\gamma_{j,t}\\}_{j\in\mathbb{N}}$ be
a sequence of paths of bounded $1$-variations converging almost surely and in
$C^{0,1/2-}$ to $B_{t}$. Consider the reflected paths

$\displaystyle\hat{\gamma}_{j,t}:=\left(\gamma^{(1)}_{j,t},-\gamma^{(2)}_{j,t},\cdots,-\gamma^{(d)}_{j,t}\right)^{\top}.$

Then $\\{\hat{\gamma}_{j,t}\\}$ converges in the same topology to the
reflected Brownian motion

$\hat{B}_{t}:=\left(B^{(1)}_{t},-B^{(2)}_{t},\cdots,-B^{(d)}_{t}\right)^{\top}.$
(38)

If $B_{t}$ starts at some point $\mathbf{r}=(r,0,\cdots,0)^{\top}$, we may
further require that $\hat{B}_{0}=\gamma_{j,0}=\hat{\gamma}_{j,0}=\mathbf{r}$
for each $j$. Also

$\displaystyle
H\left(S\left(\lambda\gamma_{j,\tau}\right)\right)^{(k)}=\sum_{n\geq
0}\sum_{w\in\mathcal{E}_{2n}^{(k)}}\lambda^{2n+1}\mathbf{Proj}_{w}\left(S(\hat{\gamma}_{j,\tau})\right).$

Taking the expectation, we get

$\displaystyle\mathbb{E}^{\mathbf{r}}\left[\sum_{n\geq
0}\sum_{w\in\mathcal{E}_{2n}^{(k)}}\mathbf{Proj}_{w}(S(\lambda\gamma_{j,\tau}))\right]=\mathbb{E}^{\mathbf{r}}\left[\sum_{n\geq
0}\sum_{w\in\mathcal{E}_{2n}^{(k)}}\mathbf{Proj}_{w}\left(S(\lambda\hat{\gamma}_{j,\tau})\right)\right].$

On the other hand, for every $w\in\mathcal{E}_{2n}^{(k)}$ and
$k\in\\{2,\cdots,d\\}$, Lemma 2.10 implies that

$\mathbf{Proj}_{w}(S(\lambda\gamma_{j,\tau}))=-\mathbf{Proj}_{w}\left(S(\lambda\hat{\gamma}_{j,\tau})\right)$

and that

$\mathbb{E}^{\mathbf{r}}\left[\sum_{n\geq
0}\sum_{w\in\mathcal{E}_{2n}^{(k)}}\mathbf{Proj}_{w}(S(\lambda\gamma_{j,\tau}))\right]=-\mathbb{E}^{\mathbf{r}}\left[\sum_{n\geq
0}\sum_{w\in\mathcal{E}_{2n}^{(k)}}\mathbf{Proj}_{w}(S(\lambda(\hat{\gamma}_{j,\tau})))\right].$

Therefore, for $k\in\\{2,\cdots,d\\}$,

$\displaystyle\mathcal{H}^{(k)}_{\lambda,\mathbf{D}}(\mathbf{r})=\mathbb{E}^{\mathbf{r}}\left[\sum_{n\geq
0}\sum_{w\in\mathcal{E}_{2n}^{(k)}}\mathbf{Proj}_{w}(S(\lambda\gamma_{j,\tau}))\right]=0.$

Now we can pass to the limit $j\to\infty$ to conclude the same for $B_{\tau}$.
∎

Theorem 4.7 remains valid in $d$-dimensional unit ball. It is straightforward
to check that $\mathcal{H}_{\lambda,\mathbf{D}}(z)$ satisfies the trivial
boundary condition for $z\in\partial\mathbf{D}$. Thus we get

######  Lemma 6.2.

There exists $\lambda^{*}>0$ such that for any $\lambda\in\mathbb{C}$ with
$|\lambda|<\lambda^{*}$, $\mathcal{H}_{\lambda,\mathbf{D}}$ is $C^{2}$ on
$\mathbf{D}$. Moreover, it satisfies

$\Delta\mathcal{H}_{\lambda,\mathbf{D}}(z)=-2\lambda\sum_{i=1}^{d}He_{i}\frac{\partial\mathcal{H}_{\lambda,\mathbf{D}}}{\partial
z^{i}}(z)-\lambda^{2}\left(\sum_{i=1}^{d}\left(He_{i}\right)^{2}\right)\mathcal{H}_{\lambda,\mathbf{D}}(z)$

for each $z\in\mathbf{D}$, and it is subject to the boundary condition
$\mathcal{H}_{\lambda,\mathbf{D}}=[0,\ldots,0,1]^{\top}$ on
$\partial\mathbf{D}$. This PDE boundary value problem has a unique solution.

The separation of variables and symmetrization arguments in §4 reduce the PDE
in Lemma 6.2 to an ODE system in the radial variable $r$ only. Moreover, in
view of Lemma 6.1, this ODE system is trivial except for the first and last
components of $\mathcal{H}_{\lambda,\mathbf{D}}$. Here and hereafter, $r=|z|$
is the radial co-ordinate on $\mathbf{D}$ and $\chi$ is the normalised Haar
measure on $SO(d)$.

######  Proposition 6.3 (ODE system for $\mathcal{H}_{\lambda,\mathbf{D}}$).

Write $h_{\lambda}^{(1)}:=\overline{\mathcal{H}^{(1)}_{\lambda,\mathbf{D}}}$
and $h_{\lambda}^{(d+1)}:=\overline{\mathcal{H}^{(d+1)}_{\lambda,\mathbf{D}}}$
where

$\overline{\mathcal{H}_{\lambda,\mathbf{D}}}(r):=\int_{SO(d)}\mathcal{H}_{\lambda,\mathbf{D}}\left(R\cdot\left[r,\underbrace{0,\ldots,0}_{d-1\,\text{zeros}}\right]^{\top}\right)\,{\rm
d}\chi(R).$

Then for $d\in\\{2,3,4,\ldots\\}$ it holds that

$\displaystyle\left(h^{(1)}_{\lambda}\right)^{\prime\prime}+\frac{d-1}{r}\left(h^{(1)}_{\lambda}\right)^{\prime}-\frac{(d-1)h^{(1)}_{\lambda}}{r^{2}}=-2\lambda\left(h^{(d+1)}_{\lambda}\right)^{\prime}-\lambda^{2}h^{(1)}_{\lambda},$
(39)
$\displaystyle\left(h^{(d+1)}_{\lambda}\right)^{\prime\prime}+\frac{d-1}{r}\left(h^{(d+1)}_{\lambda}\right)^{\prime}=-2\lambda\left\\{\frac{d-1}{r}h^{(1)}_{\lambda}+\left(h^{(1)}_{\lambda}\right)^{\prime}\right\\}-d\lambda^{2}h^{(d+1)}_{\lambda},$
(40) $\displaystyle h^{(1)}_{\lambda}(1)=0,\qquad h^{(d+1)}_{\lambda}(1)=1.$
(41)

###### Proof.

By Lemma 4.4 we have
${\mathcal{H}_{\lambda,\mathbf{D}}}(z)=\left(R\oplus\mathbf{id}\right)\overline{\mathcal{H}_{\lambda,\mathbf{D}}}(r)$
for $z=R\cdot(r,0,\ldots,0)^{\top}$. Also, Lemma 6.1 gives us
$\overline{\mathcal{H}_{\lambda,\mathbf{D}}}(z)\in\left\\{w\in\mathbb{R}^{d+1}:\,w^{(k)}=0\text{
for }k\in\\{2,\cdots,d\\}\text{ and }w^{(d+1)}\geq 1\right\\}.$ Thus

$\displaystyle{\mathcal{H}_{\lambda,\mathbf{D}}}(z)=\begin{bmatrix}R\cdot\begin{bmatrix}h_{\lambda}^{(1)}(r)\\\
0\\\ \vdots\\\ 0\end{bmatrix}\\\
h_{\lambda}^{(d+1)}(r)\end{bmatrix}=\begin{bmatrix}\frac{z^{1}}{r}h_{\lambda}^{(1)}(r)\\\
\frac{z^{2}}{r}h_{\lambda}^{(1)}(r)\\\ \vdots\\\
\frac{z^{d}}{r}h_{\lambda}^{(1)}(r)\\\ h^{(d+1)}_{\lambda}(r)\end{bmatrix},$

since the unit sphere ${\mathbf{S}^{d-1}}$ is invariant under $R$.

Consider now the PDE system in Lemma 6.2:

$\Delta\mathcal{H}_{\lambda,\mathbf{D}}=-2\lambda\sum_{i=1}^{d}He_{i}\frac{\partial\mathcal{H}_{\lambda,\mathbf{D}}}{\partial
z^{i}}-\lambda^{2}\left(\sum_{i=1}^{d}\left(He_{i}\right)^{2}\right)\mathcal{H}_{\lambda,\mathbf{D}}.$
(42)

The coefficients can be easily computed: since $H(e_{i})$ is the matrix whose
only nonzero entries are the $(i,d+1)^{\text{th}}$ and $(d+1,i)^{\text{th}}$
ones (both equal to $1$), we find that

$\left(\sum_{i=1}^{d}\left(He_{i}\right)^{2}\right)={\rm diag}(1,\cdots,1,d).$
(43)

Moreover, the $\alpha^{\text{th}}$ component of
$\sum_{i=1}^{d}He_{i}\frac{\partial\mathcal{H}_{\lambda,\mathbf{D}}}{\partial
z^{i}}$ equals ${\partial\mathcal{H}_{\lambda,\mathbf{D}}^{(d+1)}}/{\partial
z^{\alpha}}$ if $\alpha\in\\{1,2,\ldots,d\\}$ and
$\sum_{i=1}^{d}\partial\mathcal{H}_{\lambda,\mathbf{D}}^{(i)}/\partial z^{i}$
if $\alpha=d+1$. In the latter case we further have

$\displaystyle\sum_{i=1}^{d}\frac{\partial\mathcal{H}_{\lambda,\mathbf{D}}^{(i)}}{\partial
z^{i}}(z)$
$\displaystyle=\sum_{i=1}^{d}\left\\{\frac{1}{r}h_{\lambda}^{(1)}(r)-\frac{z^{i}z^{i}}{r^{3}}h_{\lambda}^{(1)}(r)+\frac{z^{i}}{r}\left(h_{\lambda}^{(1)}(r)\right)^{\prime}(r)\frac{\partial
r}{\partial z_{i}}\right\\}$
$\displaystyle=\frac{d-1}{r}h_{\lambda}^{(1)}(r)+\left(h_{\lambda}^{(1)}\right)^{\prime}(r).$
(44)

On the other hand, $z^{1}/r=\cos\theta_{1}$ in the usual spherical co-
ordinates $\omega=z/r=(\theta_{1},\ldots,\theta_{d-1})$ on $\mathbf{S}^{d-1}$.
Note also that $\Delta$ on the left-hand side of Eq. (42) is the Euclidean
Laplacian on $\mathbb{R}^{d+1}=\mathbb{R}^{d}\times\mathbb{R}$; when acting on
the $\mathbb{R}^{d}$-component, in the spherical polar co-ordinates

$\displaystyle\Delta=\partial_{rr}+\frac{d-1}{r}\partial_{r}+\frac{1}{r^{2}}\Delta_{\mathbf{S}^{d-1}},$
(45)

where $\Delta_{\mathbf{S}^{d-1}}$ is the Laplace–Beltrami operator on the
sphere $\mathbf{S}^{d-1}$.

To proceed, let us project Eq. (42) onto the first and the last components. We
shall prove the blowup of the resulting ODE system for $h_{\lambda}^{(1)}$ and
$h_{\lambda}^{(d+1)}$.

For the projection onto the $(d+1)^{\text{th}}$ component, note that
$\left[\mathbf{Proj}_{d+1}\mathcal{H}_{\lambda,\mathbf{D}}\right](z)=h_{\lambda}^{(d+1)}(r)$
by Eq. (68), which is independent of $\omega\in\mathbf{S}^{d-1}$. So Identity
(45) leads to

$\displaystyle\Delta\mathbf{Proj}_{d+1}\mathcal{H}_{\lambda,\mathbf{D}}(z)=\left(h_{\lambda}^{(d+1)}\right)^{\prime\prime}(r)+\frac{d-1}{r}\left(h_{\lambda}^{(d+1)}\right)^{\prime}(r).$

The projection of the right-hand side of Eq. (42) can be found via Eqs. (43)
and (6.1). Thus we obtain Eq. (40).

It remains to prove Eq. (39). To this end, note that Eqs. (42) – (45) yield
that

$\displaystyle\Delta\left(\frac{z^{1}}{r}h_{\lambda}^{(1)}\right)=-2\lambda\frac{\partial
h_{\lambda}^{(d+1)}}{\partial z^{1}}-\lambda^{2}\frac{z^{1}}{r}h_{1}.$

Writing in the spherical polar co-ordinates, we have $z^{1}/r=\cos\theta_{1}$,
$\partial h_{\lambda}^{(d+1)}/{\partial
z^{1}}=\cos\theta_{1}\left(h_{\lambda}^{(d+1)}\right)^{\prime}$, as well as

$\displaystyle\Delta\left(\frac{z^{1}}{r}h_{\lambda}^{(1)}\right)=\cos\theta_{1}\left\\{\left(h_{\lambda}^{(1)}\right)^{\prime\prime}+\frac{d-1}{r}\left(h_{\lambda}^{(1)}\right)^{\prime}\right\\}+\frac{h_{\lambda}^{(1)}}{r^{2}}\Delta_{\mathbb{S}^{d-1}}\cos\theta_{1}.$

In addition, the Laplace–Beltrami $\Delta_{\mathbf{S}^{d-1}}$ on
$\mathbf{S}^{d-1}$ can be expressed in terms of the Laplace–Beltrami
$\Delta_{\mathbf{S}^{d-2}}$ on $\mathbf{S}^{d-2}$ — for
$\omega=(\theta_{1},\omega^{\prime})$ where
$\omega^{\prime}=(\theta_{2},\cdots,\theta_{d-1})\in\mathbf{S}^{d-2}$, one has

$\displaystyle\Delta_{\mathbf{S}^{d-1}}=\left(\sin\theta_{1}\right)^{2-d}\frac{\partial}{\partial\theta_{1}}\left(\left(\sin\theta_{1}\right)^{d-2}\frac{\partial}{\partial\theta_{1}}\right)+\frac{1}{\sin^{2}\theta_{1}}\Delta_{\mathbf{S}^{d-2}},$

from which it follows that

$\Delta_{\mathbf{S}^{d-1}}\cos\theta_{1}=-(d-1)\cos\theta_{1}.$

We thus obtain Eq. (39).

The boundary conditions $h^{(1)}_{\lambda}(1)=0$, $h^{(d+1)}_{\lambda}(1)=1$
follow from $\Phi_{\mathbf{D}}(1)=1$, which holds since the stopping time is
zero on the boundary $\partial\mathbf{D}$. ∎

######  Remark 6.4.

When $d=2$, Proposition 6.3 agrees with Theorem 4.8
($\bar{A}_{\lambda}=h_{\lambda}^{(1)}$ and
$\bar{C}_{\lambda}=h_{\lambda}^{(3)}$).

To proceed, let us first introduce a few further notations. Define the
dimensional constants

$\displaystyle\eta_{\pm}:=d+3\pm\sqrt{d^{2}-10d+9}$

and

$\displaystyle\beta_{\pm}:=\sqrt{2d-6\pm 2\sqrt{d^{2}-10d+9}}.$

In each dimension $d\geq 2$, the Wrönskian determinant $\mathcal{W}$ is a
function of $\lambda$:

$\displaystyle\mathcal{W}\equiv\mathcal{W}[\lambda]:=\det\begin{bmatrix}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}}{2}\right)&J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}}{2}\right)\\\
\eta_{+}\beta_{+}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}}{2}\right)&\eta_{-}\beta_{-}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}}{2}\right)\end{bmatrix}.$
(46)

We can express the symmetrised development
$\overline{\mathcal{H}_{\lambda,\mathbf{D}}}(r)$ on the $d$-dimensional unit
disc as follows.

######  Theorem 6.5.

Let $\mathbf{D}=\mathbf{D}^{d}$ with $d\in\mathbb{N}\setminus\\{1,9\\}$. Let
$\lambda_{\star}$ denote the radius of convergence of
$\overline{\mathcal{H}_{\lambda,\mathbf{D}}}$. Then for all
$[0,\lambda_{\star}[$ we have

$\displaystyle\overline{\mathcal{H}_{\lambda,\mathbf{D}}}(z)=\left[\frac{z^{1}}{r}h_{\lambda}^{(1)}(r),\frac{z^{2}}{r}h_{\lambda}^{(1)}(r),\,\cdots,\,\frac{z^{d}}{r}h_{\lambda}^{(1)}(r),\,h^{(d+1)}_{\lambda}(r)\right]^{\top},$
(47)

where

$\displaystyle
h_{\lambda}^{(1)}(r)=r^{1-\frac{d}{2}}\frac{8d}{{\mathcal{W}}}\left\\{J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}}{2}\right)J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}r}{2}\right)-J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}}{2}\right)J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}r}{2}\right)\right\\},$
(48) $\displaystyle
h_{\lambda}^{(d+1)}(r)=r^{1-\frac{d}{2}}\frac{1}{{\mathcal{W}}}\bigg{\\{}-\beta_{+}\eta_{+}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}}{2}\right)J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}r}{2}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+\beta_{-}\eta_{-}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}}{2}\right)J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}r}{2}\right)\bigg{\\}}.$
(49)

For $d\in\\{2,3,\cdots,8\\}$ one may further express

$\displaystyle h_{\lambda}^{(1)}(r)=8d\,r^{1-\frac{d}{2}}\,\left(\frac{{\rm
Im}\left\\{\left[J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}}{2}\right)\right]^{\dagger}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}r}{2}\right)\right\\}}{{\rm
Im}\left\\{\eta_{-}\beta_{-}\left[J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}}{2}\right)\right]^{\dagger}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}}{2}\right)\right\\}}\right),$
(50) $\displaystyle h_{\lambda}^{(d+1)}=r^{1-\frac{d}{2}}\,\left(\frac{{\rm
Im}\left\\{\eta_{-}\beta_{-}\left[J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}}{2}\right)\right]^{\dagger}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}r}{2}\right)\right\\}}{{\rm
Im}\left\\{\eta_{-}\beta_{-}\left[J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}}{2}\right)\right]^{\dagger}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}}{2}\right)\right\\}}\right).$
(51)

###### Proof.

The ODE system (39), (40) for $h^{(1)}_{\lambda}$ and $h^{(d+1)}_{\lambda}$
can be solved, _e.g._ , by $\textsc{Maple}^{\text{\textregistered}}$. Denote
$\eta_{\pm}:=d+3\pm\sqrt{d^{2}-10d+9}$ and $\beta_{\pm}:=\sqrt{2d-6\pm
2\sqrt{d^{2}-10d+9}}$. For each fixed $\lambda$, $h^{(1)}_{\lambda}(r)$ and
$h^{(d+1)}_{\lambda}(r)$ are linear combinations of Bessel functions of the
first and second kinds:

$\displaystyle
h^{(1)}_{\lambda}(r)=\left(C_{1}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}r}{2}\right)+C_{2}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}r}{2}\right)+C_{3}Y_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}r}{2}\right)+C_{4}Y_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}r}{2}\right)\right)r^{-\frac{d}{2}+1},$
(52) $\displaystyle
h^{(d+1)}_{\lambda}(r)=-\frac{1}{8d}\Bigg{\\{}C_{1}\eta_{+}\beta_{+}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}r}{2}\right)+C_{2}\eta_{-}\beta_{-}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}r}{2}\right)$
$\displaystyle\qquad\qquad\qquad\qquad+C_{3}\eta_{+}\beta_{+}Y_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}r}{2}\right)+C_{4}\eta_{-}\beta_{-}Y_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}r}{2}\right)\Bigg{\\}}r^{-\frac{d}{2}+1}.$
(53)

In view of the boundary conditions for $h^{(1)}_{\lambda}$,
$h^{(d+1)}_{\lambda}$, that $\lim_{r\searrow 0}J_{\nu}(r)=0$, and that

$\displaystyle
Y_{\nu}(r)\sim-\frac{1}{\pi}\Gamma(\nu)\left(\frac{z}{2}\right)^{-\nu}\quad\text{
as }r\searrow 0$

for $\nu=d/2$ or $d/2-1$ ([18], §§10.7.3, 10.7.4), we get $C_{3}=C_{4}=0$;
_i.e._ , the solutions $(B_{\lambda},D_{\lambda})$ contains no branches of
$Y_{\nu}$. Therefore, we arrive at a simplified ODE system:

$\displaystyle
h^{(1)}_{\lambda}(r)=\left(C_{1}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}r}{2}\right)+C_{2}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}r}{2}\right)\right)r^{-\frac{d}{2}+1},$
(54) $\displaystyle
h^{(d+1)}_{\lambda}(r)=-\frac{1}{8d}\left(C_{1}\eta_{+}\beta_{+}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}r}{2}\right)+C_{2}\eta_{-}\beta_{-}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}r}{2}\right)\right)r^{-\frac{d}{2}+1}$
(55)

with boundary conditions

$h^{(1)}_{\lambda}(1)=0\qquad\text{ and }\qquad h^{(d+1)}_{\lambda}(1)=1.$
(56)

Solving from the algebraic system (54), (55), and (56) the constants $C_{1}$
and $C_{2}$, we get

$\displaystyle
C_{1}=8d\,\frac{J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}}{2}\right)}{\mathcal{W}}\quad\text{
and }\quad
C_{2}=-8d\,\frac{J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}}{2}\right)}{\mathcal{W}},$

with the Wrönskian determinant

$\displaystyle\mathcal{W}\equiv\mathcal{W}[\lambda]:=\det\begin{bmatrix}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}}{2}\right)&J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}}{2}\right)\\\
\eta_{+}\beta_{+}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}}{2}\right)&\eta_{-}\beta_{-}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}}{2}\right)\end{bmatrix}$
(57)

provided that it is non-vanishing. Thus we arrive at

$\displaystyle
h^{(1)}_{\lambda}(r)=r^{1-\frac{d}{2}}\frac{8d}{{\mathcal{W}}}\left\\{J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}}{2}\right)J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}r}{2}\right)-J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}}{2}\right)J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}r}{2}\right)\right\\},$
(58) $\displaystyle
h^{(d+1)}_{\lambda}(r)=\frac{r^{1-\frac{d}{2}}}{{\mathcal{W}}}\left\\{-\beta_{+}\eta_{+}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}}{2}\right)J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}r}{2}\right)+\beta_{-}\eta_{-}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}}{2}\right)J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}r}{2}\right)\right\\}.$
(59)

For $3\leq d\leq 8$ we set

$\Upsilon=\Upsilon(d):=\sqrt{-(d-1)(d-9)}.$

Then $\eta_{\pm}=d+3\pm i\Upsilon$ and $\beta_{\pm}=\sqrt{2}\sqrt{d-3\pm
i\Upsilon}$. A direct computation further leads to

$\displaystyle\left[{\rm
Re}\left({\beta_{\pm}}\right)\right]^{2}=\sqrt{(d-3)^{2}+\Upsilon^{2}}+(d-3)=2\sqrt{d}+d-3,$
(60) $\displaystyle\left[{\rm
Im}\left({\beta_{\pm}}\right)\right]^{2}=\sqrt{(d-3)^{2}+\Upsilon^{2}}-(d-3)=2\sqrt{d}-d+3.$
(61)

In the above, ${\rm Re}\left({\beta_{+}}\right)$ and ${\rm
Im}\left({\beta_{+}}\right)$ have the same sign, and ${\rm
Re}\left({\beta_{-}}\right)$ and ${\rm Im}\left({\beta_{-}}\right)$ have
opposite signs. In view of the explicit formulae (57), (58), and (59), we may
choose without loss of generality that ${\rm Re}\left({\beta_{\pm}}\right)>0$.
That is,

$\displaystyle\beta_{\pm}$ $\displaystyle=\sqrt{2\sqrt{d}+d-3}\pm
i\sqrt{2\sqrt{d}-d+3}$
$\displaystyle=\sqrt{\left(\sqrt{d}+3\right)\left(\sqrt{d}-1\right)}\pm
i\sqrt{\left(\sqrt{d}+1\right)\left(3-\sqrt{d}\right)}.$ (62)

With the above choice one can further simplify the Wrönskian in Eq. (57).
Indeed, as $\eta_{\pm}$ and $\beta_{\pm}$ are complex conjugates and, by [18],
§10.11.9, $J_{\nu}\left(z^{\dagger}\right)=\left[J_{\nu}(z)\right]^{\dagger}$,
it holds that

$\mathcal{W}=2i\,{\rm
Im}\left\\{\eta_{-}\beta_{-}\left[J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}}{2}\right)\right]^{\dagger}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}}{2}\right)\right\\}.$
(63)

Similarly,

$\displaystyle
J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}}{2}\right)J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}r}{2}\right)-J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}}{2}\right)J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}r}{2}\right)=2i\,{\rm
Im}\left\\{\left[J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}}{2}\right)\right]^{\dagger}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}r}{2}\right)\right\\}.$

Eq. (58) now yields closed-form expressions for $h_{\lambda}^{(1)}$ and
$h_{\lambda}^{(d+1)}$, namely Eqs. (50) and (51). ∎

######  Remark 6.6.

[1], Lemma 7 for 2D unit disc is a special case of the above theorem. One can
verify that the constants $\alpha,\zeta$ in [1] are given by
$\alpha=-\frac{16\beta_{-}\eta_{-}}{\left|\beta_{-}\eta_{-}\right|^{2}}$ and
$\zeta=\frac{\beta_{+}}{2}$.

######  Remark 6.7.

For $d=9$, $\mathcal{W}$ is zero and Eqs. (39) and (40) admit a fundamental
solution $\overline{\mathcal{H}_{\lambda,\mathbf{D}}}$ in stark contrast with
the solutions for $d\neq 9$ given in Eqs. (52) and (53). See Appendix A, Lemma
A.3.

The finiteness of radius of convergence of the expected signature on
$\mathbf{D}=\mathbf{D}^{d}$, $3\leq d\leq 8$ is deduced from the ODE system in
Proposition 6.3.

######  Theorem 6.8.

The radius of convergence of the expected signature of the stopped Brownian
motion on $\mathbf{D}=\mathbf{D}^{d}$ is finite for $d\in\\{2,\ldots,8\\}$.

###### Proof.

By Theorem 6.5, for $d\in\\{2,\ldots,8\\}$ we have

$\displaystyle h_{\lambda}^{(d+1)}(r)$ $\displaystyle=$ $\displaystyle
r^{1-\frac{d}{2}}\,\left(\frac{{\rm
Im}\left\\{\eta_{-}\beta_{-}\left[J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}}{2}\right)\right]^{\dagger}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}r}{2}\right)\right\\}}{{\rm
Im}\left\\{\eta_{-}\beta_{-}\left[J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}}{2}\right)\right]^{\dagger}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}}{2}\right)\right\\}}\right).$

We shall prove that there exists a $\lambda_{\star}>0$ such that
$h_{\lambda}^{(d+1)}(0)$ blows up as $\lambda\to\lambda_{\star}$.

Let us now analyse the zeros of the denominator

$\displaystyle\Theta(\lambda):={\rm
Im}\left\\{\eta_{-}\beta_{-}\left[J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}}{2}\right)\right]^{\dagger}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}}{2}\right)\right\\}.$
(64)

Consider the Taylor expansion for $J_{\nu}(z)$;
$\nu\in\left\\{\frac{d}{2}-1,\frac{d}{2}\right\\}$. As in the appendix we
write

$\displaystyle J_{\nu}(z)=\mathring{J}_{\nu}^{n}(z)+\mathscr{R}_{\nu}^{n}(z),$

where

$\displaystyle\mathring{J}_{\nu}^{n}(z):=\left(\frac{z}{2}\right)^{\nu}\sum_{k=0}^{n-1}(-1)^{k}\frac{\left(\frac{z^{2}}{4}\right)^{k}}{k!\Gamma(k+\nu+1)},$

and the remainder $\mathscr{R}_{\nu}^{n}(z)$ consists of summation from $k=n$
to $\infty$. Similarly, denote

$\displaystyle\mathring{\Theta}^{n}(\lambda):={\rm
Im}\left\\{\eta_{-}\beta_{-}\left[\mathring{J}_{\frac{d}{2}-1}^{n}\left(\frac{\lambda\beta_{+}}{2}\right)\right]^{\dagger}\mathring{J}_{\frac{d}{2}}^{n}\left(\frac{\lambda\beta_{+}}{2}\right)\right\\}.$

The error $\left|\Theta(\lambda)-\mathring{\Theta}^{n}(\lambda)\right|$ can be
estimated via Lemma A.1 and the triangle inequality:

$\displaystyle\left|\Theta(\lambda)-\mathring{\Theta}^{n}(\lambda)\right|$
$\displaystyle\quad\leq\left|\eta_{-}\beta_{-}\right|\Bigg{\\{}\left|\mathscr{R}_{\frac{d}{2}-1}^{n}\left(\frac{\lambda\beta_{-}}{2}\right)\right|\left|\mathring{J}_{\frac{d}{2}}^{n}\left(\frac{\lambda\beta_{+}}{2}\right)\right|+\left|\mathscr{R}_{\frac{d}{2}-1}^{n}\left(\frac{\lambda\beta_{-}}{2}\right)\right|\left|\mathscr{R}_{\frac{d}{2}}^{n}\left(\frac{\lambda\beta_{+}}{2}\right)\right|$
$\displaystyle\qquad+\left|\mathring{J}_{\frac{d}{2}-1}^{n}\left(\frac{\lambda\beta_{-}}{2}\right)\right|\left|\mathscr{R}_{\frac{d}{2}}^{n}\left(\frac{\lambda\beta_{+}}{2}\right)\right|\Bigg{\\}}$
$\displaystyle\quad\leq\left|\eta_{-}\beta_{-}\right|\Bigg{\\{}\left[\frac{2^{\frac{d-1}{2}}}{\sqrt{\pi}\cdot{n!}}\left(\frac{\left|\lambda\beta_{-}\right|}{4}\right)^{2n+\frac{d}{2}-1}\right]\cdot\left[\frac{1}{1-\frac{\left|{\lambda\beta_{-}}\right|^{2}}{16(2n+d-2)}}\right]\left|\mathring{J}_{\frac{d}{2}}^{n}\left(\frac{\lambda\beta_{+}}{2}\right)\right|$
$\displaystyle\qquad+\left[\frac{2^{\frac{d-1}{2}}}{\sqrt{\pi}\cdot{n!}}\left(\frac{\left|\lambda\beta_{-}\right|}{4}\right)^{2n+\frac{d}{2}-1}\right]\cdot\left[\frac{1}{1-\frac{\left|{\lambda\beta_{-}}\right|^{2}}{16(2n+d-2)}}\right]\cdot\left[\frac{\left(\frac{|\lambda\beta_{+}|}{4}\right)^{2n+\frac{d}{2}}}{\left[\left(n+\frac{d}{2}\right)!\right]^{2}}\right]\cdot\left[\frac{1}{1-\frac{|\lambda\beta_{+}|^{2}}{64(2n+d+2)^{2}}}\right]$
$\displaystyle\quad+\left|\mathring{J}_{\frac{d}{2}-1}^{n}\left(\frac{\lambda\beta_{-}}{2}\right)\right|\left[\frac{\left(\frac{|\lambda\beta_{+}|}{4}\right)^{2n+\frac{d}{2}}}{\left[\left(n+\frac{d}{2}\right)!\right]^{2}}\right]\cdot\left[\frac{1}{1-\frac{|\lambda\beta_{+}|^{2}}{64(2n+d+2)^{2}}}\right]\Bigg{\\}}$
$\displaystyle\quad=:{\rm Err}_{\Theta}^{n}(\lambda).$

Both ${\rm Err}_{\Theta}^{n}(\lambda)$ and $\mathring{\Theta}^{n}(\lambda)$
involve only finitely many terms; hence, with each given $\lambda$, they can
be computed explicitly by hand.

We are now at the stage of concluding:

$\text{There exists $\lambda_{\star}\in]2.5,3[$ such that
$\Theta(\lambda_{\star})=0$}.$ (65)

Indeed, by the triangle inequality and continuity of $\Theta$, it suffices to
prove that

$\left[\mathring{\Theta}+{\rm
Err}_{\Theta}^{n}\right](\lambda=2.5)<0<\left[\mathring{\Theta}-{\rm
Err}_{\Theta}^{n}\right](\lambda=3).$ (66)

With the help of $\textsc{Maple}^{\text{\textregistered}}$, we take $n=7$ and
compute the two bounds in Eq. (66). The results are tabulated below, which
verify Eq. (66) and hence Eq. (65).

Table 1. The denominator $\Theta$ has a root in $]2.5,3[$ Dimension | Value for $\left[\mathring{\Theta}+{\rm Err}_{\Theta}^{n}\right](2.5)$ | Value for $\left[\mathring{\Theta}-{\rm Err}_{\Theta}^{n}\right](3)$
---|---|---
3 | -2.072008 | 6.951356
4 | -1.682841 | 8.366543
5 | -1.315936 | 6.921044
6 | -1.107269 | 4.734511
7 | -0.693811 | 2.530460
8 | -0.408603 | 1.115177

Next let us investigate $h^{(d+1)}(0)$, defined in the limiting sense:

$\displaystyle h^{(d+1)}(0)=\lim_{r\rightarrow 0}r^{1-\frac{d}{2}}\frac{{\rm
Im}\left\\{\eta_{-}\beta_{-}\left[J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}r}{2}\right)\right]^{\dagger}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}}{2}\right)\right\\}}{\Theta(\lambda)}=:\frac{\mathcal{N}(\lambda)}{\Theta(\lambda)}.$

As

$\displaystyle\lim_{r\rightarrow
0}\frac{\left[J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}r}{2}\right)\right]^{\dagger}}{r^{\frac{d}{2}-1}}=\frac{1}{\Gamma(\frac{d}{2})}\left(\frac{\lambda\beta_{-}}{4}\right)^{\frac{d}{2}-1},$

we have

$\displaystyle\mathcal{N}(\lambda)=\frac{1}{\Gamma(\frac{d}{2})}{\rm
Im}\left(\eta_{-}\beta_{-}\left(\frac{\lambda\beta_{-}}{4}\right)^{\frac{d}{2}-1}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}}{2}\right)\right).$

To show that the numerator $\mathcal{N}(\lambda)$ is non-vanishing for
$\lambda\in[2.5,3]$, we split $\mathcal{N}(\lambda)$ into the approximating
polynomial $\mathring{\mathcal{N}}^{n}(\lambda)$ and the remainder
$\text{Err}^{n}_{\mathcal{N}}$, where

$\displaystyle\mathring{\mathcal{N}}^{n}(\lambda)={\rm
Im}\left(\eta_{-}\beta_{-}\left(\frac{\lambda\beta_{-}}{4}\right)^{\frac{d}{2}-1}\mathring{J}^{n}_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}}{2}\right)\right).$

Here again we use the polynomial
$\mathring{J}^{n}_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}}{2}\right)$ to
approximate ${J}^{n}_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}}{2}\right)$ in
the numerator, with $n=5$. Note that $\mathring{\mathcal{N}}^{n}(\lambda)$ is
an explicit polynomial in $\lambda$ of degree $2n+d$, which is monotone
decreasing on $[2.5,3]$. One can bound the error uniformly on $[2.5,3]$:

$\displaystyle\text{Err}_{\mathcal{N}}^{n}$
$\displaystyle:=\sup_{\lambda\in[2.5,3]}\left|\mathcal{N}(\lambda)-\mathring{\mathcal{N}}^{n}(\lambda)\right|$
$\displaystyle\leq\left\\{\left|\eta_{-}\beta_{-}\left(\frac{3\beta_{-}}{4}\right)^{\frac{d}{2}-1}\right|\right\\}\cdot\left\\{\sup_{\lambda\in[2.5,3]}\left|J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}}{2}\right)-\mathring{J}^{n}_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}}{2}\right)\right|\right\\}$
$\displaystyle\leq\left\\{\left|\eta_{-}\beta_{-}\left(\frac{3\beta_{-}}{4}\right)^{\frac{d}{2}-1}\right|\right\\}\left\\{\left|E\left(\frac{3\beta_{+}}{2},n,\frac{d}{2}\right)\right|\right\\},$

where $E(z,n,\nu)$ is given below as in Appendix, Lemma A.1:

$\displaystyle
E(z,n,\nu):=\begin{cases}\left\\{\frac{\left(\frac{|z|}{2}\right)^{2n+\nu}}{n!(n+\nu)!}\right\\}\cdot\left\\{\frac{1}{1-\frac{|z|^{2}}{4(n+1)^{2}}}\right\\}&\text{
if $\nu$ is an integer},\\\
\left\\{\frac{2^{n}\sqrt{\pi}}{{n!(2n+2\nu+1)!!}}\left(\frac{|z|}{2}\right)^{2n+\nu}\right\\}\cdot\left\\{\frac{1}{1-\frac{|z|^{2}}{4(n+1)^{2}}}\right\\}&\text{
if $\nu$ is a half-integer}.\end{cases}$

Table 2. The numerator $\mathcal{N}(\lambda)$ is non-vanishing for $\lambda\in]2.5,3[$ Dimension | Value for $\mathring{\mathcal{N}}^{n}(2.5)$ | Error bound $\text{Err}^{n}_{\mathcal{N}}$
---|---|---
3 | -48.656672 | 1.266852
4 | -55.063129 | 1.265336
5 | -51.368007 | 5.982517
6 | -40.528560 | 3.647551
7 | -26.851665 | 12.270795
8 | -13.908808 | 5.810051

Table 2 summarises the values for $\mathring{\mathcal{N}}(2.5)$ and
$\text{Err}_{\mathcal{N}}^{n}$ for $d\in\\{3,\cdots,8\\}$ with accuracy up to
6 decimal points. This together with Lemma A.2 implies that

$\displaystyle\sup_{\lambda\in[2.5,3]}\mathcal{N}(\lambda)\leq\mathring{\mathcal{N}}^{n}(2.5)+\text{Err}^{n}_{\mathcal{N}}<0,$

hence proves the thesis. ∎

###  6.2. General $d$-dimensional $C^{2,\alpha}$ bounded domains

Let $\Omega\Subset\mathbb{R}^{d}$ be a $C^{2,\alpha}$-domain and $x\in\Omega$
be an arbitrary point. Denote by $\mathcal{H}_{\lambda,\Omega}(z)$ be the
hyperbolic development of the expected signature of a $d$-dimensional Brownian
motion starting from $z$ and stopped at the first exit time from $\Omega$
(scaled by $\lambda$). We are concerned with the domain-averaging hyperbolic
development. Without loss of generality we may assume $x=0$, as
$\Phi_{\Omega}(z)=\Phi_{\Omega-x}(z-x)$.

Recall from Definition 4.1 the domain-averaging hyperbolic development (with
integration understood entry-wise):

$\displaystyle\overline{\mathcal{H}_{\lambda,\epsilon,\Omega}}(z):=\int_{SO(d)}\mathcal{H}_{\lambda,\Omega_{R}}(z)\,{\rm
d}\chi(R).$

Here
$\overline{\mathcal{H}_{\lambda,\epsilon,\Omega}}(z)\in\mathfrak{gl}(d+1;\mathbb{R})$
for each $z\in\overline{\mathbf{B}(0,\epsilon)}$, with $\epsilon>0$ chosen so
small that $\mathbf{B}(0,\epsilon)\subset\bigcap_{S\in SO(d)}\Omega_{S}$.
Compare with Proposition 6.3 where $\Omega=\mathbf{D}$.

By Theorem 4.7, $\overline{\mathcal{H}_{\lambda,\epsilon,\Omega}}$ satisfies
the same PDE as in Lemma 6.2 on $\mathbf{B}(0,\epsilon)$:

######  Theorem 6.9 (PDE for
$\overline{\mathcal{H}_{\lambda,\epsilon,\Omega}}$).

Let $\epsilon>0$ and $\mathbf{B}(0,\epsilon)\subset\bigcap_{R\in
SO(d)}\Omega_{R}$. There exists $\lambda^{*}>0$ such that for every
$\lambda\in[0,\lambda^{*}]$,

$\Delta\overline{\mathcal{H}_{\lambda,\epsilon,\Omega}}(z)=-2\lambda\sum_{i=1}^{d}He_{i}\frac{\partial\overline{\mathcal{H}_{\lambda,\epsilon,\Omega}}}{\partial
z^{i}}(z)-\lambda^{2}\left(\sum_{i=1}^{d}\left(He_{i}\right)^{2}\right)\overline{\mathcal{H}_{\lambda,\epsilon,\Omega}}(z)\qquad\text{
for each }z\in\mathbf{B}(0,\epsilon).$

The domain-averaging hyperbolic development
$\overline{\mathcal{H}_{\lambda,\epsilon,\Omega}}(z)$ on a general domain
$\Omega$ does not have the sparseness property as in Lemma 6.1, in contrast to
the case of the unit disc. To overcome this difficulty, we shall symmetrize
$\overline{\mathcal{H}_{\lambda,\epsilon,\Omega}}(z)$ with a reflected version
of it. More precisely, consider

$\displaystyle\Omega_{\ast}:=\\{(x_{1},-x_{2},\cdots,-x_{d}):\,(x_{1},x_{2},\cdots,x_{d})\in\Omega\\}.$

Assume as before $0\in\Omega$; thus, there is an $\epsilon>0$ such that
$\mathbf{D}_{\epsilon}$, the disc of radius $\epsilon$, lies in $\Omega$. Set

$\displaystyle\widetilde{\mathcal{H}_{\lambda,\epsilon,\Omega}}(z):=\frac{\overline{\mathcal{H}_{\lambda,\epsilon,\Omega}}(z)+\overline{\mathcal{H}_{\lambda,\epsilon,\Omega_{\ast}}}(z)}{2}\qquad\text{
for }z\in\mathbf{D}_{\epsilon}.$

######  Lemma 6.10 (Sparseness of
$\widetilde{\mathcal{H}_{\lambda,\epsilon,\Omega}}$).

$\widetilde{\mathcal{H}_{\lambda,\epsilon,\Omega}}$ evaluated at ${\bf
r}=(r,0,\cdots,0)$ for $r\in]0,\epsilon[$ has all components being zero,
except possibly for the first and the last ones.

###### Proof.

The proof is similar to that of Lemma 6.1 for the unit disc case. Let
$(B_{t})_{t}$ denote the standard $d$-dimensional Brownian path. Consider the
reflected path

$\hat{B}_{t}:=\left(B^{(1)}_{t},-B^{(2)}_{t},\cdots,-B^{(d)}_{t}\right)^{\top},$
(67)

which is still a Brownian motion. If $B_{t}$ starts at
$\mathbf{r}=(r,0,\cdots,0)^{\top}$, we may further require that
$\hat{B}_{0}=\mathbf{r}$. By the definition of
$\widetilde{\mathcal{H}_{\lambda,\epsilon,\Omega}}$, for
$k\in\\{2,\cdots,d\\}$ we have

$\displaystyle\widetilde{\mathcal{H}_{\lambda,\epsilon,\Omega}}^{(k)}(r,0,\cdots,0)$
$\displaystyle\quad:=\overline{\mathcal{H}_{\lambda,\epsilon,\Omega}}^{(k)}(r)+\overline{\mathcal{H}_{\lambda,\epsilon,\Omega_{\ast}}}^{(k)}(r)$
$\displaystyle\quad=\int_{{\bf
S}^{d-1}}\left\\{\mathbb{E}^{r}\left[\sum_{n\geq
0}\sum_{\omega\in\mathcal{E}_{2n}^{k}}\mathbf{Proj}_{\omega}\left(S\left(B_{\tau_{\Omega_{\alpha}}}\right)\right)\right]+\mathbb{E}^{r}\left[\sum_{n\geq
0}\sum_{\omega\in\mathcal{E}_{2n}^{k}}\mathbf{Proj}_{\omega}\left(S\left(\hat{B}_{\tau_{\Omega_{\ast,\alpha}}}\right)\right)\right]\right\\}\,{\rm
d}\mu(\alpha)$ $\displaystyle\quad=0.$

The last line holds as
$\mathbf{Proj}_{\omega}\left(S\left(B_{\tau_{\Omega_{\alpha}}}\right)\right)+\mathbf{Proj}_{\omega}\left(S\left(\hat{B}_{\tau_{\Omega_{\ast,\alpha}}}\right)\right)=0$
almost surely for $\omega\in\mathcal{E}_{2n}^{k}$. ∎

As before, write $\eta_{\pm}:=d+3\pm\sqrt{d^{2}-10d+9}$,
$\beta_{\pm}:=\sqrt{2d-6\pm 2\sqrt{d^{2}-10d+9}}$, and $r=|z|$. We have the
following analogue of Theorem 6.5.

######  Theorem 6.11.

Let $\mathbf{D}_{\epsilon}\subset\Omega$ and
$d\in\mathbb{N}\setminus\\{1,9\\}$. Denote the boundary condition for
$\widetilde{\mathcal{H}_{\lambda,\epsilon,\Omega}}$ at $r=\epsilon$ as
$\widetilde{\mathcal{H}_{\lambda,\epsilon,\Omega}}(\epsilon)=[\mathfrak{p},0,\cdots,0,\mathfrak{q}]^{\top}$.
Then there exists $\lambda_{\star}>0$ such that for every
$\lambda\in]0,\lambda_{\star}[$,

$\displaystyle\widetilde{\mathcal{H}_{\lambda,\epsilon,\Omega}}(z)=\left[\frac{z^{1}}{r}h_{\lambda}^{(1)}(r),\,\frac{z^{2}}{r}h_{\lambda}^{(1)}(r),\,\cdots,\,\frac{z^{d}}{r}h_{\lambda}^{(1)}(r),\,h^{(d+1)}_{\lambda}(r)\right]^{\top}.$
(68)

The functions $h^{(1)}_{\lambda}$, $h^{(d+1)}_{\lambda}$ and the Wrönskian
determinant $\mathcal{W}_{\epsilon}$ are given by

$\displaystyle
h^{(1)}_{\lambda}(r)=\frac{\left(\frac{r}{\epsilon}\right)^{1-\frac{d}{2}}}{\mathcal{W}_{\epsilon}}\cdot\Bigg{\\{}\left[\eta_{-}\beta_{-}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}\epsilon}{2}\right)J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}r}{2}\right)-\eta_{+}\beta_{+}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}r}{2}\right)J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)\right]\mathfrak{p}$
$\displaystyle\quad+\left[-J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}\epsilon}{2}\right)J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}r}{2}\right)+J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}r}{2}\right)J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)\right]8d\mathfrak{q}\Bigg{\\}};$
$\displaystyle
h^{(d+1)}_{\lambda}(r)=\frac{\left(\frac{r}{\epsilon}\right)^{1-\frac{d}{2}}}{\mathcal{W}_{\epsilon}}\cdot\Bigg{\\{}\left[J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}\epsilon}{2}\right)J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}r}{2}\right)-J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}r}{2}\right)J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)\right]\frac{\mathfrak{p}\beta_{+}\beta_{-}\eta_{+}\eta_{-}}{8d}$
$\displaystyle\quad+\left[\eta_{-}\beta_{-}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}\epsilon}{2}\right)J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}r}{2}\right)-\eta_{+}\beta_{+}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}r}{2}\right)\right]\mathfrak{q}\Bigg{\\}};$
$\displaystyle\mathcal{W}_{\epsilon}=\det\begin{bmatrix}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)&J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}\epsilon}{2}\right)\\\
\eta_{+}\beta_{+}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)&\eta_{-}\beta_{-}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}\epsilon}{2}\right)\end{bmatrix}.$

######  Remark 6.12.

For $2\leq d\leq 8$ we can further simplify $h^{(1)}_{\lambda}$ and
$h^{(d+1)}_{\lambda}$:

$\displaystyle
h^{(1)}_{\lambda}(r)=\left(\frac{r}{\epsilon}\right)^{1-\frac{d}{2}}\left\\{\frac{\mathfrak{p}\cdot{\rm
Im}\,\left[\eta_{-}\beta_{-}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}\epsilon}{2}\right)J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}r}{2}\right)\right]+8d\mathfrak{q}\cdot{\rm
Im}\,\left[J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}\epsilon}{2}\right)J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}r}{2}\right)\right]}{{\rm
Im}\left[\eta_{-}\beta_{-}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}\epsilon}{2}\right)J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)\right]}\right\\};$
$\displaystyle
h^{(d+1)}_{\lambda}(r)=\left(\frac{r}{\epsilon}\right)^{1-\frac{d}{2}}\left\\{\frac{\frac{1}{8d}\mathfrak{p}\cdot{\rm
Im}\,\left[\eta_{+}\beta_{+}\eta_{-}\beta_{-}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}\epsilon}{2}\right)J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}r}{2}\right)\right]+\mathfrak{q}\cdot{\rm
Im}\,\left[\eta_{-}\beta_{-}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}r}{2}\right)\right]}{{\rm
Im}\left[\eta_{-}\beta_{-}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}\epsilon}{2}\right)J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)\right]}\right\\}.$

###### Proof.

Since both $\overline{\mathcal{H}_{\lambda,\epsilon,\Omega}}$ and
$\overline{\mathcal{H}_{\lambda,\epsilon,\Omega_{\ast}}}$ satisfy the same PDE
as in Theorem 6.9, $\widetilde{\mathcal{H}_{\lambda,\epsilon,\Omega}}$
satisfies also the same PDE on $\mathbf{B}(0,\epsilon)\subset\bigcap_{S\in
SO(d)}\Omega_{S}$:

$\Delta\widetilde{\mathcal{H}_{\lambda,\epsilon,\Omega}}(z)=-2\lambda\sum_{i=1}^{d}He_{i}\frac{\partial\widetilde{\mathcal{H}_{\lambda,\epsilon,\Omega}}(z)}{\partial
z^{i}}-\lambda^{2}\left(\sum_{i=1}^{d}\left(He_{i}\right)^{2}\right)\widetilde{\mathcal{H}_{\lambda,\epsilon,\Omega}}(z)\qquad\text{
for each }z\in\mathbf{B}(0,\epsilon).$

The sparseness Lemma 6.10 implies that components of
$\overline{\mathcal{H}_{\lambda,\epsilon,\Omega}}$ are all zero expect for the
first and the last ones, just as in the case of unit disc. Let
$\widetilde{\mathcal{H}_{\lambda,\epsilon,\Omega}}(r)=\left(h^{(1)}_{\lambda}(r),0,\cdots
0,h^{(d+1)}_{\lambda}(r)\right)^{\top}$. It then satisfies the PDE system
(39), (40) in Proposition 6.3, which differs from the unit disc case only in
terms of boundary data.

Adapting the computations in the proof for Theorem 6.5, we get

$\displaystyle\widetilde{\mathcal{H}^{(1)}_{\lambda,\epsilon,\Omega}}(r)=r^{1-\frac{d}{2}}\left(C_{1}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}r}{2}\right)+C_{2}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}r}{2}\right)\right),$
$\displaystyle\widetilde{\mathcal{H}^{(d+1)}_{\lambda,\epsilon,\Omega}}(r)=-\frac{1}{8d}r^{1-\frac{d}{2}}\left\\{C_{1}\eta_{+}\beta_{+}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}r}{2}\right)+C_{2}\eta_{-}\beta_{-}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}r}{2}\right)\right\\}.$

for $C_{1}$ and $C_{2}$ to be specified. One may determine these constants
from the boundary condition at $r=\epsilon$. Indeed, we infer from Eq. (70)
that

$\epsilon^{1-\frac{d}{2}}\begin{bmatrix}J_{\frac{d}{2}}\cdot\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)&J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}\epsilon}{2}\right)\\\
-\frac{1}{8d}\eta_{+}\beta_{+}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)&-\frac{1}{8d}\eta_{-}\beta_{-}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}\epsilon}{2}\right)\end{bmatrix}\cdot\begin{bmatrix}C_{1}\\\
C_{2}\end{bmatrix}=\begin{bmatrix}\mathfrak{p}\\\ \mathfrak{q}\end{bmatrix}.$
(69)

This algebraic system can be explicitly solved:

$\displaystyle
C_{1}=\frac{\epsilon^{\frac{d}{2}-1}}{\mathcal{W}_{\epsilon}}\left\\{\eta_{-}\beta_{-}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}\epsilon}{2}\right)\mathfrak{p}-8dJ_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}\epsilon}{2}\right)\mathfrak{q}\right\\},$
$\displaystyle
C_{2}=\frac{\epsilon^{\frac{d}{2}-1}}{\mathcal{W}_{\epsilon}}\left\\{-\eta_{+}\beta_{+}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)\mathfrak{p}+8dJ_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)\mathfrak{q}\right\\},$

where (comparing with Eq. (63))

$\displaystyle\mathcal{W}_{\epsilon}$
$\displaystyle:=\det\begin{bmatrix}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)&J_{\frac{d}{2}}\left(\frac{\lambda\beta_{-}\epsilon}{2}\right)\\\
\eta_{+}\beta_{+}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)&\eta_{-}\beta_{-}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}\epsilon}{2}\right)\end{bmatrix}$
$\displaystyle=2i\,{\rm
Im}\left\\{\eta_{-}\beta_{-}\left[J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)\right]^{\dagger}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)\right\\}.$

This completes the proof. ∎

Finally, we arrive at the stage of proving Theorem 1.1 (reproduced below).

######  Theorem.

Let $\Omega$ be a bounded $C^{2,\alpha}$-domain in $\mathbb{R}^{d}$, for some
$d\in\\{2,\ldots,8\\}$ and $\alpha>0$. The expected signature $\Phi$ of a
Brownian motion stopped upon the first exit time from $\Omega$ has finite
radius of convergence everywhere on $\Omega$.

###### Proof.

Our arguments will essentially be an adaptation of the proof of Theorem 6.8.
We shall only indicate necessary modifications.

First of all, Step 1 in the proof of Theorem 1.1 ($d=2$ case) in §5 remains
valid for $\Omega\Subset\mathbb{R}^{d}$. This is the only place that we need
the $C^{2,\alpha}$-regularity of the domain. We conclude that

$\displaystyle\left\|\mathbf{Proj}_{n}\Phi\right\|_{C^{2}(\Omega)}\leq C^{n}$

for $C$ depending only on the geometry of $\Omega$, the Hölder index $\alpha$,
and the dimension $d$.

Consider now
$\widetilde{\mathcal{H}_{\lambda,\epsilon,\Omega}}=\left(h^{(1)}_{\lambda},\cdots,h^{(d+1)}_{\lambda}\right)^{\top}$.
We show that there exists $\lambda_{\star}>0$ for which
$\lim_{\lambda\rightarrow\lambda_{\star}}h^{(d+1)}_{\lambda}(0)=\infty$. Label
as before the boundary conditions at $r=\epsilon$ by

$h^{(1)}_{\lambda}(\epsilon)=\mathfrak{p}\quad\text{ and }\quad
h^{(d+1)}_{\lambda}(\epsilon)=\mathfrak{q},$ (70)

where, as a precaution, $\mathfrak{p}$ and $\mathfrak{q}$ depend in general on
$\lambda$ and $\epsilon$.

By Theorem 6.11 and the ensuing remark, we have the closed-form expression

$\displaystyle h^{(d+1)}_{\lambda}(r)$
$\displaystyle=\left(\frac{r}{\epsilon}\right)^{1-\frac{d}{2}}\left\\{\frac{\frac{1}{8d}\mathfrak{p}\cdot{\rm
Im}\,\left[\eta_{+}\beta_{+}\eta_{-}\beta_{-}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}\epsilon}{2}\right)J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}r}{2}\right)\right]+\mathfrak{q}\cdot{\rm
Im}\,\left[\eta_{-}\beta_{-}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}r}{2}\right)\right]}{{\rm
Im}\left[\eta_{-}\beta_{-}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}\epsilon}{2}\right)J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)\right]}\right\\}$

for $d\in\\{2,\dots,8\\}$. Set $h^{(d+1)}_{\lambda}(0):=\lim_{r\searrow
0}h^{(d+1)}_{\lambda}(r)$; by properties of the Bessel function,

$\displaystyle
h^{(d+1)}_{\lambda}(0)=\frac{\mathcal{N}_{\epsilon}(\lambda)}{\Theta_{\epsilon}(\lambda)}$

with the numerator

$\displaystyle\mathcal{N}_{\epsilon}(\lambda)$
$\displaystyle=\underset{r\rightarrow
0}{\lim}\left(\frac{r}{\epsilon}\right)^{1-\frac{d}{2}}\Bigg{\\{}\frac{\mathfrak{p}}{8d}\cdot{\rm
Im}\,\left[\eta_{+}\beta_{+}\eta_{-}\beta_{-}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}\epsilon}{2}\right)J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{+}r}{2}\right)\right]$
$\displaystyle\qquad\qquad\qquad\qquad+\mathfrak{q}\cdot{\rm
Im}\,\left[\eta_{-}\beta_{-}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}r}{2}\right)\right]\Bigg{\\}}$
$\displaystyle=\mathfrak{q}\cdot\underbrace{\frac{1}{\Gamma(\frac{d}{2})}\cdot{\rm
Im}\,\left[\eta_{-}\beta_{-}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)\left(\frac{\lambda\beta_{-}\epsilon}{4}\right)^{\frac{d}{2}-1}\right]}_{\mathcal{N}^{(1)}_{\epsilon}(\lambda)}$
$\displaystyle\qquad\qquad\qquad\qquad+\mathfrak{p}\cdot\underbrace{\frac{1}{\Gamma(\frac{d}{2})}\cdot{\rm
Im}\,\left[\frac{1}{8d}\eta_{+}\beta_{+}\eta_{-}\beta_{-}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}\epsilon}{2}\right)\left(\frac{\lambda\beta_{+}\epsilon}{4}\right)^{\frac{d}{2}-1}\right]}_{\mathcal{N}^{(2)}_{\epsilon}(\lambda)}$
(71)

and the denominator

$\Theta_{\epsilon}(\lambda)={\rm
Im}\left[\eta_{-}\beta_{-}J_{\frac{d}{2}-1}\left(\frac{\lambda\beta_{-}\epsilon}{2}\right)J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)\right].$

From Lemma 4.9 and Corollary 4.10 we infer that

$\left[h^{(1)}_{\lambda}(r),0,\cdots,0,h^{(d+1)}_{\lambda}(r)\right]^{\top}\in\left\\{x\in\mathbb{R}^{d+1}:\,x^{d+1}\geq\left|x^{1}\right|\text{
and }x^{d+1}\geq 1\right\\}\qquad\text{for all }r\in]0,\epsilon].$

In particular, when $\mathfrak{p}\neq 0$ we have
$\frac{\mathfrak{q}}{|\mathfrak{p}|}\geq 1$ and $\mathfrak{q}\geq 1$ in the
almost sure sense, thanks to Eq. (70).

The proof of Theorem 6.8 shows that there is a root $\lambda_{\star}$ for the
denominator $\Theta_{\epsilon}$, with
$\lambda_{\star}\in\left]\frac{2.5}{\epsilon},\frac{3}{\epsilon}\right[$. It
thus suffices to check that the numerator $\mathcal{N}_{\epsilon}(\lambda)$ is
non-vanishing.

Case 1: $\mathfrak{p}=0$. Then we have

$\mathcal{N}_{\epsilon}(\lambda)=\mathfrak{q}\cdot{\rm
Im}\left[\eta_{-}\beta_{-}J_{\frac{d}{2}}\left(\frac{\lambda\beta_{+}\epsilon}{2}\right)\left(\frac{\lambda\beta_{-}\epsilon}{4}\right)^{\frac{d}{2}-1}\right],$

which is essentially the same as the numerator for the unit ball case (modulo
scaling). By the proof of Theorem 6.8, for $\mu\in]2.5,3[$ one has
$\eta_{-}\beta_{-}J_{\frac{d}{2}}\left(\frac{\mu\beta_{+}}{2}\right)\left(\frac{\mu\beta_{-}}{4}\right)^{\frac{d}{2}-1}\neq
0$. Using a change of variable and the geometrical constraint
$\mathfrak{q}\geq 1$, one may easily show that
$\mathcal{N}_{\epsilon}(\lambda)$ is bounded away from zero for all
$\lambda\in\left]\frac{2.5}{\epsilon},\frac{3}{\epsilon}\right[$ .

Case 2: $\mathfrak{p}\neq 0$. Since the 2-dimensional case has been treated
separately, we assume from now on $d\in\\{3,4,\ldots,8\\}$. Once we establish
the following _claim_ :

$\left|\mathcal{N}_{\epsilon}^{(1)}(\lambda)\right|-\left|\mathcal{N}_{\epsilon}^{(2)}(\lambda)\right|>0\qquad\text{
for all }\lambda\in\left[\frac{2.5}{\epsilon},\frac{3}{\epsilon}\right],$ (72)

we can immediately conclude the proof using the triangle inequality plus the
geometrical constraint $\mathfrak{q}\geq|\mathfrak{p}|>0$. Indeed, it holds
that

$\displaystyle\left|\mathcal{N}_{\epsilon}(\lambda)\right|$
$\displaystyle=\left|\mathfrak{q}\mathcal{N}_{\epsilon}^{(1)}(\lambda)+\mathfrak{p}\mathcal{N}_{\epsilon}^{(2)}(\lambda)\right|$
$\displaystyle=\mathfrak{p}\left|\frac{\mathfrak{q}}{\mathfrak{p}}\mathcal{N}_{\epsilon}^{(1)}(\lambda)+\mathcal{N}_{\epsilon}^{(2)}(\lambda)\right|$
$\displaystyle\geq\mathfrak{p}\left(\left|\frac{\mathfrak{q}}{\mathfrak{p}}\mathcal{N}_{\epsilon}^{(1)}(\lambda)\right|-\left|\mathcal{N}_{\epsilon}^{(2)}(\lambda)\right|\right)$
$\displaystyle\geq\mathfrak{p}\left(\left|\mathcal{N}_{\epsilon}^{(1)}(\lambda)\right|-\left|\mathcal{N}_{\epsilon}^{(2)}(\lambda)\right|\right)$
$\displaystyle>0.$

The proof of _claim_ (72) will be presented in the appendix. See Lemma A.5. ∎

###  6.3. Concluding remarks

We have proved in this paper the finiteness of radius of convergence of the
expected signature for stopped Brownian motions in dimensions $2\leq d\leq 8$.
The restriction on the range of $d$ mainly arises from the ODE system for the
hyperbolic development $\mathcal{H}_{\lambda,\Omega}$ derived in Proposition
6.3 (see also Theorem 6.5). This ODE has explicit solution involving key
parameters $\eta_{\pm}:=d+3\pm\sqrt{d^{2}-10d+9}$ and
$\beta_{\pm}:=\sqrt{2d-6\pm 2\sqrt{d^{2}-10d+9}}$, which are real for $d\geq
9$, whence the solutions demonstrate qualitatively different behaviours. We
shall leave this point for future investigations.

##  Appendix A

In this appendix, we first collect a (crude) upper bound for the Bessel
functions:

######  Lemma A.1.

Consider Bessel functions of the first kind, $J_{\nu}(z)$, and their Taylor
expansion:

$\displaystyle{J}_{\nu}(z):=\left(\frac{z}{2}\right)^{\nu}\sum_{k=0}^{\infty}(-1)^{k}\frac{\left(\frac{z^{2}}{4}\right)^{k}}{k!\Gamma(k+\nu+1)}\qquad\text{
for $z\in\mathbb{C}$}.$

Write
$J_{\nu}(z)=\mathring{J}_{\nu}^{n}(z)+\mathscr{R}_{\nu}^{n}(z):=\sum_{k=0}^{n-1}+\sum_{k=n}^{\infty}$.
The remainder term $\mathscr{R}_{\nu}^{n}(z)$ can be estimated by

$\displaystyle\left|\mathscr{R}_{\nu}^{n}(z)\right|\leq
E(z,n,\nu)\qquad\text{for $|z|<2(n+1)$},$

where

$\displaystyle
E(z,n,\nu):=\begin{cases}\left\\{\frac{\left(\frac{|z|}{2}\right)^{2n+\nu}}{n!(n+\nu)!}\right\\}\cdot\left\\{\frac{1}{1-\frac{|z|^{2}}{4(n+1)^{2}}}\right\\}&\text{
if $\nu$ is an integer},\\\
\left\\{\frac{2^{n}}{{n!(2n+2\nu+1)!!}}\left(\frac{|z|}{2}\right)^{2n+\nu}\right\\}\cdot\left\\{\frac{1}{1-\frac{|z|^{2}}{4(n+1)^{2}}}\right\\}&\text{
if $\nu$ is a half-integer}.\end{cases}$

###### Proof.

From the Taylor expansion for $J_{\nu}$ we get

$\displaystyle\left|\mathscr{R}_{\nu}^{n}(z)\right|\leq\left(\frac{|z|}{2}\right)^{\nu}\sum_{k=n}^{\infty}\frac{\left(\frac{|z|}{2}\right)^{2k}}{k!\Gamma(k+\nu+1)}=\left(\frac{|z|}{2}\right)^{2n+\nu}\sum_{k=0}^{\infty}\frac{\left(\frac{|z|}{2}\right)^{2k}}{(n+k)!\Gamma(n+k+\nu+1)}.$

Then, we deduce from the functional identity $\Gamma(z+1)=z\Gamma(z)$ that

$\displaystyle\left|\mathscr{R}_{\nu}^{n}(z)\right|$
$\displaystyle\leq\left(\frac{|z|}{2}\right)^{2n+\nu}\frac{1}{n!\Gamma(n+\nu+1)}\sum_{k=0}^{\infty}\frac{\left(\frac{|z|}{2}\right)^{2k}}{(n+1)^{2k}}$
$\displaystyle=\left(\frac{|z|}{2}\right)^{2n+\nu}\frac{1}{n!\Gamma(n+\nu+1)}\cdot\frac{1}{1-\frac{|z|^{2}}{4(n+1)^{2}}}\qquad\text{
for }|z|<2(n+1).$

The proof is complete in view of the special values for $\Gamma(z)$ at
integers and half-integers. ∎

The following simple lemma is rather helpful when estimating errors in Taylor
expansions:

######  Lemma A.2.

Let $f:[a,b]\rightarrow\mathbb{R}$ be a non-increasing function. Let
$g:[a,b]\rightarrow\mathbb{R}$ be such that $\sup_{r\in[a,b]}|f(r)-g(r)|\leq
C$ for some constant $C$. Then $\sup_{r\in[a,b]}g(r)\leq f(a)+C$.

###### Proof.

By the monotonicity of $f$, for any $r\in[a,b]$,

$\displaystyle g(r)=g(r)-f(r)+f(r)\leq g(r)-f(r)+f(a)\leq|g(r)-f(r)|+f(a).$

Therefore, taking the supremum over $[a,b]$, we get

$\displaystyle\sup_{r\in[a,b]}g(r)\leq\sup_{r\in[a,b]}|g(r)-f(r)|+f(a)\leq
C+f(a).$

∎

In the computer-assisted proof below, the supplementary
$\textsc{Maple}^{\text{\textregistered}}$ codes can be found in the Github
repository https://github.com/hello0630/ESigStoppedBM.

######  Lemma A.3.

Let $\mathbf{D}=\mathbf{D}^{d}$ with $d=9$. Let $\lambda_{\star}$ denote the
radius of convergence of $\overline{\mathcal{H}_{\lambda,\mathbf{D}}}$. Then
for all $[0,\lambda_{\star}[$ we have

$\displaystyle\overline{\mathcal{H}_{\lambda,\mathbf{D}}}(z)=\left[\frac{z^{1}}{r}h_{\lambda}^{(1)}(r),\frac{z^{2}}{r}h_{\lambda}^{(1)}(r),\,\cdots,\,\frac{z^{d}}{r}h_{\lambda}^{(1)}(r),\,h^{(d+1)}_{\lambda}(r)\right]^{\top},$
(A.1)

where

$\displaystyle
h^{(1)}(r)=\frac{1}{{t}^{8}}\Bigg{\\{}-3\,\lambda\,t\left({\lambda}^{2}\left({\lambda}^{2}{\it
C_{4}}+{\it C_{2}}\right){t}^{4}+\left(-35\,{\lambda}^{2}{\it C_{4}}-5\,{\it
C_{2}}\right){t}^{2}+105\,{\it
C_{4}}\right)\cos\left(\lambda\,\sqrt{3}t\right)$
$\displaystyle\quad+6\,\sqrt{3}\sin\left(\lambda\,\sqrt{3}t\right)\left({\lambda}^{2}\left(\frac{5}{2}\,{\lambda}^{2}{\it
C_{4}}+{\it C_{2}}\right){t}^{4}+\left(-{\frac{70\,{\lambda}^{2}{\it
C_{4}}}{3}}-\frac{5}{6}\,{\it C_{2}}\right){t}^{2}+{\frac{35\,{\it
C_{4}}}{2}}\right)\Bigg{\\}},$ $\displaystyle
h^{(d+1)}(r)=\frac{1}{\lambda\,{t}^{7}}\Bigg{\\{}-\left({t}^{4}{\lambda}^{6}{\it
C_{4}}+{t}^{2}\left({\it C_{2}}\,{t}^{2}-13\,{\it
C_{4}}\right){\lambda}^{4}+\left(5\,{\it C_{2}}\,{t}^{2}+10\,{\it
C_{4}}\right){\lambda}^{2}\mbox{}-5\,{\it
C_{2}}\right)\sqrt{3}\sin\left(\lambda\,\sqrt{3}t\right)$
$\displaystyle\qquad-9\,t\cos\left(\lambda\,\sqrt{3}t\right)\lambda\,\left({t}^{2}{\lambda}^{4}{\it
C_{4}}-10/3\,{\lambda}^{2}{\it C_{4}}+5/3\,{\it C_{2}}\right)\Bigg{\\}},$

with the constants

$\displaystyle
C_{2}=\frac{3\lambda}{\mathcal{W}}\left(\left({\lambda}^{5}-35\,{\lambda}^{3}+105\,\lambda\right)\cos\left(\lambda\,\sqrt{3}\right)-5\,\left({\lambda}^{4}-{\frac{28\,{\lambda}^{2}}{3}}+7\right)\sin\left(\lambda\,\sqrt{3}\right)\sqrt{3}\right),$
$\displaystyle
C_{4}=\frac{-3}{\mathcal{W}}\lambda\,\left(\left({\lambda}^{3}-5\,\lambda\right)\cos\left(\lambda\,\sqrt{3}\right)-2\,\left({\lambda}^{2}-5/6\right)\sin\left(\lambda\,\sqrt{3}\right)\sqrt{3}\right)$

and the Wrönskian

$\displaystyle\mathcal{W}\equiv\mathcal{W}[\lambda]$
$\displaystyle=\left(-324\,{\lambda}^{6}+4410\,{\lambda}^{4}-8550\,{\lambda}^{2}+1575\right)\left(\cos\left(\lambda\,\sqrt{3}\right)\right)^{2}$
$\displaystyle\qquad-18\,\lambda\,\sqrt{3}\sin\left(\lambda\,\sqrt{3}\right)\left({\lambda}^{6}-50\,{\lambda}^{4}+250\,{\lambda}^{2}-175\right)\cos\left(\lambda\,\sqrt{3}\right)$
$\displaystyle\qquad+27\,{\lambda}^{8}+54\,{\lambda}^{6}-2385\,{\lambda}^{4}+3825\,{\lambda}^{2}-1575.$

###### Proof.

The proof is similar to that of Theorem 6.5. We use
$\textsc{Maple}^{\text{\textregistered}}$ to obtain the fundamental solution
and use the boundary condition to determine the constants. One can refer the
calculation details to CheckSolutionGeneralDomain_d=9.mw in the Github
repository. ∎

Next, let us prove Eq. (37) in Step 5 of the proof for Theorem 1.1 —

$\displaystyle\mathcal{N}(\mu;\mathfrak{e}):={\rm
Im}\left\\{\mathfrak{e}\alpha J_{1}(\mu\zeta)-J_{0}(\mu\zeta)\right\\}\neq
0\quad\text{ for all }\mu\in[2.5,3],\,|\mathfrak{e}|\geq 1.$

######  Lemma A.4.

Let $\zeta=\frac{\sqrt{-2+2\sqrt{7}i}}{2}$ (which is a root for
$p(z)=z^{4}+z^{2}+2$) and $\alpha=\frac{1}{2}\zeta^{3}+\zeta$. Then

$\mathcal{N}(\mu;\mathfrak{e}):={\rm Im}\left\\{\mathfrak{e}\alpha
J_{1}(\mu\zeta)-J_{0}(\mu\zeta)\right\\}\neq 0\quad\text{ for all
}\mu\in[2.5,3]\text{ and }|\mathfrak{e}|\geq 1.$

###### Proof.

It has been established along the proof of Lemma 8 in [1] that

$\displaystyle{\rm
Im}\left\\{\bar{\alpha}J_{1}\left(\mu\bar{\zeta}\right)\right\\}<-1.3\quad\text{
for all }\mu\in[2.5,3].$

We shall prove the following inequalities:

$\displaystyle 0<{\rm
Im}\left\\{J_{0}\left(\mu\bar{\zeta}\right)\right\\}<-{\rm
Im}\left\\{\bar{\alpha}J_{1}\left(\mu\bar{\zeta}\right)\right\\}\quad\text{
for all }\mu\in[2.5,3].$ (A.2)

Indeed, as $|\mathfrak{e}|\geq 1$, we can conclude by estimating

$\displaystyle\left|\mathcal{N}(\mu;\mathfrak{e})\right|$
$\displaystyle=\left|-\mathfrak{e}\cdot{\rm
Im}\left\\{\bar{\alpha}J_{1}\left(\mu\bar{\zeta}\right)\right\\}+{\rm
Im}\left\\{J_{0}\left(\mu\bar{\zeta}\right)\right\\}\right|$
$\displaystyle\geq\bigg{|}\left|\mathfrak{e}\cdot{\rm
Im}\left\\{\bar{\alpha}J_{1}\left(\mu\bar{\zeta}\right)\right\\}\right|-\left|{\rm
Im}\left\\{J_{0}\left(\mu\bar{\zeta}\right)\right\\}\right|\bigg{|}$
$\displaystyle=\left|\underbrace{\left(\left|\mathfrak{e}\right|-1\right)}_{\geq
0}\cdot\underbrace{{\rm
Im}\left\\{\bar{\alpha}J_{1}\left(\mu\bar{\zeta}\right)\right\\}}_{<-1.3}+\underbrace{{\rm
Im}\left\\{J_{0}\left(\mu\bar{\zeta}\right)\right\\}+{\rm
Im}\left\\{\bar{\alpha}J_{1}\left(\mu\bar{\zeta}\right)\right\\}}_{<0}\right|$
$\displaystyle>0.$

To prove the second inequality in Eq. (A.2), we write

$\displaystyle{\rm
Im}\left\\{\bar{\alpha}J_{1}\left(\mu\bar{\zeta}\right)\right\\}+{\rm
Im}\left\\{J_{0}\left(\mu\bar{\zeta}\right)\right\\}$
$\displaystyle\qquad=\underbrace{{\rm
Im}\left\\{\bar{\alpha}\mathring{J}^{n}_{1}\left(\mu\bar{\zeta}\right)\right\\}+{\rm
Im}\left\\{\mathring{J}^{n}_{0}\left(\mu\bar{\zeta}\right)\right\\}}_{=:\mathring{\mathcal{N}}^{n}(\mu)}+\underbrace{{\rm
Im}\left\\{\bar{\alpha}\mathcal{R}^{n}_{1}\left(\mu\bar{\zeta}\right)\right\\}+{\rm
Im}\left\\{\mathcal{R}^{n}_{0}\left(\mu\bar{\zeta}\right)\right\\}}_{=:\,{\mathscr{R}}^{n}(\mu)}.$

The notation on the right-hand side agrees with the decomposition of
$J_{\nu}^{n}$ into $\mathring{J}^{n}_{\nu}+\mathscr{R}_{\nu}^{n}$ via Taylor
expansions. $\mathring{\mathcal{N}}^{n}$ is a polynomial in $\mu$ of degree
$2n+1$. In the remaining parts of the proof we shall choose $n=6$.

The bounds in Lemma A.1 yield that

$\displaystyle\sup_{\mu\in[0,3]}\max\left(\left|\mathscr{R}_{0}^{n}\left(\mu\bar{\zeta}\right)\right|,\left|\mathscr{R}_{1}^{n}\left(\mu\bar{\zeta}\right)\right|\right)\leq
0.0006367,$

which gives us

$\displaystyle\sup_{\mu\in[0,3]}\left|\mathscr{R}^{n}(\lambda)\right|\leq(|\alpha|+1)\max\left(\left|\mathscr{R}_{0}^{n}\left(\mu\bar{\zeta}\right)\right|,\left|\mathscr{R}_{1}^{n}\left(\mu\bar{\zeta}\right)\right|\right)\leq
0.0011395.$

To estimate $\mathring{\mathcal{N}}^{n}$, let us write it as a polynomial of
degree $n$ based at the point $\mu=2.5$:

$\displaystyle\mathring{\mathcal{N}}^{n}(\mu)=\sum_{i=0}^{2n+1}C_{i}(\mu-2.5)^{i}.$

The coefficients can be found, _e.g._ , using
$\textsc{Maple}^{\text{\textregistered}}$. The first four $C_{i}$ are
negative, with $C_{0}<-0.119150$, $C_{1}<-0.169$, $C_{2}<-0.184$,
$C_{3}<-0.049$, but $C_{4}>0$. Let
$\mathscr{T}^{3}(\mu)=\sum_{i=0}^{3}C_{i}(\mu-2.5)^{i}$. For $\mu\in[2.5,3]$,
one may bound

$\displaystyle\left|\mathring{\mathcal{N}}^{n}(\mu)-\mathscr{T}^{3}(\mu)\right|\leq\sum_{i=4}^{12}|C_{i}|\cdot\left(\frac{1}{2}\right)^{i}\leq
0.00036.$

Note that $\mathscr{T}^{3}$ is a decreasing function. By Lemma A.2 we can
further bound $\mathscr{T}^{n}(\mu)$ by

$\displaystyle\sup_{\mu\in[2.5,3]}\mathscr{T}^{n}(\mu)\leq\mathscr{T}^{n}(2.5)+\sup_{\mu\in[2.5,3]}\left|\mathscr{T}^{n}(\mu)-\mathscr{T}^{3}(\mu)\right|.$

By triangle inequality and the explicit value
$\mathscr{T}^{n}(2.5)=-0.1181564882$, we notice that

$\displaystyle\sup_{\mu\in[2.5,3]}\left\\{\mathscr{T}^{n}(\mu)+{\mathscr{R}}^{n}(\mu)\right\\}\leq-0.1181564882+0.00038+0.0011395<0.$

The proof is now complete. ∎

Finally, let us verify the _claim_ (72) towards the end of the proof of
Theorem 1.1.

######  Lemma A.5.

For $d\in\\{2,3,\cdots,8\\}$, as in Eq. (6.2) (with $\mu=\epsilon\lambda$) we
set

$\displaystyle\mathcal{N}^{(1)}(\mu):=\frac{1}{\Gamma(\frac{d}{2})}{\rm
Im}\,\left[\eta_{-}\beta_{-}J_{\frac{d}{2}}\left(\frac{\mu\beta_{+}}{2}\right)\left(\frac{\mu\beta_{-}}{4}\right)^{\frac{d}{2}-1}\right],$
$\displaystyle\mathcal{N}^{(2)}(\mu):=\frac{1}{\Gamma(\frac{d}{2})}{\rm
Im}\,\left[\frac{1}{8d}\eta_{+}\beta_{+}\eta_{-}\beta_{-}J_{\frac{d}{2}-1}\left(\frac{\mu\beta_{-}}{2}\right)\left(\frac{\mu\beta_{+}}{4}\right)^{\frac{d}{2}-1}\right].$

Then it holds that

$\displaystyle\left|\mathcal{N}^{(1)}(\mu)\right|-\left|\mathcal{N}^{(2)}(\mu)\right|>0\qquad\text{
for each }\mu\in[2.5,3].$

###### Proof.

The case $d=2$ has already been treated in Lemma A.4; we only consider
$d\in\\{3,4,\cdots,8\\}$ from now on. The notations in the proof are
consistent with Lemma A.1. Also, as shown in the proof of Theorem 6.8,
$\mathcal{N}^{(1)}(\mu)<0$ for any $\mu\in[2.5,3]$. It is enough to prove that

  1. (1)

$\mathcal{N}^{(2)}(\mu)>0$;

  2. (2)

$\mathcal{N}(\mu):=\mathcal{N}^{(1)}(\mu)+\mathcal{N}^{(2)}(\mu)<0$ for each
$\mu\in[2.5,3]$.

For the first claim (1), we use $\frac{1}{\Gamma\left(\frac{d}{2}\right)}{\rm
Im}\,\left[\frac{1}{8d}\eta_{+}\beta_{+}\eta_{-}\beta_{-}\mathring{J}^{n}_{\frac{d}{2}-1}\left(\frac{\mu\beta_{-}}{2}\right)\left(\frac{\mu\beta_{+}}{4}\right)^{\frac{d}{2}-1}\right]$
to approximate $\mathcal{N}^{(2)}$. The remainder is bounded by
$\frac{1}{\Gamma(\frac{d}{2})}\left|\frac{1}{8d}\eta_{+}\beta_{+}\eta_{-}\beta_{-}\left(\frac{\mu\beta_{+}}{4}\right)^{\frac{d}{2}-1}\right|\cdot
E\left(\frac{\mu\beta_{-}}{2},n,\frac{d}{2}-1\right)$.

Similarly, we use Taylor approximation to approximate $\mathcal{N}(\mu)$. The
truncated polynomial up to the $n^{\text{th}}$ term is

$\displaystyle\mathring{\mathcal{N}}^{n}(\mu)$
$\displaystyle:=\frac{1}{\Gamma(\frac{d}{2})}{\rm
Im}\,\left[\eta_{-}\beta_{-}\mathring{J}^{n}_{\frac{d}{2}}\left(\frac{\mu\beta_{+}}{2}\right)\left(\frac{\mu\beta_{-}}{4}\right)^{\frac{d}{2}-1}\right]$
$\displaystyle\qquad\qquad+\frac{1}{\Gamma(\frac{d}{2})}{\rm
Im}\,\left[\frac{1}{8d}\eta_{+}\beta_{+}\eta_{-}\beta_{-}\mathring{J}^{n}_{\frac{d}{2}-1}\left(\frac{\mu\beta_{-}}{2}\right)\left(\frac{\mu\beta_{+}}{4}\right)^{\frac{d}{2}-1}\right].$

The error can be estimated uniformly as follows:

$\displaystyle\sup_{\mu\in[2.5,3]}\,\left|\mathring{\mathcal{N}}^{n}(\mu)-{\mathcal{N}}(\mu)\right|$
$\displaystyle\quad\leq\text{Err}^{n}_{\mathcal{N}}:=\sup_{\mu\in[2.5,3]}\,\Bigg{\\{}\frac{1}{\Gamma\left(\frac{d}{2}\right)}\left|\eta_{-}\beta_{-}\left(\frac{\mu\beta_{-}}{4}\right)^{\frac{d}{2}-1}\right|E\left(\frac{\mu\beta_{+}}{2},n,\frac{d}{2}\right)$
$\displaystyle\qquad\qquad\qquad\qquad+\frac{1}{\Gamma\left(\frac{d}{2}\right)}\left|\frac{1}{8d}\eta_{+}\beta_{+}\eta_{-}\beta_{-}\left(\frac{\mu\beta_{+}}{4}\right)^{\frac{d}{2}-1}\right|E\left(\frac{\mu\beta_{-}}{2},n,\frac{d}{2}-1\right)\Bigg{\\}}$
$\displaystyle\quad\qquad\qquad=\frac{1}{\Gamma\left(\frac{d}{2}\right)}\left|\eta_{-}\beta_{-}\left(\frac{3\beta_{-}}{4}\right)^{\frac{d}{2}-1}\right|E\left(\frac{3\beta_{+}}{2},n,\frac{d}{2}\right)$
$\displaystyle\quad\qquad\qquad\qquad\quad+\frac{1}{\Gamma\left(\frac{d}{2}\right)}\left|\frac{1}{8d}\eta_{+}\beta_{+}\eta_{-}\beta_{-}\left(\frac{3\beta_{+}}{4}\right)^{\frac{d}{2}-1}\right|E\left(\frac{3\beta_{-}}{2},n,\frac{d}{2}-1\right)$

as $|z|\mapsto\left|E(z,n,\nu)\right|$ is an increasing function, thanks to
Lemma A.1.

Similar arguments as for Lemma A.4 show that $\mathring{\mathcal{N}}^{n}(\mu)$
is decreasing on $[2.5,3]$. Table 3 summarises the values
$\mathring{\mathcal{N}}^{n}(2.5)$ and $\text{Err}^{n}_{\mathcal{N}}$ for
$n=6$. By Lemma A.2, we can conclude that

$\displaystyle\sup_{\lambda\in[2.5,3]}\mathcal{N}(\lambda)\leq\mathring{\mathcal{N}}^{n}(2.5)+\text{Err}^{n}_{\mathcal{N}}<0.$

Table 3. The numerator $\mathcal{N}(\mu)$ is non-vanishing on $]2.5,3[$ for general domains Dimension | Value for $\mathring{\mathcal{N}}^{n}(2.5)$ | Value for $\text{Err}^{n}_{\mathcal{N}}$
---|---|---
3 | -3.487949 | 0.366411
4 | -5.367159 | 0.317482
5 | -6.082985 | 1.552336
6 | -5.465016 | 0.811930
7 | -3.964234 | 2.802239
8 | -2.193441 | 1.131338

Thus we have proved the second claim (2). ∎

Acknowledgement. HN is supported by the EPSRC under the program grant
EP/S026347/1 and the Alan Turing Institute under the EPSRC grant EP/N510129/1.
The authors extend their gratitude to Terry Lyons, Weijun Xu, and Guangyu Xi
for kind communications. SL also thanks the mathematics department at Rice
University, Houston for the nice working atmosphere, where this work was
initiated when SL worked as a G. C. Evans Instructor. Both authors are deeply
indebted to Horatio Boedihardjo for very insightful and constructive
discussions.

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