# Feedback enhanced Dyakonov–Shur instability in Graphene-FET

Pedro Cosme<EMAIL_ADDRESS>GoLP/Instituto de Plasmas
e Fusão Nuclear, Instituto Superior Técnico, 1049-001 Lisboa, Portugal  Diogo
Simões  Instituto Superior Técnico, 1049-001 Lisboa, Portugal

###### Abstract

Graphene devices are known to have the potential to operate THz signals. In
particular, graphene field-effect transistors have been proposed as devices to
host plasmonic instabilities in the THz realm; for instance, Dyakonov-Shur
instability which relies upon dc excitation. In this work, starting from a
hydrodynamical description of the charge carriers, we extend the transmission
line description of graphene field-effect transistors to a scheme with a
positive feedback loop, also considering the effects of delay, which leads to
the transcendental transfer function with terms of the form
$e^{as}\operatorname{sech}^{k}(s)/s$. Applying the conditions for the
excitation of Dyakonov–Shur instability, we report an enhanced voltage gain in
the linear regime that is corroborated by our simulations of the nonlinear
hydrodynamic model for the charge carriers. This translates to both greater
saturation amplitude – often up to 50% increase – and fastest growth rate of
the self-oscillations. Thus, we bring forth a prospective concept for the
realization of a THz oscillator suitable for future plasmonic circuitry.

##  I Introduction

Present day technology – and thus society itself – relies heavily on high
frequency communication and nanoscale devices; yet, despite all the advances
in this subject the THz range of the spectrum, with the potential to bridge
the realms of electronics and photonics, still proves to be a challenge. Thus,
new and efficient ways to generate, detect, or manipulate electromagnetic
radiation in this range are ever sought after.

In this context, several designs of graphene nano-devices have been proposed
[1, 2, 3, 4, 5, 6, 7, 8, 9] where an instability triggers the current to self-
oscillate. Amongst the various mechanisms, Dyakonov–Shur (DS) instability [10,
11], driven by the large asymmetry of impedance at the contacts, has been
frequently brought forth as a way to induce self-oscillations from dc current.
However, such instability has a modest amplitude gain, to circumvent that we
study the use of a feedback loop as a way to improve the gain [12] as well as
the growth rate.

In this work, we consider the effect of positive feedback on a graphene field-
effect transistor (gFET) under the conditions for DS instability. For the
analytical characterization, we resort to a transmission line description [13,
14] that can be obtained from the hydrodynamic description of charge carriers.
Alongside fully nonlinear simulations performed by tethys simulation tool [15]

A growing number of both theoretical [16, 17, 18, 19, 20] and experimental
works [21, 22, 23, 24, 25, 24, 26] have shown that the hydrodynamic regime of
carriers in graphene and similar 2D materials is achievable even around room
temperature. Therefore, the mean-field description of the carriers number
density $n$ and momentum density $\mathbf{p}=nm^{\star}\mathbf{v}$ can be
written as a set of conservation equation in the form [2, 1]:

$\frac{\partial n}{\partial
t}+\mathbf{\nabla}\cdot\frac{\mathbf{p}}{m^{\star}}=0\text{ and}$ (1a)
$\frac{\partial\mathbf{p}}{\partial
t}+\mathbf{\nabla}\cdot\left(\frac{\mathbf{p}\otimes\mathbf{p}}{nm^{\star}}+\frac{v^{2}_{F}}{3}n^{3/2}+\frac{c^{2}}{2}n^{2}\right)=0,$
(1b)

where $v_{F}=10^{6}{\rm\,ms^{-1}}$ is the Fermi velocity and $c=$ the
characteristic velocity of the plasma oscillations in the gradual-channel
approximation[27]; moreover, due to the linear nature of the bands in graphene
the mass is defined via the Drude weight $m^{\star}=\hbar\sqrt{\pi n}/v_{F}$.
This system undergoes DS instability when the boundary conditions

$n(t,0)=n_{0}\text{ and }n(L,t)v(L,t)=n_{0}v_{0},$ (2)

corresponding to a ac short ac open [10, 11], are applied. Assuming uniformity
along the direction transverse to the charge flow and in the gradual-channel
approximation, we can convert to the local description to the macroscopic
quantities of interest of current and voltage at source $(x=0)$ and drain
$(x=L)$ by $I(x)=-eWv(x)n(x)$ and $V(x)=-en(x)/C_{g}$, with $W$ the gFET width
and $C_{g}$ its gate capacitance per unit area.

##  II Transfer matrix description of the Graphene transistor

###  II.1 Open-Loop Transfer Function

Figure 1: Left panel: Schematic rendition of the positive feedback loop. Right
panel: Simplified model of the gFET considered, highlighting the gate-channel
capacitance $C_{\rm ox}$ as well as the main parasitic impedances represented.

From the hydrodynamical description given in (1b) one can derive a transfer
matrix description (see details in Refs. [13, 14]) in the form

$\begin{bmatrix}V_{\scriptscriptstyle S}\\\ I_{\scriptscriptstyle
S}\end{bmatrix}\\!=\hat{T}\begin{bmatrix}V_{\scriptscriptstyle D}\\\
I_{\scriptscriptstyle D}\end{bmatrix}\text{with }\hat{T}=e^{\theta
s}\begin{bmatrix}\cosh s&Z_{0}\sinh s\\\ Z_{0}^{-1}\sinh s&\cosh
s\end{bmatrix}\\!,$ (3)

akin to a transmission line, with $\theta=v_{0}/v_{p}$ the ratio between the
drift velocity and the plasmons velocity, the characteristic impedance
$Z_{0}^{-1}=C_{g}Wv_{p}$, and $s=i\frac{Lv_{p}\omega}{v_{0}^{2}-v_{p}^{2}}$
the normalized complex frequency. Thus, we can retrieve the voltage gain

$\frac{V_{\scriptscriptstyle D}}{V_{\scriptscriptstyle S}}=e^{-\theta
s}\left(\cosh s+\frac{Z_{0}}{Z_{\scriptscriptstyle D}(s)}\sinh
s\right)^{-1}=g(s)$ (4)

making use of the drain impedance $Z_{\scriptscriptstyle
D}=V_{\scriptscriptstyle D}/I_{\scriptscriptstyle D}$, which for an ideal ac
open circuit, as required for Dyakonov–Shur, $Z_{\scriptscriptstyle
D}=\infty$. For non-ideal impedance matching, one can consider a typical a
common gate configuration where $Z_{\scriptscriptstyle D}^{-1}\approx sC_{\rm
gd}+r_{o}^{-1}$ and append such poles as $|V_{\scriptscriptstyle
D}/V_{\scriptscriptstyle S}|\approx\left|\frac{e^{\theta
s}\operatorname{sech}s}{1-Z_{0}/Z_{\scriptscriptstyle D}}\right|$ for
$|Z_{0}/Z_{\scriptscriptstyle D}|\ll 1$, thus, for the sake of simplicity, in
the remainder of this work we assume $Z_{\scriptscriptstyle D}\to\infty$.

As we can see, all the poles of $g(s)$ are located along the imaginary axis,
rendering the system marginally stable and in fact any perturbation leads to
its instability. Moreover, note that, the unit step response of (4) is a
square wave [28] as

$\frac{\operatorname{sech}s}{s}\xrightarrow{\mathcal{L}^{-1}}\begin{cases}2,&1<t<3\pmod{4}\\\
0,&\text{otherwhise}\end{cases},$ (5)

a trait that is clear in our simulations during the linear regime.

###  II.2 Closed-Loop Transfer Function

Let us now assume that a feedback loop is imposed on our system, as
represented in Fig.1. In general, we have

$\begin{bmatrix}V_{\scriptscriptstyle S}\\\ I_{\scriptscriptstyle
S}\end{bmatrix}=(\hat{T}-\hat{H})\begin{bmatrix}V_{\scriptscriptstyle D}\\\
I_{\scriptscriptstyle D}\end{bmatrix}$ (6)

where the matrix $\hat{H}$ encodes the proportion of voltage and/or current
that is looped. solving for $V_{\scriptscriptstyle D}$ leads to the closed
loop gain:

$\frac{V_{\scriptscriptstyle D}}{V_{\scriptscriptstyle S}}\equiv
G(s)=\frac{1}{e^{\theta s}\cosh s-\beta e^{-\tau s}},$ (7)

with $\beta<1$ the voltage fraction and $\tau$ a delay factor. Since for any
experimental implementation of feedback, the source and drain of the gFET will
have to be connected by conductive material, its effect can not be
instantaneous, but rather to take a given time delay $\tau$ to occur, such
delay is estimated to be of the order of the travel time of a a plasmon, i.e.
$\tau\gtrsim L/v_{p}$.

The feedback drastically modifies the structure of the transfer function, and
the location of the poles start to wander over the complex plane, as can be
seen in Fig.2, for moderate values of delay the odd poles drift towards the
unstable right-half plane while the even ones recede becoming decaying modes –
a feature that might be exploited for upconversion.

Figure 2: Location of the first six poles of the closed-loop transfer function
$G(s)$ on the $s$-plane as function of feedback ratio, with $\theta\equiv
v_{0}/v_{p}=1/20$. Ticks along the curves indicate the ratio $\beta$ in $0.2$
increments, and the values of delay are $\tau=0$ (solid black lines);
$\tau=0.4$ (red dashed lines); and $\tau=1$ (blue dot-dashed lines).

Furthermore, as expected the feedback pushes the overall gain up, as is made
clear by the Bode plots on Fig.3; the delay, however, appears to curb the
self-amplification.

Figure 3: Bode plot for the closed-loop transfer function $G(s=i\omega)$ with
$\theta\equiv v_{0}/v_{p}=1/20$. Top panel: Instantaneous feedback comparing
the effect of feedback ratio. Bottom panel: Effect of delay time $\tau$ for
fixed ratio of $\beta=0.2$ .

Regarding the unit step response of the looped system, we expand the transfer
function in powers of the feedback amplification, which leads to

$\frac{G(s)}{s}=\frac{e^{-\theta
s}\operatorname{sech}s}{s}+\beta\frac{e^{-s(2\theta+\tau)}\operatorname{sech}^{2}s}{s}+\mathcal{O}(\beta^{2}).$
(8)

So, while the first term yields the usual square wave, the subsequent terms in
the series return unbounded functions (cf. appendix A and Refs. [28, 29]), as
$\mathcal{L}^{-1}[\operatorname{sech}^{k}(s)/s]\asymp t^{k-1}$, this leads to
a faster growth rate for the instability but evidently cannot represent
physical modes of the solutions for $t\to\infty$. Nonetheless, one must bear
in mind that insofar we have discussed but linear analysis of a nonlinear
system, and the transient growing terms of (8) saturate at a certain point.

##  III Nonlinear Simulations

Thus far, the transfer function analysis provided crucial insight for the
stability and early stages of the gFET operation under DS instability.
However, aiming for continuous the THz generation, the time evolution for
later times is critical. To surpass the limitations of the linear analysis,
the hydrodynamic system (1b) needs to be numerically simulated so that the
full nonlinear effects could manifest. We resorted to Tethys computer program
[15], a finite-volume solver designed for the simulation of graphene
electrohydrodynamics.

Under the DS conditions, we obtain the typical results (cf. [1, 30, 31, 32,
33]) for the voltage, which are presented on Fig.4 and where is clear the
positive impact of the feedback albeit the modest $\beta$ ratio.

Even in the absence of feedback, one can distinguish three regimes, viz.
linear evolution, transient time, and saturation. For the first few periods
the system roughly follows a square wave, as predicted by (5), before reaching
saturation, which for the $\beta=0$ is dictated by Rankine–Hugoniot jump
considerations that limit the maximum amplitude – and thus the power diverted
to the ac response.

Figure 4: Nonlinear oscillations due to DS instability with delayed feedback.
Direct simulation of the fluid model (1b) with $S/v_{0}=20$. Note the growth
of the saturation amplitude and the quicker onset of the saturated regime.

To the contrary, closing the feedback loop leads to a significant increase in
the saturation amplitude and to a fastest saturation itself. However, the
waveform is distorted into a new shape due to the interplay of the higher
order modes in the expansion (8).

##  IV Conclusion

Since the discovery of graphene and its first applications to integrated
circuit devices the number of uses and designs is ever-growing, boosted also
by the discovery and development of new 2D materials.

In that context, our study – of a graphene transistor under a feedback loop –
paves the way for future self-driven oscillators in the THz range. We report
that the effect of the closed loop unto Dyakonov Shur instability mitigates
the usual technical problems, since it enhances both the growth rate and the
amplitude of oscillations. The former is essential to overcome the eventual
damping mechanisms ever present in real devices; while the latter improves the
power and resilience of the signal.

Furthermore, resorting to the transfer matrix description, we characterized
the delayed closed loop system. Firstly, the trajectory of the poles on the
complex plane show that the feedback leads to further destabilization of the
odd spectral modes – moving rightwards – in detriment of the even ones – that
move to the left half plane. Such effect explain the reported third harmonic
intensification [12] and can be further tuned to place this third harmonic on
the goal frequency. Secondly, expanding the transfer function in powers of the
feedback ratio evinces the structure of the unit step response as a sum of
square waves whose amplitude grows with a power law, since the inverse Laplace
transform of $G(s)/s$ scales as $t^{2}$ up to second order in the feedback
ratio $\beta$. Nonetheless, as expected, the delay must be tailored to avoid
destructive interference between the waves imposed by the loop and the source
terminal and the counter-propagating ones inside the channel.

The simulation of the nonlinear stage complemented our linear analysis and
proved the increase of the saturation amplitude, e.g. from a voltage ratio
$V_{\scriptscriptstyle D}/V_{\scriptscriptstyle S}\approx 2$ to
$V_{\scriptscriptstyle D}/V_{\scriptscriptstyle S}\approx 3$, by applying a
20% feedback. However, this saturation stage and its limit cycle is still open
for further inquiries, namely: which Liénard equation would accurately
describe it, and how does the limit cycle vary with the system parameters.

Lastly, throughout this work, by using the classic transmission line model, we
implicitly assumed galilean invariance of the plasmons. This constitutes a
good approximation, since for most systems the drift velocities are well below
the Fermi velocity. Yet, in future works the deviation from this assumption
could be an interesting line of research to pursue.

###### Acknowledgements.

Lastly, one of the authors (PC) gratefully acknowledges the financial support
of FCT – Fundação para a Ciência e a Tecnologia (FCT–Portugal) through grant
PB/DB/150415/2019

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##  Appendix A Inverse Laplace transform

The computation of the unit step response for the closed-loop system resorst
to inverse Laplace transforms of powers of the form
$\mathcal{L}^{-1}[\operatorname{sech}^{k}(s)/s]$. Here, in Tab.1, we provide
closed form expressions for the first few terms, that attest the polynomial
growth for the envelope of the base square wave.

Table 1: First six inverse Laplace transform of $\operatorname{sech}^{k}(s)/s$. $\displaystyle F(s)$ | $\displaystyle f(t)=\mathcal{L}^{-1}[F(s)]$
---|---
$\displaystyle\frac{\operatorname{sech}s}{s}$ | $\displaystyle 1-(-1)^{\left\lfloor\frac{t+1}{2}\right\rfloor}$
$\displaystyle\frac{\operatorname{sech}^{2}s}{s}$ | $\displaystyle 1+(-1)^{\left\lfloor\frac{t}{2}\right\rfloor}\left(\\!-2\left\lfloor\frac{t}{2}\right\rfloor-1\right)$
$\displaystyle\frac{\operatorname{sech}^{3}s}{s}$ | $\displaystyle 1+(-1)^{\left\lfloor\frac{t+1}{2}\right\rfloor}\left(2\left\lfloor\frac{t+1}{2}\right\rfloor^{2}-1\right)$
$\displaystyle\frac{\operatorname{sech}^{4}s}{s}$ | $\displaystyle 1+(-1)^{\left\lfloor\frac{t}{2}\right\rfloor}\left(\frac{4}{3}\left\lfloor\frac{t}{2}\right\rfloor^{3}+2\left\lfloor\frac{t}{2}\right\rfloor^{2}-\frac{4}{3}\left\lfloor\frac{t}{2}\right\rfloor-1\right)$
$\displaystyle\frac{\operatorname{sech}^{5}s}{s}$ | $\displaystyle 1+(-1)^{\left\lfloor\\!\frac{t+1}{2}\\!\right\rfloor}\\!\left(\\!-\frac{2}{3}\left\lfloor\frac{t+1}{2}\right\rfloor^{4}+\frac{8}{3}\left\lfloor\frac{t+1}{2}\right\rfloor^{2}-1\right)$
$\displaystyle\frac{\operatorname{sech}^{6}s}{s}$ | $\displaystyle 1+(-1)^{\left\lfloor\frac{t}{2}\right\rfloor}\bigg{(}\\!\\!-\frac{4}{15}\left\lfloor\frac{t}{2}\right\rfloor^{5}-\frac{2}{3}\left\lfloor\frac{t}{2}\right\rfloor^{4}+\frac{4}{3}\left\lfloor\frac{t}{2}\right\rfloor^{3}+$
|
$\displaystyle+\frac{8}{3}\left\lfloor\frac{t}{2}\right\rfloor^{2}-\frac{16}{15}\left\lfloor\frac{t}{2}\right\rfloor-1\bigg{)}$


#  An arbitrage driven price dynamics of Automated Market Makers in the
presence of fees

Joseph Najnudel  School of Mathematics, University of Bristol, BS8 1TW,
Bristol, United Kingdom<EMAIL_ADDRESS>,  Shen-Ning Tung
Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan
<EMAIL_ADDRESS>,  Kazutoshi Yamazaki  School of Mathematics and
Physics, the University of Queensland, Brisbane QLD 4072 Australia
<EMAIL_ADDRESS>and  Ju-Yi Yen  Department of Mathematics, University of
Cincinnati, Cincinnati, OH 45221, USA<EMAIL_ADDRESS>
###### Abstract.

We present a model for price dynamics in the Automated Market Makers (AMM)
setting. Within this framework, we propose a reference market price following
a geometric Brownian motion. The AMM price is constrained by upper and lower
bounds, determined by constant multiplications of the reference price. Through
the utilization of local times and excursion-theoretic approaches, we derive
several analytical results, including its time-changed representation and
limiting behavior.

##  1\. Introduction

Automated Market Makers (AMMs) [2, 3] are innovative algorithms that utilize
blockchain technology to automate the process of pricing and order matching on
decentralized exchanges. Their foundation on blockchain and the employment of
smart contracts enable users to buy and sell crypto assets securely, peer-to-
peer, without the dependency on intermediaries or custodians.

A critical distinction between AMMs and traditional centralized limit order
book models is their mechanism for price determination. While an order book
model derives prices through the intentions of individual buyers and sellers,
an AMM determines prices based on the available liquidity in its pool. This
pool, called the “liquidity pool”, consists of funds deposited by users, known
as liquidity providers (LPs). These providers “lock in” varying amounts of two
or more tokens into a smart contract, making them available as liquidity for
other users’ trades. The barrier to entry for becoming an LP is low; one
merely needs a self-custody wallet and a stock of compatible tokens.

A critical cost encountered by LPs in AMMs is adverse selection. This
challenge primarily stems from arbitrageurs capitalizing on disparities
between the lagging prices within AMMs and the real-time market prices often
observed on centralized exchanges. Prominent studies, including [5, 4], delve
into this concern by quantifying the losses experienced by LPs due to
arbitrage. They employ a metric known as ‘loss-versus-rebalancing’ (LVR) to
measure the impact.

This paper presents a model of AMM pricing dynamics based on the following
assumptions:

  1. a)

A reference market with infinite liquidity and no trading costs exists, where
the reference market price $p=(p_{t})_{t\geq 0}$ follows a geometric Brownian
motion.

  2. b)

The AMM applies a fixed trading fee tier of $(1-\gamma)\%$, proportional to
the trading volume. This fee can range from 1 basis point (bp) to 100 basis
points (bps). Notably, Uniswap v3 offers options such as 1 bp, 5 bps, 30 bps,
and 100 bps. Our focus is on the worst-case scenario, assuming that all trades
within the pool are motivated by arbitrage.

  3. c)

Arbitrageurs actively monitor the market, initiating trades whenever they
identify arbitrage opportunities.

These assumptions integrate elements from two sources: [5], which focuses on
continuous arbitrage without fees, and [4], which considers discrete arbitrage
with fees. In our model, we address continuous arbitrage while incorporating
the impact of fees.

From assumptions a)–c), we deduce the following relationships between the
reference market price $p$ and AMM price $\tilde{p}$:

  * •

Under Assumptions a) and b), an arbitrage opportunity exists when
$p_{t}<\gamma\tilde{p}_{t}$ or $p_{t}>\frac{1}{\gamma}\tilde{p}_{t}$ (see [1,
§2.4]).

  * •

According to Assumption b), the AMM price $\tilde{p}_{t}$ remains stable
within the range of $\gamma p_{t}\leq\tilde{p}_{t}\leq\frac{1}{\gamma}p_{t}$,
termed the no-arbitrage interval. This follows immediately from the first
assertion.

  * •

Assumption c) implies that arbitrage actions occur only when
$p_{t}=\gamma\tilde{p}_{t}$ or $p_{t}=\frac{1}{\gamma}\tilde{p}_{t}$, hence
$\tilde{p}$ changes only at these times.

Figure 1. Figure 1

In Figure 1, a sample path of the logarithm of $\tilde{p}$ is provided, along
with the logarithm of $p$ and the lower and upper bounds $\log(\gamma p)$ and
$\log(\gamma^{-1}p)$, respectively. It is pushed upward (resp., downward) at
times when it coincides with the lower (resp., upper) bound, and it stays
constant at all other times.

Our work is inspired by the unpublished paper by Tassy and White [7]. While
the context in the cited work specifically addresses Constant Product Market
Makers, our model can be directly applied to all AMMs. The key contribution of
this paper is the detailed probabilistic description of the considered AMM
price process, underpinned by the local time of the Brownian motion. The
logarithm of $\tilde{p}$ can be seen as a concatenation of the running infimum
and supremum processes of the market price, with appropriate shifting. By
leveraging the classical results on the Skorokhod equation of a reflected
Brownian motion, expressed in terms of Brownian local times, several
analytical results concerning $\tilde{p}$ can be derived. Of particular
interest is the well-posedness of such a process, its time-changed
representation, and its asymptotic behavior. To the best of our knowledge,
this paper represents the first study of this stochastic process, motivated by
its applications in AMM.

The remainder of the paper is structured as follows: First, we present a
precise mathematical construction of our AMM price process. Subsequently, we
explore two distinct approaches to analyze the process under consideration in
this study.

##  2\. AMM price process

We assume the market price $p=(p_{t})_{t\geq 0}$ (normalized in such a way
that it is equal to $1$ at time $0$, and with the appropriate unit of time) is
given by a geometric Brownian motion $p=\exp B$ for a standard Brownian motion
$B=(B_{t})_{t\geq 0}$, and the AMM price $\tilde{p}$ is constrained by the
inequalities

$\gamma p\leq\tilde{p}\leq\gamma^{-1}p$

where the parameter $\gamma\in(0,1)$ is related to the fees of the AMM. These
inequalities are equivalent to

(2.1) $B_{t}-c\leq U_{t}\leq B_{t}+c$

where, for $t\geq 0$, $U_{t}$ is $\log\tilde{p}$ at time $t$, $B_{t}$ is $\log
p$ at time $t$, and $c:=\log(\gamma^{-1})>0$; see Figure 1.

The process $U=(U_{t})_{t\geq 0}$ is chosen in such a way that
$U_{0}=B_{0}=0$, and $U$ remains constant at any time where this is compatible
with the inequalities (2.1). More precisely, it is constant except when it is
pushed up by the process $(B_{t}-c)_{t\geq 0}$ or pushed down by
$(B_{t}+c)_{t\geq 0}$ in order to preserve the inequalities (2.1).

The precise description is described as follows, using a sequence of stopping
times. If $B$ reaches $c$ before $-c$ (we call this event $A$), then we
define, by induction:

  * •

$T_{0}$ as the infimum of $t\geq 0$ such that $B_{t}=c$.

  * •

For all $k\geq 0$, $T_{2k+1}$ is the first time $t\geq T_{2k}$ such that
$B_{t}-\sup_{T_{2k}\leq s\leq t}B_{s}=-2c$.

  * •

For all $k\geq 0$, $T_{2k+2}$ is the first time $t\geq T_{2k+1}$ such that
$B_{t}-\inf_{T_{2k+1}\leq s\leq t}B_{s}=2c$.

By the well-known property of the reflected Brownian motion, $T_{m+1}-T_{m}$
is finite, and so is $T_{m}$ almost surely (a.s.) for each $m\geq 0$. Then, we
define $(U_{t})_{t\geq 0}$ as follows:

  * •

$U_{t}=0$ for $0\leq t\leq T_{0}$.

  * •

For all $k\geq 0$ and $T_{2k}<t\leq T_{2k+1}$, $U_{t}=\sup_{T_{2k}\leq s\leq
t}B_{s}-c$.

  * •

For all $k\geq 0$ and $T_{2k+1}<t\leq T_{2k+2}$, $U_{t}=\inf_{T_{2k+1}\leq
s\leq t}B_{s}+c$.

Informally, $U$ is pushed up by $B-c$ in the intervals of time
$[T_{2k},T_{2k+1}]$ and pushed down by $B+c$ in the intervals of time
$[T_{2k+1},T_{2k+2}]$. Similarly, if $B$ reaches $-c$ before $c$ (event
$A^{c}$), then we define:

  * •

$T_{0}$ as the infimum of $t\geq 0$ such that $B_{t}=-c$.

  * •

For all $k\geq 0$, $T_{2k+1}$ is the first time $t\geq T_{2k}$ such that
$B_{t}-\inf_{T_{2k}\leq s\leq t}B_{s}=2c$.

  * •

For all $k\geq 0$, $T_{2k+2}$ is the first time $t\geq T_{2k+1}$ such that
$B_{t}-\sup_{T_{2k+1}\leq s\leq t}B_{s}=-2c$.

Then,

  * •

$U_{t}=0$ for $0\leq t\leq T_{0}$.

  * •

For all $k\geq 0$ and $T_{2k}<t\leq T_{2k+1}$, $U_{t}=\inf_{T_{2k}\leq s\leq
t}B_{s}+c$.

  * •

For all $k\geq 0$ and $T_{2k+1}<t\leq T_{2k+2}$, $U_{t}=\sup_{T_{2k+1}\leq
s\leq t}B_{s}-c$.

######  Proposition 1.

The process $U:=(U_{t})_{t\geq 0}$ is the only continuous process with finite
variation satisfying the following properties

  1. (a)

$U_{0}=0$,

  2. (b)

$B_{t}-c\leq U_{t}\leq B_{t}+c$ for all $t\geq 0$,

  3. (c)

$U$ is nondecreasing in all intervals where it remains strictly smaller than
$B+c$,

  4. (d)

$U$ is nonincreasing in all intervals where it remains strictly larger than
$B-c$.

Notice that the last two properties imply that $U$ is constant on intervals
where $B-c<U<B+c$.

###### Proof.

The process $U$ is clearly continuous on $\mathbb{R}\backslash(T_{m})_{m\geq
0}$. A careful look at the definition above shows that $U$ has no
discontinuity at times $(T_{m})_{m\geq 0}$. Indeed, on $A$,

(2.2) $\displaystyle\begin{split}\lim_{t\uparrow
T_{2k+1}}U_{t}&=\lim_{t\uparrow T_{2k+1}}\left(\sup_{T_{2k}\leq s\leq
t}B_{s}-B_{t}\right)+\lim_{t\uparrow
T_{2k+1}}B_{t}-c=2c+B_{T_{2k+1}}-c=U_{T_{2k+1}},\\\ \lim_{t\uparrow
T_{2k+2}}U_{t}&=\lim_{t\uparrow T_{2k+2}}\left(\inf_{T_{2k+1}\leq s\leq
t}B_{s}-B_{t}\right)+\lim_{t\uparrow
T_{2k+2}}B_{t}+c=-2c+B_{T_{2k+2}}+c=U_{T_{2k+2}},\end{split}$

where the second equalities hold because the supremum and infimum are attained
at the hitting times $T_{2k+1}$ and $T_{2k+2}$ of reflected Brownian motion.
The continuity holds in the same way on $A^{c}$ as well.

Since $U$ is monotone on each interval of the form $[T_{m},T_{m+1}]$ for
$m\geq 0$, it has finite variation.

We now show properties (a)-(d) focusing on the case $A$. The proof for the
other case is similar. We have

  * (i)

$U$ is equal to zero on $[0,T_{0}]$.

  * (ii)

For all $k\geq 0$, $U$ is nondecreasing in $[T_{2k},T_{2k+1}]$ and
nonincreasing in $[T_{2k+1},T_{2k+2}]$.

  * (iii)

For all $k\geq 0$, $U$ is equal to $B-c$ at time $T_{2k}$ and to $B+c$ at time
$T_{2k+1}$.

Indeed, (i) holds by the definition of $U$ and (iii) is clear from the
definitions of the stopping times $(T_{m})_{m\geq 0}$ (see also (2.2)). (ii)
holds by the definition of $U$ in terms of a piecewise supremum and infimum
process, which is piecewise monotone. Now (a) holds by (i) and (b) holds by
(ii) and (iii).

It remains to show (c) and (d); we only provide proof for (d) as that for (c)
is similar. Let $J:=\\{t\geq 0:U_{t}>B_{t}-c\\}$. By (iii),
$J\subset[0,\infty)\backslash(T_{2k})_{k\geq 0}$, which is the union of
$[0,T_{0})$ and $(T_{2k},T_{2k+2}),k\geq 0$. We know from (i) that $U$ is
constant on $[0,T_{0}]$, and from (ii), that $U$ is nonincreasing in
$[T_{2k+1},T_{2k+2})$. It is then enough to consider intervals

$I\subset J\cap\left(\bigcup_{k\geq 0}(T_{2k},T_{2k+1}]\right)$

a sub-interval of $(T_{2k},T_{2k+1}]_{k\geq 0}$ at which $U_{t}>B_{t}-c$. For
$t\in I$, for some $k\geq 0$, $U_{t}=\sup_{T_{2k}\leq s\leq t}B_{s}-c$ and
$U_{t}>B_{t}-c$, thus, the supremum of $B$ on $[T_{2k},t]$ is not reached at
$t$. This implies that $\sup_{T_{2k}\leq s\leq t}B_{s}$ remains constant for
$t\in I$, and thus $U$ is constant, and then nonincreasing, in $I$, completing
the proof of (d).

It remains to prove that the properties stated in Proposition 1 uniquely
determine $U$.

Let $U^{1}$ and $U^{2}$ be two distinct processes satisfying the conditions
(a)-(d), and let

(2.3) $\displaystyle S:=\inf\\{t\geq 0:U^{1}_{t}\neq U^{2}_{t}\\}.$

We assume $S<\infty$ and derive contradiction.

Necessarily, by (a), (c) and (d), $U^{1}=U^{2}=0$ until the first time where
$B$ reaches $\\{-c,c\\}$ which is strictly positive, and thus $S>0$. By the
definition of $S$ as in (2.3), $U^{1}_{t}=U^{2}_{t}$ for all $t\in[0,S)$, and
by the continuity of $U^{1}$ and $U^{2}$, we also have
$U^{1}_{S}=U^{2}_{S}=:U_{S}$.

Suppose

(2.4) $\displaystyle U^{1}_{S}-B_{S}=U^{2}_{S}-B_{S}=U_{S}-B_{S}\geq 0.$

By this, continuity of $B$ and $U$ and (b), we have $B_{t}-c<U^{j}_{t}\leq
B_{t}+c$ for $j=1,2$ on the interval $[S,S+u]$ for sufficiently small $u>0$.
For $j\in\\{1,2\\}$, let us define the following processes on $[S,S+u]$:

$\displaystyle A_{t}^{j}$ $\displaystyle:=U_{S}-U^{j}_{t},$ $\displaystyle
Y_{t}$ $\displaystyle:=B_{t}+c-U_{S},$ $\displaystyle Z_{t}^{j}$
$\displaystyle:=Y_{t}+A_{t}^{j}=B_{t}+c-U^{j}_{t}.$

By (b), $Y_{S}\geq 0$ and $Z_{t}^{j}\geq 0$, and $A_{t}^{j}$ is continuous and
vanishes at the starting time $S$.

Moreover, the variation of $A^{j}$ is carried by the set of times where
$Z^{j}=0$. Indeed, $A^{j}$ changes only when $U^{j}$ changes, or equivalently,
only when $U^{j}$ is equal to $B-c$ or $B+c$, and then only when $U^{j}=B+c$
since we know that $U^{j}>B-c$ on the interval $[S,S+u]$. From this last
property and (d), $U^{j}$ is nonincreasing in $[S,S+u]$, which implies that
$A^{j}$ is nondecreasing in this interval. By Skorokhod’s lemma (see Lemma
2.1, Chapter VI of [6]; Lemma 3.1.2, Chapter 3 of [8]), $A^{j}$ is uniquely
determined by $Y$ on the interval $[S,S+u]$, and thus $A^{1}=A^{2}$, which
implies that $U^{1}=U^{2}$ on the interval $[S,S+u]$. This contradicts the
fact that $S$ is the infimum of times where $U^{1}\neq U^{2}$ as in (2.3).
This contradiction shows that $S=\infty$, i.e. one cannot have two distinct
processes satisfying the assumptions of Proposition 1. Similar argument holds
for the complement case of (2.4).

∎

Now, let $W=(W_{t})_{t\geq 0}$ be another standard Brownian motion, and let
$F:\mathbb{R}\to[-c,c]$ be the triangle wave function with period $4c$ such
that

$F(x)=\left\\{\begin{array}[]{ll}\cdots&\cdots\\\ -2c-x&\textrm{for
}x\in[-3c,-c]\\\ x&\textrm{for }x\in[-c,c]\\\ 2c-x&\textrm{for }x\in[c,3c]\\\
\cdots&\cdots\end{array}\right.$

The derivative of $F$ is $1$ in the interval $((4k-1)c,(4k+1)c)$ and $-1$ in
the interval $((4k+1)c,4k+3)c)$, for all $k\in\mathbb{Z}$. The second
derivative of $F$ in the sense of the distribution is

$2\sum_{k\in\mathbb{Z}}(\delta_{(4k-1)c}-\delta_{(4k+1)c})$

where $\delta_{x}$ denotes the Dirac measure at $x$. By Itô-Tanaka formula,

$F(W_{t})=\beta_{t}-V_{t},\quad t\geq 0,$

where $(\beta_{t})_{t\geq 0}$ is a standard Brownian motion,

$V_{t}:=\sum_{k\in\mathbb{Z}}(L^{(4k+1)c}_{t}-L^{(4k-1)c}_{t}),\quad t\geq 0,$

with $L^{x}_{t}$ denoting the local time of $W$ at time $t$ and level
$x\in\mathbb{R}$.

######  Remark 1.

The processes $(\beta_{t})_{t\geq 0}$ and $(V_{t})_{t\geq 0}$ are continuous
and enjoy the following properties:

  1. (a)

$V_{0}=0$.

  2. (b)

$(V_{t})_{t\geq 0}$ has paths of finite variation, since it is the difference
of two nondecreasing processes, given by sums of local times at different
levels.

  3. (c)

For all $t\geq 0$, $V_{t}-\beta_{t}=-F(W_{t})\in[-c,c]$, so $\beta_{t}-c\leq
V_{t}\leq\beta_{t}+c$.

  4. (d)

In an interval
$K:=\\{t:V_{t}<\beta_{t}+c\\}=[0,\infty)\backslash\\{t:F(W_{t})=-c\\}$, $W$
does not reach levels congruent to $-c$ modulo $4c$ (these levels
corresponding to values of $x\in\mathbb{R}$ such that $F(x)=-c$). Local times
at levels $(4k-1)c$ for $k\in\mathbb{Z}$ are constant on $K$, which implies,
from the definition of $V$, that $V$ is nondecreasing on $K$.

  5. (e)

In an interval
$K^{\prime}:=\\{t:V_{t}>\beta_{t}-c\\}=[0,\infty)\backslash\\{t:F(W_{t})=c\\}$,
$W$ does not reach levels congruent to $c$ modulo $4c$. Local times at these
levels are constant on $K^{\prime}$, which implies that $V$ is nonincreasing
on $K^{\prime}$.

From Proposition 1, and the corresponding uniqueness property, the process
$(V_{t})_{t\geq 0}$ can be recovered from $(\beta_{t})_{t\geq 0}$ in the same
way as $(U_{t})_{t\geq 0}$ has been constructed from $(B_{t})_{t\geq 0}$.
Since $(B_{t})_{t\geq 0}$ and $(\beta_{t})_{t\geq 0}$ are both Brownian
motions, $(B_{t},U_{t})_{t\geq 0}$ has the same joint distribution as
$(\beta_{t},V_{t})_{t\geq 0}$.

We provide two descriptions of the processes discussed here using different
time changes, the inverse local times and the hitting times.

### Time change with inverse local times

We define the following linear combination of local times:

$\left(\mathcal{L}_{t}:=\sum_{k\in\mathbb{Z}}L^{(2k+1)c}_{t}\right)_{t\geq
0},$

and its inverse process $(\sigma_{\ell})_{\ell\geq 0}$ given by

$\sigma_{\ell}:=\inf\\{t\geq 0,\mathcal{L}_{t}>\ell\\}.$

Note that $\mathcal{L}_{t}\nearrow\infty$ as $t\to\infty$, thus
$\sigma_{\ell}<\infty$ for all $\ell\geq 0$ a.s.

Since $\mathcal{L}$ is an additive functional of the Brownian motion $W$ (see
[6], Chapter X), by the strong Markov property, the time-changed Brownian
motion $(W_{\sigma_{\ell}})_{\ell\geq 0}$ is a continuous-time Markov chain.
Moreover, since the variation of $(\mathcal{L}_{t})_{t\geq 0}$ is supported on
the set of times where $W$ is in $E:=\\{(2k+1)c,k\in\mathbb{Z}\\}$, we have
that the Markov chain $(W_{\sigma_{\ell}})_{\ell\geq 0}$ takes its value only
in $E$. This process can only jump between consecutive values in $E$. Indeed,
if $(W_{\sigma_{\ell}})_{\ell\geq 0}$ jumps between non-consecutive values $x$
and $y$ in $E$ at some $\ell\geq 0$, then the path
$(W_{t})_{\sigma_{\ell-}\leq t\leq\sigma_{\ell}}$ starts at $x$, ends at $y$,
without accumulating local time at levels in $E$ strictly between $x$ and $y$:
this event occurs with probability zero. Similarly, the starting point
$W_{\sigma_{0}}$ can only be $c$ or $-c$. Thus, the following results are
immediate.

######  Proposition 2.

The process $(W_{\sigma_{\ell}})_{\ell\geq 0}$ satisfies the following
properties:

  1. (a)

It starts at $c$ with probability $1/2$, and at $-c$ with probability $1/2$.

  2. (b)

It jumps by $2c$ or $-2c$, the rate of all possible jumps being the same,
again by symmetry.

Now, let us compute the rate of jumps. By translation invariance and the
strong Markov property, the probability of having no jump during an interval
of length $\ell$ is equal, for a Brownian motion $B$, its local time $L^{B}$
at level zero and its inverse local time $\tau_{\ell}:=\inf\\{t\geq
0:L^{B}>\ell\\}$, $\ell\geq 0$ :

$\displaystyle\mathbb{P}[\max\\{|B_{s}|,s\leq\tau_{\ell}\\}<2c]$
$\displaystyle=\left(\mathbb{P}[\max\\{B_{s},s\leq\tau_{\ell}\\}<2c]\right)^{2}\
\text{since the negative and positive excursions of $B$ are independent}$
$\displaystyle=\left(\mathbb{P}[L^{B}_{\inf\\{t\geq
0,B_{t}=2c\\}}>\ell]\right)^{2}=\left(\mathbb{P}[\text{BESQ}_{2}(2c)>\ell]\right)^{2}\
\text{by Ray-Knight theorem}$

where $\text{BESQ}_{2}$ denotes a squared Bessel process of dimension $2$.
Hence, for a standard exponential variable ${\bf e}$,

$\displaystyle\mathbb{P}[\max\\{|B_{s}|,s\leq\tau_{\ell}\\}<2c]$
$\displaystyle=\left(\mathbb{P}[2\times{\bf e}\times 2c>\ell]\right)^{2}$
$\displaystyle=\left(e^{-\ell/4c}\right)^{2}=e^{-\ell/2c}.$

Hence, the sum of the rates of jump by $2c$ and $-2c$ is equal to $1/2c$, and
then each of the two possible jumps has a rate $1/4c$ by symmetry.

We have

$V_{\sigma_{\ell}}=\sum_{k\in\mathbb{Z}}(L_{\sigma_{\ell}}^{(4k+1)c}-L_{\sigma_{\ell}}^{(4k-1)c})$

and

$\mathcal{L}_{\sigma_{\ell}}=\sum_{k\in\mathbb{Z}}(L_{\sigma_{\ell}}^{(4k+1)c}+L_{\sigma_{\ell}}^{(4k-1)c}).$

In an interval of values of $\ell$ where $W_{\sigma_{\ell}}$ remains constant
and equal to $(4k-1)c$ for a given $k\in\mathbb{Z}$, the variations of
$V_{\sigma_{\ell}}$ are opposite to the variations of
$\mathcal{L}_{\sigma_{\ell}}=\ell$, since the only local time which varies in
the corresponding sums is the local time at level $(4k-1)c$. Similarly, in an
interval of values of $\ell$ where $W_{\sigma_{\ell}}$ remains constant and
equal to $(4k+1)c$ for a given $k\in\mathbb{Z}$, the variations of
$V_{\sigma_{\ell}}$ are equal to the variations of $\ell$. Hence,
$(V_{\sigma_{\ell}})_{\ell\geq 0}$ is the difference of time spent by the
Markov process $(W_{\sigma_{\ell}})_{\ell\geq 0}$ at levels congruent to $c$
modulo $4c$, minus the time spent at levels congruent to $-c$ modulo $4c$. If
we reduce $(W_{\sigma_{\ell}})_{\ell\geq 0}$ modulo $4c$ and divide it by $c$,
we get a Markov chain, $(Y_{\ell})_{\ell\geq 0}$, on the two-state space
$\\{-1,1\\}$, with its starting point uniform in this space, and rate of jumps
equal to $1/2c$. Then, $(V_{\sigma_{\ell}})_{\ell\geq 0}$ is the difference of
time spent at state $1$ by this Markov process, minus the time spent at state
$-1$, i.e. the integral of the Markov chain:
$V_{\sigma_{\ell}}=\int_{0}^{\ell}(1_{\\{Y_{s}=1\\}}-1_{\\{Y_{s}=-1\\}})ds$.

### Time change with hitting times

Let us define the sequence of stopping times $(H_{k})_{k\geq 0}$, as follows:

  * •

If $W$ hits $c$ before $-c$ (we call this event $\mathcal{A}$), $H_{0}=H_{1}$
is the first hitting time of $c$, and for $k\geq 1$,

$H_{k+1}=\inf\\{t>H_{k},|W_{t}-W_{H_{k}}|\geq 2c\\}.$

  * •

If $W$ hits $-c$ before $c$ (or $\mathcal{A}^{c}$), $H_{0}$ is the first
hitting time of $-c$, and for $k\geq 0$,

$H_{k+1}=\inf\\{t>H_{k},|W_{t}-W_{H_{k}}|\geq 2c\\}.$

Between $H_{0}$ and $H_{1}$, $V$ varies as the opposite of the local time of
$W$ at $-c$; for $k\geq 1$, between $H_{2k-1}$ and $H_{2k}$, $V$ varies as the
local time of $W$ at $W_{H_{2k-1}}$; for $k\geq 1$, between $H_{2k}$ and
$H_{2k+1}$, it varies as the opposite of the local time of $W$ at
$W_{H_{2k}}$. We deduce, for all $m\geq 1$,

$V_{H_{m}}=\sum_{j=0}^{m-1}(-1)^{j-1}\left(L^{W_{H_{j}}}_{H_{j+1}}-L^{W_{H_{j}}}_{H_{j}}\right),$

where we recall $L^{x}_{t}$ denoting the local time of $W$ at time $t$ and
level $x$. Notice that the term $j=0$ vanishes on $\mathcal{A}$, since
$H_{0}=H_{1}$ in this case.

All these increments of local time, except the first one if $H_{0}=H_{1}$ in
the event $\mathcal{A}$, are independent and have the distribution of the
local time $L^{B}_{T^{*}_{2c}}$ of $B$ at level zero, where $B$ is a Brownian
motion and

$T^{*}_{2c}=\inf\\{t>0,|B_{t}|\geq 2c\\}.$

Now, we show that using this approach, one can compute this distribution via
Ray-Knight Theorem: for $\ell\geq 0$, $L^{B}_{T^{*}_{2c}}>\ell$ if and only if
no excursion of $B$ before the inverse local time $\tau_{\ell}$ reaches $2c$
or $-2c$, which implies

$\displaystyle\mathbb{P}(L^{B}_{T^{*}_{2c}}>\ell)=\mathbb{P}(\text{no
excursion of $B$ reaches }\pm 2c\text{ before }\tau_{\ell})$
$\displaystyle=\left(\mathbb{P}(\text{no excursion of $B$ reaches }2c\text{
before }\tau_{\ell})\right)^{2}\ \text{by independence of excursions}$
$\displaystyle=\left(\mathbb{P}(\inf\\{t\geq
0,B_{t}=2c\\}>\tau_{\ell})\right)^{2}$
$\displaystyle=\left(\mathbb{P}(L^{B}_{\inf\\{t\geq
0,B_{t}=2c\\}}>\ell)\right)^{2}$ $\displaystyle=\left(\mathbb{P}(4c{\bf
e}>\ell)\right)^{2}\text{ by Ray-Knight Theorem }$
$\displaystyle=e^{-\ell/2c}.$

Hence, $L^{B}_{T^{*}_{2c}}$ has the same distribution as $2c{\bf e}$. We
deduce the equality in distribution, for $m\geq 2$,

$V_{H_{m}}=2c\left(-X{\bf e}_{0}+\sum_{j=1}^{m-1}(-1)^{j-1}{\bf
e}_{j}\right),$

where $({\bf e}_{j})_{j\geq 0}$ is a sequence of i.i.d. standard exponential
variables, and $X$ is an independent, Bernoulli random variable with parameter
$1/2$: $X=0$ if and only if $W$ hits $c$ before $-c$ (i.e. $\mathcal{A}$).

For $k\geq 1$, $V_{H_{2k+1}}$ is a sum of $-2cX{\bf e}_{0}$ and of $k$ i.i.d.
variables with the same distribution as $2c({\bf e}_{1}-{\bf e}_{2})$. These
$k$ variables are centered, with variance equal to $8c^{2}$, since ${\bf
e}_{1}$ and ${\bf e}_{2}$ are independent with variance $1$. We get the
following central limit theorem:

######  Proposition 3.

When $k$ tends to infinity,

$\frac{V_{H_{2k+1}}}{c\sqrt{8k}}\longrightarrow\mathcal{N}(0,1)$

in distribution.

The increment $H_{2k+1}-H_{1}$ is itself the sum of $2k$ i.i.d. random
variables with the same distribution as $T^{*}_{2c}$. By the law of large
numbers, we have a.s.

$\frac{H_{2k+1}}{2k}\underset{k\rightarrow\infty}{\longrightarrow}\mathbb{E}[T^{*}_{2c}].$

Stopping the martingale $(B^{2}_{t}-t)_{t\geq 0}$ at time $T^{*}_{2c}$, we
deduce for all $t\geq 0$, by optional sampling, that

$\mathbb{E}[B^{2}_{\min(t,T^{*}_{2c})}]=\mathbb{E}[\min(t,T^{*}_{2c})].$

For $t\rightarrow\infty$, $B^{2}_{\min(t,T^{*}_{2c})}\leq 4c^{2}$ almost
surely converges to $4c^{2}$, and $\min(t,T^{*}_{2c})$ converges to
$T^{*}_{2c}$ from below. Applying dominated convergence to the left-hand side
and monotone convergence to the right-hand side, we deduce

$4c^{2}=\mathbb{E}[T^{*}_{2c}],$

and then almost surely

(2.5)
$\displaystyle\frac{H_{2k+1}}{2k}\underset{k\rightarrow\infty}{\longrightarrow}4c^{2}.$

We get

$\frac{V_{H_{2k+1}}}{\sqrt{H_{2k+1}}}=\frac{V_{H_{2k+1}}}{c\sqrt{8k}}\cdot
2c\left(\frac{H_{2k+1}}{2k}\right)^{-1/2}$

where the first factor converges to a standard normal distribution by
Proposition 3 and the second factor converges almost surely to $1$ by (2.5).
By Slutsky’s thorem, we get the following:

######  Proposition 4.

When $k$ tends to infinity,

$\frac{V_{H_{2k+1}}}{\sqrt{H_{2k+1}}}\longrightarrow\mathcal{N}(0,1)$

in distribution.

This result can be compared to the central limit theorem

$\frac{V_{t}}{\sqrt{t}}\underset{t\rightarrow\infty}{\longrightarrow}\mathcal{N}(0,1),$

which is a direct consequence of the inequality

$\beta_{t}-c\leq V_{t}\leq\beta_{t}+c,$

where $(\beta_{t})_{t\geq 0}$ is a Brownian motion.

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