# Sojourn times of Markov symmetric processes in continuous time.

Pechersky E.A.1, Presman E.L.2, Yambartsev A.A.3 111AY thanks FAPESP for the
finantial support via the grant 2017/10555-0.

###### Abstract

The symmetric birth and death process in the integers $\\{1,\ldots,N\\}$ with
linear rates is studied. The process moves slowly and spends more time in the
neighborhood of the state 1. It represents our attempt at explaining the
asymmetry between amounts of matter and antimatter by the inhomogeneity of the
sojourn time. The state of the process reflects a relative frequency of an
antimatter amount. The observed matter-antimatter imbalance in the Universe we
consider a result of stochastic competition between them. The amount of matter
significantly exceeds the amount of antimatter, which corresponds to the lower
states of the process, and the path toward matter-antimatter equilibrium can
be very long.

1Dobrushin´s Laboratory, Institute of Information Transmission Problems of
Russian Academy of Science, Moscow, Russia.
E-mail<EMAIL_ADDRESS>
2Central Economics and Mathematics Institue of Russian Academy of Sciences
(CEMI RAS), Moscow, Russia.
E-mail<EMAIL_ADDRESS>
3Department of Statistics, Institute of Mathematics and Statistics, University
of São Paulo, São Paulo, Brazil.
E-mail<EMAIL_ADDRESS>
Key words: birth-death process; large deviations; rate function; matter-
antimatter imbalance

MSC2020: Primary 60F10; secondary 60J27; 60K40

In memory of Nikita D. Vvedenskaya

##  1 Introduction

###  1.1 Markov process

Markov Process. We study a continuous time Markov process $X^{N}(t)$ with the
state space $\mathcal{N}=\\{1,...,N\\}$ and piecewise constant paths. We
consider two kinds of the initial state, it is either a point from
$\mathcal{N}$ or the stationary distribution (2.3) on $\mathcal{N}$. The jumps
of the process with non-zero transition rates are $k\to k+1$ and $k\to k-1$
with the rates $\lambda_{k,k+1},\lambda_{k,k-1}$ if $1<k<N$, and $1\to 2,N\to
N-1$ with the rates $\lambda_{1,2}=1,\lambda_{N,N-1}=N$. The dynamics of the
process is described by the site-symmetry generator with rates
$\lambda_{k,k+1}=\lambda_{k,k-1}=k$ for $k\neq 1,N$, thus, in this case we
have the edge-asymmetric, $\lambda_{k,k+1}\neq\lambda_{k+1,k}$, see (2.2). It
means that the considered processes $X^{N}$ are space non-homogeneous.

Invariant measures. Note that the corresponding embedded discrete-time Markov
chain $\widehat{X}^{N}$ is a symmetric simple random walk over $\mathcal{N}$
with reflections at the boundaries $1$ and $N$ (see Section 2.2). However, the
continuous-time process $X^{N}$ is far from any symmetry. As will be shown,
despite the property of symmetry $\lambda_{k,k+1}=\lambda_{k,k-1}$ the process
$X^{N}$ spends most of its time in the vicinity of the point $0$. That is, the
distribution of $X^{N}(t)$ is concentrated in the neighborhood of $0$, while
the embedded process $\widehat{X}^{N}$ has a uniform distribution over the set
$\mathcal{N}\setminus\\{1,N\\}$. Both $X^{N}$ and $\widehat{X}^{N}$ satisfy
detailed balance with respect to their stationary distribution.

Law of Large Numbers. We study a sequence $\xi^{N}$ of the Markov processes
obtained by the scaling by $N$ of the original processes $X^{N}$, that is
$\xi^{N}:=\frac{X^{N}}{N}\in{\mathfrak{N}}^{N}=\\{\frac{1}{N},\frac{2}{N},...,1\\}$.
We show that if the initial state is one-pointed then the mean value of
$\xi^{N}$ tends to the constant path equal to the limiting initial value
$\xi^{N}(0)$ with growing $N$. And if the initial state is sampled according
to the stationary distribution then the limit of the stationary distribution
of $\xi^{N}$ shows the effect of the concentration near the path identically
equal to 0.

Large Deviations. Finally, the rate function of the large deviations is
obtained by the method described in [9]. For any given $x_{0},x_{T}\in[0,1]$
we find the optimal path $\gamma(t),t\in[0,T]$ starting at $\gamma(0)=x_{0}$
and taking value $x_{T}$ at the end of time interval $\gamma(T)=x_{T}$. We
show that the optimal path is a parabola.

###  1.2 Motivations

The vast majority of laws in physics are related to a certain symmetry. Often,
symmetries imply the existence of conservation laws, such as the well-known
conservation laws in mechanics and the conservation of the electric charge.

However, the most impressive discoveries are associated with asymmetric
manifestations in nature. Many asymmetries have been found in the microcosm
that obeys quantum laws.

One well-known phenomenon is the asymmetry between matter and antimatter or
the matter-antimatter imbalance. It is an impressive problem that has not yet
been solved. The attempt to find an answer to this problem is based on the
search for a violation of some symmetry in physical law, such as the violation
of the law of conservation of baryon charges (see, for example, [1], [2],
[16], [17]). According to [3] this asymmetry is a reason for the matter-
antimatter imbalance.

There is another reason for the asymmetry: randomness inherent in the system.
Randomness can cause fluctuations, often quite large.

Very often, the asymmetry of physical processes caused by the random behavior
of the system expresses itself through a phenomenon of phase transition. The
phase transition is associated with very different behaviors of the same
physical system on opposite sides of a “critical point” of some system
parameter. Usually, a phase transition is related to the randomness created by
fluctuations in the system or the system environment.

Here we propose another reason for the asymmetry. It is based in fluctuations
modeled as a spatially inhomogeneous continuous-time Markov process. The
asymmetry arises due to distinctions in the occupation time spent by the
Markov process in different states.

The state-space of the process under consideration is a finite set
$\mathcal{N}=\\{1,...,N\\}$, and the process dynamics creates piecewise
constant trajectories with jumps $+1$ or $-1$. The transition probabilities to
the left and right neighbors from any state $2<i<N-1$ are equal, but the
transition rate linearly increases with the growth of $i$. The process
exhibits the asymmetry related to the sojourn time. In the area close to
$1\in\mathcal{N}$ the process spends much more time than in the states near
the $N$. Assuming the process describes the particle dynamics, it means that
if a particle enters a region close to $1\in\mathcal{N}$ due to the
fluctuations, it takes a long time before leaving this region.

This work is the first step toward explaining the asymmetry between matter and
antimatter due to the fluctuations created by a stochastic symmetric
competition between matter and antimatter. We assume that violations of
baryon-antibaryon symmetry arise in the course of Markovian dynamics. The
breaking of symmetry occurs when the process path is constant during some
time. The longer the constant period, the higher the probability of violation.

This effect can be sharper if we consider Markov dynamics out of the average
in the corresponding direction. Such an event may have a small probability.
However, it may strengthen the studied effect. This study belongs to the area
of theory probability called large deviations theory.

Another motivation comes from our study of large deviations for birth-death
processes with polynomial transition rates. In the series of papers, [15],
[14], [13], [12], we consider the birth-death processes where the transition
rates $\lambda_{i,i+1}$ and $\lambda_{i,i-1}$ depends on the state $i$
polynomially: $\lambda_{i,i+1}=Pi^{\alpha}$ and $\lambda_{i,i+1}=Qi^{\beta}$.
The papers [14], [13], [12] explore the method of exchange of measures
suggested in [15] in the cases when $\alpha\neq\beta$ or $\alpha=\beta,P\neq
Q$. But all our attempts have failed to extend the method for the case
$\alpha=\beta,P=Q$.

The model considered here corresponds to the birth-death processes (on the
interval $0,1,\ldots,N$) with $\alpha=\beta=1,P=Q$, where the method of [9]
was employed.

##  2 Model. Generator

The main subject of our research is the sequence (in $N$, as $N\to\infty$) of
Markov processes $(X^{N}(t),t\in[0,T])$ for some fixed $T>0$. The upper index
means that the states of the process belong to the set
$\mathcal{N}=\\{1,\ldots,N\\}$. The stochastic dynamics of $X^{N}(t)$ is
controlled by the generators $G^{N}$, which is an operator on the set of probe
functions $\mathcal{F}^{N}=\\{f:\>\mathcal{N}\to{\mathbb{R}}\\}$:

$G^{N}f(m)=\lambda m\bigl{(}f(m+1)-f(m)\bigr{)}{\mathbbm{1}}_{m<N}(m)+\lambda
m\bigl{(}f(m-1)-f(m)\bigr{)}{\mathbbm{1}}_{m>1}(m),$ (2.1)

where ${\mathbbm{1}}_{\mathcal{B}}(\cdot)$ is an indicator for subset
$\mathcal{B}\subset\mathcal{N}$

${\mathbbm{1}}_{\mathcal{B}}(m)=\begin{cases}1,\mbox{if }m\in\mathcal{B},\\\
0,\mbox{otherwise},\end{cases}$

and the rates $\lambda_{m,m+1}$ and $\lambda_{m,m-1}$ are

$\displaystyle\lambda_{m,m+1}=\lambda_{m,m-1}=\lambda m,\ m=2,\ldots,N-1,$
(2.2) $\displaystyle\lambda_{1,2}=\lambda,\ \lambda_{N,N-1}={\lambda}{N}.$

Note that the endpoints of $\mathcal{N}$ are reflective. The initial state
$X^{N}(0)$ has some distribution on $\mathcal{N}$. In particular, it can be a
measure concentrated at the one point state $m\in\mathcal{N}$.

######  Remark 2.1.

It can be seen from the form of $G^{N}$ that the Markov process has jump-wise
paths. The jumps are equal to $\pm 1$. The intensities of jumps from the state
$m$ depend on the value of $m$, and the intensities to the left and right are
equal if $m\in\\{2,...,N-1\\}$. Thus, we have symmetrical jumps except for the
endpoints of $\mathcal{N}$.

###  2.1 Stationary distribution

Detailed balance allows finding stationary distributions of the processes. The
stationary distribution concentrates near 0 as $N$ increases.

Denote by ${\pi^{N}(m),m\in\mathcal{N}}$, the stationary measure of
$X^{N}(t)$.

######  Theorem 2.2.

$X^{N}(t)$ satisfies the detailed balance with respect to the stationary
distribution. $\pi^{N}$ is given by

$\pi^{N}(m)=\frac{\frac{1}{m}}{\sum_{k=1}^{N}\frac{1}{k}},\ \
m\in\mathcal{N}.$ (2.3)

Let ${\mathcal{M}}=\\{1,...,M\\}\subseteq\mathcal{N}$. Let $h(N)$ be a
sequence of integers such that

$\frac{\ln(h(N))}{\ln(N)}\to 1,\mbox{ and
}\limsup_{N\to\infty}\frac{N}{h(N)}>1.$

Assume that $M=h(N)$. Then

$\pi^{N}(\mathcal{M})\to 1,\mbox{ as }N\to\infty,$

###### Proof.

The proof uses the detailed balance. The balance equations have the following
shape (see (2.1))

$\pi^{N}(m)m=\pi^{N}(m+1)(m+1),\mbox{ for all }1\leq m\leq N-1.$

For $m>1$ we obtained

$\pi^{N}(m)=\pi^{N}(1)\frac{1}{m}.$

Since $\sum_{m=1}^{N}\pi^{N}(m)=1$ we have

$\pi^{N}(1)=\frac{1}{\sum_{m=1}^{N}\frac{1}{m}},$

and

$\pi^{N}(m)=\frac{\frac{1}{m}}{\sum_{m=1}^{N}\frac{1}{m}},\ \
m\in\mathcal{N}.$ (2.4)

The probability of the set $\mathcal{M}=\\{1,...,M\\}$ is

$\pi^{N}(\mathcal{M})=\sum_{m=1}^{M}\pi^{N}(m)=\frac{\sum_{m=1}^{M}\frac{1}{m}}{\sum_{m=1}^{N}\frac{1}{m}}.$
(2.5)

Here we use the Euler representation of the partial sum of the Harmonic series
$H_{K}=\sum_{m=1}^{K}\frac{1}{m}=\ln K+\gamma+\varepsilon_{K}$, where $\gamma$
is the Euler–Mascheroni constant and $K\varepsilon_{K}\to\frac{1}{2}$ as
$K\to\infty$. Then

$\pi^{N}(\mathcal{M})=\frac{\ln M+\gamma+\varepsilon_{M}}{\ln
N+\gamma+\varepsilon_{N}}=\frac{\ln h(N)+\gamma+\varepsilon_{M}}{\ln
N+\gamma+\varepsilon_{N}}\to 1$ (2.6)

as $N\to\infty$ $\blacksquare$

###  2.2 Embedded chain

The stationary distribution of the embedded Markov chain $\widehat{X}$ is
fundamentally different from the stationary distribution of the process
$X^{N}$.

Consider the embedded chain $\widehat{X}^{N}(k)$ corresponding to $X^{N}(t)$
with initial value $\widehat{X}^{N}(0)=m$. The chain $\widehat{X}^{N}(k)$
instantly changes its state immediately upon the occurence of jumps
$X^{N}(t)$, regardless of the time the process spends in the state before the
jump. The discrete-time dynamics is determined by the following probability
transition matrix

$\begin{pmatrix}0&\displaystyle{\frac{\lambda_{1,2}}{R_{1}}}&0&0&0&...&0&0\\\
\displaystyle\frac{\lambda_{2,1}}{R_{2}}&0&\displaystyle\frac{\lambda_{2,3}}{R_{2}}&0&0&...&0&0\\\
0&\displaystyle\frac{\lambda_{3,2}}{R_{3}}&0&\displaystyle\frac{\lambda_{3,4}}{R_{3}}&0&...&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&0&0&...&0&\displaystyle\frac{\lambda_{N-1,N}}{R_{N-1}}\\\
0&0&0&0&0&...&\displaystyle\frac{\lambda_{N,N-1}}{R_{N}}&0\end{pmatrix},$

where

$R_{m}=\begin{cases}\lambda_{m,m+1}+\lambda_{m,m-1}=2\lambda m,\mbox{ if
}m\in\\{2,...,N-1\\},\\\ \lambda_{1,2}=\lambda,\mbox{ if }m=1,\\\
\lambda_{N,N-1}=\lambda N,\mbox{ if }m=N.\end{cases}$

The value of $R_{m}$ is the rate at which $X^{N}(t)$ leaves the state $m$. For
the case $m\in\\{2,...,N-1\\}$ the transition probability is
$\displaystyle\frac{\lambda_{m,m+1}}{R_{m}}=\displaystyle\frac{\lambda_{m,m-1}}{R_{m}}=\frac{1}{2}$,
and for $m=1$ and $m=N$ the transition probabilities are
$\displaystyle{\frac{\lambda}{R_{1}}=\frac{\lambda N}{R_{N}}}=1$. The two
transition probabilities equal to 1 represent the reflection from states 1 and
$N$.

Using the detailed balance equations, we obtain the stationary distribution
$\widehat{\pi}$ of $\widehat{X}(k)$

$\widehat{\pi}(1)=\widehat{\pi}(N)=\frac{1}{2(N-1)},\
\widehat{\pi}(m)=\frac{1}{N-1},\ m=2,...,N-1.$

######  Remark 2.3.

The chain $\widehat{X}(k)$, with initial state $\widehat{X}(0)$ is a simple
random walk in $\mathcal{N}=\\{1,\ldots,N\\}$ with reflections on the
boundaries. If the current state $\widehat{X}(k-1)\in\\{2,...,N-1\\}$, then
the next state $\widehat{X}(k)=\widehat{X}(k-1)+Y_{k}$, where the independent
increments $Y_{k}$ take values $\pm 1$ uniformly
${\mathbf{P}}(Y_{k}=1)={\mathbf{P}}(Y_{k}=-1)=1/2$, and
${\mathbf{P}}(Y_{k}=1)=1\ \left({\mathbf{P}}(Y_{k}=-1)=1\right)$ on the
boundary state, when $\widehat{X}(k-1)=1\ \left(\widehat{X}(k-1)=N\right)$.

The dynamics of the process $(X^{N}(t))$ on continuous-time changes
dramatically compared to the embedded chain $\widehat{X}(k)$. The occupation
(or sojourn) time plays a crucial role in the nature of the dynamics. Our aim
is to understand the occupation time behavior on different subsets of
$\mathcal{N}$.

For any $N$ the process $X^{N}(t)$ is homogeneous in time, but non-homogeneous
in space. Thus, the sojourn time in any subset $\mathcal{B}\subset\mathcal{N}$
may vary depending on $N$. We have shown that for any $M<N$ which is not very
close to $N$ the time of the sojourn probability in
$\mathcal{M}=\\{1,\ldots,M\\}$ tends to 1 as $N,M\to\infty$ (see Theorem 2.2).

##  3 Scaling

In this section we consider the scaled version of the processes $X^{N}$:
$\xi^{N}(t)=\frac{X^{N}(t)}{N}$. The state space of $\xi^{N}$ is
${\mathfrak{N}}^{N}=\left\\{\frac{1}{N},\frac{2}{N},...,1\right\\}$. The
generator of $\xi^{N}(t)$ is the following operator (cf. (2.1)) acting on the
function space
${\mathfrak{F}}^{N}=\\{{f}:\>{\mathfrak{N}}^{N}\to{\mathbb{R}}\\}$ for all
$\gamma\in{\mathfrak{N}}^{N}\subset[0,1]$

$g^{N}f(\mathscr{\gamma})=\lambda
N\gamma\Bigl{(}f\Bigl{(}\gamma+\frac{1}{N}\Bigr{)}-f(\gamma)\Bigr{)}{\mathbbm{1}}_{\\{\gamma<1\\}}(\gamma)+\lambda
N\gamma\Bigl{(}f\Bigl{(}\gamma-\frac{1}{N}\Bigr{)}-f(\gamma)\Bigr{)}{\mathbbm{1}}_{\\{\gamma>\frac{1}{N}\\}}(\gamma).$
(3.1)

Denote $\xi^{N}_{\gamma^{N}}$ the process $\xi^{N}$ with the initial point
$\gamma^{N}\in{\mathfrak{N}}^{N}$, that is
$\xi^{N}_{\gamma^{N}}(0)=\gamma^{N}$. Let $\mathbf{D}^{N}[0,T]$ be the set of
all trajectories of $\xi^{N}$ during the time interval $[0,T]$ and let

$\mathbf{D}[0,T]=\bigcup_{N}\mathbf{D}^{N}[0,T].$

Since $\mathbf{D}^{N}[0,T]\subset\mathbf{D}[0,T]$ then all measures
$\mu^{N}_{\gamma^{N}}$ of the prcesses $\xi^{N}_{\gamma^{N}}$ are concentrated
in $\mathbf{D}[0,T]$. However, the set $\mathbf{D}[0,T]$ does not contain all
continuous functions. Let $\bar{\mathbf{D}}[0,T]$ be completion of
$\mathbf{D}[0,T]$ with respect to the uniform metric. Then
$\mathbf{C}[0,T]\subseteq\bar{\mathbf{D}}[0,T]$.

Let $(\gamma^{N})$ be the sequence of the initial points such that
$\gamma^{N}\to\gamma\in(0,1]$, for example $\gamma^{N}=\frac{\lfloor\gamma
N\rfloor}{N}$. In what follows we will show that the measures
$\mu^{N}_{\gamma^{N}}$ converge to the measure $\mu_{\gamma}$ concentrated on
the path set
$\\{\boldsymbol{\mathfrak{\gamma}}:\>\boldsymbol{\mathfrak{\gamma}}(0)=\gamma\\}$.
Further the bold letters we reserve for the functions. For example,
$\boldsymbol{\mathfrak{\gamma}}(t)=\gamma$ means that the function
$\boldsymbol{\mathfrak{\gamma}}$ at the point $t$ takes the value $\gamma$. We
hope this notation will not confuse a reader.

##  4 The Law of Large Numbers

In this section, we study the dynamics of a sequence of the processes
$\xi^{N}$ as $N$ grows. One of the purposes of this study is to determine the
evolutionary behavior of the scaled processes $\xi^{N}$ with the initial state
from ${\mathfrak{N}}=\bigcup_{N}{\mathfrak{N}^{N}}$. Recall that the process
$\xi^{N}_{\gamma^{N}}(t)$ denotes the process $\xi^{N}(t)$ with the initial
$\xi^{N}(0)=\gamma^{N}\in{\mathfrak{N}}$. The measure $\mu_{\gamma^{N}}$ which
is the distribution of $\xi^{N}_{\gamma^{N}}(t)$ is concentrated in the path
set $\mathbf{D}^{N}[0,T]$.

We consider two cases for the law of large numbers: the single-point initial
distribution (Section 4.1) and the stationary initial distribution (Section
4.2).

###  4.1 The single-point initial distribution

We introduce a set of constant functions. Let
$\boldsymbol{\mathfrak{\zeta}}(t)\equiv\gamma\in[0,1]$, where $\gamma$ is a
fixed number. Let $\gamma^{N}$ be a sequence of the initial states,
$\xi^{N}_{\gamma^{N}}(0)=\gamma^{N}$, such that
$\lim_{N\to\infty}\gamma^{N}=\gamma\in[0,1]$ (for example, we can choose
$\gamma^{N}=\lfloor\gamma N\rfloor/N$). Here we prove the law of large numbers
for $\xi^{N}_{\gamma^{N}}(t)$ when $N$ goes to infinity.

######  Theorem 4.1.

For any $\gamma\in[0,1]$ the relation

$\lim_{N\to\infty}{\mathbf{P}}\left(\sup_{t\in[0,T]}\left|\xi^{N}_{\gamma^{N}}(t)-\boldsymbol{\mathfrak{\zeta}}(t)\right|\geq\varepsilon\right)=0.$
(4.1)

holds true.

###### Proof.

Let $J^{N}(t)$ be the number of the jumps of $\xi^{N}_{\gamma^{N}}(t)$ on
$(0,t]$. Consider an auxiliary process $Y(t)=\sum_{i=1}^{J^{N}(t)}Y_{i},$ on
the time interval $[0,T]$, where $Y_{i}$ are i.i.d. sequence of random
variables with $P(Y_{i}=1)=1-P(Y_{i}=-1)=\frac{1}{2}$. Let
$Y_{N}(t):=\frac{1}{N}Y(t)$. Observe that the following inequality holds:

${\mathbf{P}}\left(\sup_{t\in[0,T]}\left|\gamma_{N}+Y_{N}(t)-\boldsymbol{\mathfrak{\zeta}}(t)\right|<\varepsilon\right)\leq{\mathbf{P}}\left(\sup_{t\in[0,T]}\left|\xi^{N}_{\gamma^{N}}(t)-\boldsymbol{\mathfrak{\zeta}}(t)\right|<\varepsilon\right).$

Note that there exists $N_{\varepsilon}$ such that
$|\gamma^{N}-\gamma|<\varepsilon$ for all $N>N_{\varepsilon}$. For these $N$
we have

${\mathbf{P}}\left(\sup_{t\in[0,T]}\left|Y_{N}(t)\right|<\frac{\varepsilon}{2}\right)<{\mathbf{P}}\left(\sup_{t\in[0,T]}\left|\gamma^{N}+Y_{N}(t)-\boldsymbol{\mathfrak{\zeta}}(t)\right|<\varepsilon\right).$

To finish the proof of (4.1) we use Kolmogorov inequality for the maximum of
sum of i.i.d random variables (see, for example, [5], Theorem 22.4). We have

$\displaystyle{\mathbf{P}}\left(\sup_{t\in[0,T]}\left|Y_{N}(t)\right|\geq\varepsilon\right)$
$\displaystyle=\sum_{n=0}^{\infty}\left.{\mathbf{P}}\left(\sup_{t\in[0,T]}\Bigl{|}\sum_{i=1}^{J^{N}(t)}Y_{i}\Bigr{|}\geq\varepsilon
N\ \right|\ J^{N}(T)=n\right){\mathbf{P}}\left(J^{N}(T)=n\right)$
$\displaystyle=\sum_{n=0}^{\infty}\left.{\mathbf{P}}\left(\max_{1\leq k\leq
n}\Bigl{|}\sum_{i=1}^{k}Y_{i}\Bigr{|}\geq\varepsilon N\ \right|\
J^{N}(T)=n\right){\mathbf{P}}\left(J^{N}(T)=n\right)$
$\displaystyle\leq\sum_{n=0}^{\infty}\frac{n}{4\varepsilon^{2}N^{2}}{\mathbf{P}}(J^{N}(T)=n)={\frac{1}{4\varepsilon^{2}N^{2}}\mathbb{E}(J^{N}(T))\leq\frac{NT}{4\varepsilon^{2}N^{2}}}\to
0\mbox{ as }N\to\infty.$

Finally,

$\displaystyle 1$
$\displaystyle\geq{\mathbf{P}}\left(\sup_{t\in[0,T]}\left|\xi^{N}_{\gamma^{N}}(t)-\boldsymbol{\mathfrak{\zeta}}(t)\right|<\varepsilon\right)\geq
1-{\mathbf{P}}\left(\sup_{t\in[0,T]}\left|Y_{N}(t)\right|\geq\frac{\varepsilon}{2}\right)$
(4.2) $\displaystyle\geq 1-\frac{T}{\varepsilon^{2}N}\to 1,\mbox{ as
}N\to\infty.$

This finishes the proof of (4.1). $\Box$

######  Remark 4.2.

We apply the inequality $\mathbb{E}(J^{N}(T))\leq NT$ because the intensity of
the jumps is maximal near the boundary on 1.

###  4.2 Stationary initial distribution

We now turn to the case when the initial state is distributed according to the
stationary measure. To this end, we introduce the constant process
$\boldsymbol{\mathfrak{\zeta}}(t)$ as before. That is
$\boldsymbol{\mathfrak{\zeta}}(t)\equiv\gamma\in[0,1]$ for $t\in[0,T]$ if
$\boldsymbol{\mathfrak{\zeta}}(0)=\gamma$. The distribution of
$\boldsymbol{\mathfrak{\zeta}}(t)$ satisfies the following conditions: for any
$u\in(0,1]$ the probabilities
${\mathbf{P}}(\boldsymbol{\mathfrak{\zeta}}(t)<u)=1$ and
${\mathbf{P}}(\boldsymbol{\mathfrak{\zeta}}(t)>u)=0$. The described property
means that the process $\boldsymbol{\mathfrak{\zeta}}(t)$ is concentrated on
the line $\boldsymbol{\mathfrak{\gamma}}(t)\equiv 0$.

Let $\xi_{\mathbf{st}}^{N}(t)$ be the piecewise constant process on
$\mathbf{D}^{N}[0,T]$ with the initial distribution $\pi^{N}$. Note that the
distribution $\pi^{N}$ is defined on the set $\mathcal{N}$, but we will
maintain the same notation for the stationary measure on ${\mathfrak{N}}^{N}$.
Then the distribution $\xi_{\mathbf{st}}^{N}(t)$ at any $t\in[0,T]$ is also
$\pi^{N}$ since $\pi^{N}$ is stationary. Recall that the generator of
$\xi_{\mathbf{st}}^{N}(t)$ is (3.1).

######  Theorem 4.3.

$\xi^{N}_{\mathbf{st}}\Rightarrow\boldsymbol{\mathfrak{\zeta}}\mbox{ \
weakly}$ (4.3)

as $N\to\infty$

The proof of the theorem is based on the result of the paper [6] on the
convergence of integrals of finite-dimensional distributions of random
processes. The paper [6] states conditions for the weak convergence of the
functionals
$\Phi(\xi_{st}^{N})=\int_{0}^{T}\varphi(\xi_{\mathbb{st}}^{N}(u)\mathrm{d}u$
to
$\Phi(\boldsymbol{\mathfrak{\zeta}})=\int_{0}^{T}\varphi(\boldsymbol{\mathfrak{\zeta}}(u))\mathrm{d}u$,
where $\varphi$ is the continuous function from
$\mathbf{C}[0,1]\to{\mathbb{R}}$.

The mentioned theorem in the considered case is

######  Theorem 4.4 (Borovkov, Pechersky, [6]).

Let

$\lim_{N\to\infty}\int_{0}^{T}\dots\int_{0}^{T}{\mathbf{P}}\left(\bigcap_{i=1}^{\ell}(\xi^{N}_{\mathbb{st}}(t_{i})<u_{i})\right)\mathrm{d}t_{1},...,\mathrm{d}t_{\ell}=\int_{0}^{T}\dots\int_{0}^{T}{\mathbf{P}}\left(\bigcap_{i=1}^{\ell}(\boldsymbol{\mathfrak{\zeta}}(t_{i})<u_{i})\right)\mathrm{d}t_{1},...,\mathrm{d}t_{\ell}$

for any finite number $t_{1},...,t_{\ell}\in[0,T]$ then the distribution of
$\Phi(\xi^{N}_{\mathbb{st}})=\int_{0}^{T}\varphi(\xi^{N}_{\mathbb{st}}(u))\mathrm{d}u$
weakly converges to
$\Phi(\boldsymbol{\mathfrak{\zeta}})=\int_{0}^{T}\varphi(\boldsymbol{\mathfrak{\zeta}}(u))\mathrm{d}u$
for any $\varphi\in\mathbf{C}[0,1]$.

Due to the properties of the process $\boldsymbol{\mathfrak{\zeta}}$, the
convergence of the integral functionals is sufficient for weak convergence
(4.3). The result we apply to prove (4.3) reformulated for our case as the
following

######  Lemma 4.5.

For any $\\{u_{1},...,u_{\ell}\\}\subset[0,1]$ and
$\boldsymbol{\mathfrak{\zeta}}<\widetilde{u}=\min_{j=1,...,\ell}u_{j}$ the
following equality

$\lim_{N\to\infty}\int_{0}^{T}\dots\int_{0}^{T}{\mathbf{P}}\left(\bigcap_{i=1}^{\ell}(\xi^{N}_{\mathbb{st}}(t_{i})<u_{i})\right)\mathrm{d}t_{1},...,\mathrm{d}t_{\ell}=\int_{0}^{T}\dots\int_{0}^{T}{\mathbf{P}}\left(\bigcap_{i=1}^{\ell}(\boldsymbol{\mathfrak{\zeta}}(t_{i})<u_{i})\right)\mathrm{d}t_{1},...,\mathrm{d}t_{\ell}=T^{\ell}$
(4.4)

holds for any finite set of time points $t_{1},...,t_{\ell}\in[0,T]$.

Proof of Lemma 4.5. The righthand side of (4.4)

$\int_{0}^{T}\dots\int_{0}^{T}{\mathbf{P}}\left(\cap_{j=1}^{\ell}(\boldsymbol{\mathfrak{\zeta}}(t_{j})<u_{j})\right)dt_{1}\ldots
dt_{\ell}=T^{\ell},$

by the definition of $\boldsymbol{\mathfrak{\zeta}}(t)$. To find the limit on
the lefthand side, let $0<a<\widetilde{u}=\min_{j=1,...,\ell}\\{u_{j}\\}$ and
$\varepsilon>0$ be such that
$(a-\varepsilon,a+\varepsilon)\subset(0,\widetilde{u})$. For a fixed $N$ we
have

Figure 1: Illustration for the proof of Theorem 4.3

$\displaystyle\int_{0}^{T}\dots\int_{0}^{T}$
$\displaystyle{\mathbf{P}}\left(\cap_{j=1}^{\ell}\big{(}\xi^{N}_{\mathbf{st}}(t_{j})<u_{j}\big{)}\right)dt_{1}\ldots
dt_{\ell}$ (4.5)
$\displaystyle=\int_{0}^{T}\dots\int_{0}^{T}\sum_{y<a}{\mathbf{P}}\left(\cap_{j=1}^{\ell}\big{(}\xi^{N}_{y}(t_{j})<u_{j}\big{)}\right)\pi^{N}(y)dt_{1}\ldots
dt_{\ell}$ $\displaystyle+\int_{0}^{T}\dots\int_{0}^{T}\sum_{y\geq
a}{\mathbf{P}}\left(\cap_{j=1}^{\ell}\big{(}\xi^{N}_{y}(t_{j})<u_{j}\big{)}\right)\pi^{N}(y)dt_{1}\ldots
dt_{\ell}$

Probabilities in the last two sums we estimate as the following. For any
$y<a$, and sufficiently large $N$, we use the following bounds (see (4.2))

$1\geq{\mathbf{P}}\left(\cap_{j=1}^{\ell}\big{(}\xi^{N}_{y}(t_{j})<u_{j}\big{)}\right)\geq{\mathbf{P}}\left(\xi^{N}_{y}(t)\in[y-\varepsilon,y+\varepsilon]\right)\geq
1-\frac{T}{\varepsilon^{2}N}.$

For the first term of (4.5) we have

$T^{\ell}\geq\int_{0}^{T}\dots\int_{0}^{T}\sum_{y<a}{\mathbf{P}}\left(\cap_{j=1}^{\ell}\big{(}\xi^{N}_{y}(t_{j})<u_{j}\big{)}\right)\pi^{N}(y)dt_{1}\ldots
dt_{\ell}\geq T^{\ell}\pi^{N}([0,a))+O\left(\frac{1}{N}\right)\to T^{\ell},$

as $N\to\infty$, according to Theorem 2.2. In the second term for all $y\geq
x$ corresponding probabilities we can bound by 1, and

$0\leq\int_{0}^{T}\dots\int_{0}^{T}\sum_{y\geq
a}{\mathbf{P}}\left(\cap_{j=1}^{\ell}\big{(}\xi^{N}_{y}(t_{j})<u_{j}\big{)}\right)\pi^{N}(y)dt_{1}\ldots
dt_{\ell}\leq T^{\ell}\pi^{N}([a,1])\to 0,$

as $N\to\infty$, according to Theorem 2.2. This finishes the proof of the
lemma. $\Box$

Proof of Theorem 4.3. Theorem 4.4 and Lemma 4.5 prove the theorem. $\Box$

##  5 Large Deviations

This section is devoted to studying the process deviations far from average
values. This study belongs to the well-developed area of probability theory
known as the theory of Large Deviations. Here we use the techniques developed
in [9]. We apply and adapt these results with additional calculations
corresponding to the peculiarities of our model. We also refer a reader to
[11] and [4], where the method was applied to some similar Markov processes.
Thus, here we only pass through some points related to our process, and we
would like to avoid cumbersome constructions used in the proof of LDP.

######  Remark 5.1.

The one-dimensional case of the generalized Ehrenfest model studied in [11] is
very similar to the model studied here with a slight difference. Despite the
differences in the generators of Fleming semigroups the corresponding
Hamiltonians are similar. We will refer to this paper passing through the
proof of LDP.

Let $\mathbf{C}_{1}[0,T]$ be a set of continuously differentiable function on
$[0,T]$ with values from $[0,1]$. The following two terms are our main
interest.

  1. Rate function (functional) $I:\boldsymbol{\mathfrak{\gamma}}\to\mathbb{R}$ for $\boldsymbol{\mathfrak{\gamma}}\in\mathbf{C}_{1}[0,T]$, see (5.14).

  2. Hamiltonian $H(\gamma,\varkappa)$, with $\gamma\in[0,1]$ and $\varkappa\in\mathbb{R}$, see (5.5).

For any $\boldsymbol{\mathfrak{\gamma}}\in\mathbf{C}_{1}[0,T]$ the rate
function $I(\boldsymbol{\mathfrak{\gamma}})$ measures deviations of
$\boldsymbol{\mathfrak{\gamma}}$ from the process average.
$I(\boldsymbol{\mathfrak{\gamma}})$ is approximately the logarithm of the
following probability of the deviation
$\boldsymbol{\mathfrak{\gamma}}(t)-\mathbf{E}\xi^{N}(t)$:

$I(\boldsymbol{\mathfrak{\gamma}})=\lim_{\varepsilon}\lim_{N}\frac{1}{N}\ln{\mathbf{P}}\left(\xi^{N}(\cdot)\in
U_{\varepsilon}(\boldsymbol{\mathfrak{\gamma}})\right),$

where $U_{\varepsilon}(\boldsymbol{\mathfrak{\gamma}})$ is
$\varepsilon$-neighborhood of $\boldsymbol{\mathfrak{\gamma}}$. If
$\boldsymbol{\mathfrak{\gamma}}(t)=\lim_{N}\mathbf{E}\xi^{N}(t)$ then
$I(\boldsymbol{\mathfrak{\gamma}})=0$. The existence of this limit is known as
a local large deviation principle (LLDP)

######  Definition 5.2.

Let the sequence of random processes $\xi^{N}(\,\cdot\,)$ be defined as in
(4.1). We say that this sequence satisfies a local large deviation principle
with an LD functional $I\,:\mathbf{C}_{1}[0,T]\to[0,\infty]$ if for all
$f\in\mathbf{C}_{1}[0,T]$

$\lim\limits_{\varepsilon\rightarrow
0}\limsup\limits_{N\rightarrow\infty}\dfrac{1}{N}\ln\mathbf{P}(\xi^{N}(\,\cdot\,)\in
U_{\varepsilon}(f))=\lim\limits_{\varepsilon\rightarrow
0}\liminf\limits_{N\rightarrow\infty}\dfrac{1}{N}\ln\mathbf{P}(\xi^{N}(\,\cdot\,)\in
U_{\varepsilon}(f))=-I(f),$

where

$U_{\varepsilon}(f)=\\{g\in\mathbf{D}[0,T]:\ \rho(f,g)<\varepsilon\\},\ \mbox{
and }\ \rho(f,g):=\sup_{t\in[0,T]}\left|f(t)-g(t)\right|.$

Frequently, the existence of this limit together with exponential tightness
verify the large deviation principle (LDP).

######  Definition 5.3.

Let $\xi^{N}(\,\cdot\,)$ be sequence of random processes defined as in (4.1).
We say that it satisfies a large deviation principle with an LD functional
$I\,:\;\mathbf{C}_{1}[0,T]\to(0,\infty]$ if, whenever measurable set
$\mathbb{B}\subseteq\mathbf{C}_{1}[0,T]$, we have that

$\begin{array}[]{c}\limsup\limits_{T\rightarrow\infty}\dfrac{1}{N}\ln\mathbf{P}(\,\xi^{N}(\,\cdot\,)\in\mathbb{B}\,)\leq-I({\rm{Cl}}(\mathbb{B})),\\\
\liminf\limits_{T\rightarrow\infty}\dfrac{1}{N}\ln\mathbf{P}(\,\xi^{N}(\,\cdot\,)\in\mathbb{B}\,)\geq-I({\rm{Int}}(\mathbb{B})).\end{array}$

This way worked well in papers [12] – [15], where the method to obtain the
LLDP was proposed and applied for birth-and-death processes. But we couldn’t
extend the method for the process considered here, so we opted for the
operator method proposed by Fend & Kurtz [9].

It seems the shortest way to find the rate function is Bryc’s theorem. It
claims the following.

$I(\boldsymbol{\mathfrak{\gamma}})=\sup_{\mathscr{L}}\left\\{\mathscr{L}(\boldsymbol{\mathfrak{\gamma}})-\Lambda(\mathscr{L})\right\\},$
(5.1)

where $\\{\mathscr{L}\\}$ is a set of functionals on the process path set
$\mathbf{D}[0,T]$, and

$\Lambda(\mathscr{L})=\lim_{N}\frac{1}{N}\ln\int\exp\\{\mathscr{L}(\boldsymbol{\mathfrak{\gamma}})\\}{\mathbf{P}}(\xi^{N}\in\mathrm{d}\boldsymbol{\mathfrak{\gamma}})$

for any functional $\mathscr{L}$. We have omitted the conditions and other
details required for Bryc’s theorem (see Proposition 3.8 in [9]).

Unfortunately, this is not a very practical way. The main difficulty is
related to the fact that we consider Markov-dependent random variables. In
this case, $\Lambda(\mathscr{L})$ is too general, and therefore, it is very
rarely computable. Do the functionals $\Lambda(\mathscr{L})$ form a semigroup
acting on a set of functions? A positive answer to this question would
construct a generator of this semigroup, which makes it possible to create a
research method of $\Lambda(\mathscr{L})$.

###  5.1 Fleming semigroup

The above question is solved with the help of a (Fleming) semigroup, proposed
by Fleming ([10]):

$V_{t}^{N}{f}(\gamma)=\frac{1}{N}\ln\int\exp\\{N{f}(\gamma)\\}{\mathbf{P}}(\xi^{N}(t)\in\mathrm{d}{\gamma})\mid\xi^{N}(0)=\gamma_{0}),$
(5.2)

where the probe functions belongs to the set of $C_{1}[0,1]$ continuously
differentiable function on $[0,1]$ with values from $\mathbb{R}$,
$f\in\mathbf{C}_{1}[0,1]$. The problem now is to investigate the limiting
semigroup

$\mathbf{V}_{t}{f}(\gamma)=\lim_{N}V_{t}^{N}{f}(\gamma).$ (5.3)

Since $V_{t}^{N}$ is a semigroup it is reasonable to find its generator. The
generator corresponding to $V_{t}^{N}$ describes the Markov dynamics which is
called non-linear and its generator is called non-linear Hamiltonian
$\mathcal{H}^{N}$: if
$\gamma\in{\mathfrak{N}}^{N}\setminus\\{\frac{1}{N},1\\}$, then

$\displaystyle\mathcal{H}^{N}{f}(\gamma)$
$\displaystyle=\frac{1}{N}\exp\big{\\{}-N{f}(\gamma)\big{\\}}\left(\lambda
N\gamma\left(\exp\left\\{N{f}\left(\gamma+\tfrac{1}{N}\right)\right\\}-\exp\big{\\{}N{f}(\gamma)\big{\\}}\right)\right.$
$\displaystyle\phantom{=\frac{1}{N}\exp\big{\\{}-Nf(x_{N})\big{\\}}}+\lambda
N\gamma\left(\exp\left\\{N{f}\left(\gamma-\tfrac{1}{N}\right)\right\\}-\exp\big{\\{}N{f}(\gamma)\big{\\}}\right)\Big{)},$

if $\gamma=\frac{1}{N}$, then
$\mathcal{H}^{N}{f}(\tfrac{1}{N})=\frac{\lambda}{N}\exp\left\\{-N{f}(\tfrac{1}{N})\right\\}\left(\exp\left\\{N{f}\left(\tfrac{2}{N}\right)\right\\}-\exp\left\\{N{f}(\tfrac{1}{N})\right\\}\right),$
and if $\gamma=1$, then
$\mathcal{H}^{N}{f}(1)=\lambda\exp\left\\{-N{f}(1)\right\\}\left(\exp\left\\{N{f}\left(1-\tfrac{1}{N}\right)\right\\}-\exp\left\\{N{f}(1)\right\\}\right).$
It is easy to see that the limiting operator $\mathcal{H}$ should be

$\mathcal{H}f(\gamma)=\lim_{N}\mathcal{H}^{N}{f}(\gamma)=\lambda\gamma\left(\exp\left\\{\frac{\mathrm{d}f}{\mathrm{d}\gamma}(\gamma)\right\\}-1\right)+\lambda\gamma\left(\exp\left\\{-\frac{\mathrm{d}\gamma}{\mathrm{d}\gamma}(\gamma)\right\\}-1\right)=:H\left(\gamma,\frac{\mathrm{d}f}{\mathrm{d}\gamma}(\gamma)\right),$
(5.4)

where the set of probe functions is the set $\mathbf{C}_{1}^{0}[0,1]$ of
continuously differentiable functions with zero derivative at 1,
$\mathbf{C}_{1}^{0}[0,1]\subset\mathbf{C}_{1}[0,1]$. Using the notation
$\varkappa=\frac{\mathrm{d}f}{\mathrm{d}\gamma}(\gamma)$ we write the
Hamiltonian

$H(\gamma,\varkappa)=\lambda\gamma(e^{\varkappa}-1)+\lambda\gamma(e^{-\varkappa}-1).$
(5.5)

The proof of the operator convergence

$\mathcal{H}=\lim\mathcal{H}^{N}$

in the $\mathbf{C}_{1}^{0}[0,1]$ is not diffcult. First, we consider the
convergence on ${\mathfrak{N}}$, where the limit

$\lim_{N\to\infty}\sup_{\gamma\in{\mathfrak{N}}}\left|\mathcal{H}^{N}{f}(\gamma)-\mathcal{H}f(\gamma)\right|=0,$
(5.6)

is obvious. Second, consider a sequence $(\gamma_{N})$ where
$\gamma_{N}\in{\mathfrak{N}}$ such that $\gamma_{N}\to\gamma$ for an arbitrary
$\gamma\in\mathbf{[}0,1]$. Then

$\lim_{N}\frac{f(\gamma_{N}+1/N)-f(\gamma_{N})}{1/N}=\frac{\mathrm{d}f}{\mathrm{d}\gamma}(\gamma)$

and the derivative of $f$ is continuous on $[0,1]$. Therefore (5.6) can be
extended to

$\lim_{N\to\infty}\sup_{\gamma\in[0,1]}\left|\mathcal{H}^{N}{f}(\gamma)-\mathcal{H}f(\gamma)\right|=0.$

As can be seen the resulting function
$\mathcal{H}f(\gamma)=H\left(\gamma,\frac{\mathrm{d}{f}}{\mathrm{d}\gamma}\right)$
depends on the derivative $\frac{\mathrm{d}{f}}{\mathrm{d}\gamma}$, the first
variable $\gamma$ is the reason for the non-linearity of the dynamics.

Next considerations prove that $\mathcal{H}$ is the generator of the semigroup
$\mathbf{V}_{t}$.

###  5.2 Exponential compact containment

The exponential compact containment  can be obtained directly from the
definition for our case. The sequence $\xi^{N}$ satisfy exponential compact
containment condition in $\bar{\mathbf{D}}(0,T]$ iff for each $a>0$ there
exists a compact $K_{a}\subset[0,1]$ such that

$\limsup_{n\to\infty}\frac{1}{n}\ln P(X_{n}(t)\notin K_{a}\mbox{ for some
}t\leq T)\leq-a.$

In our case this condition is easily verified taking $K_{a}=[0,1]$. Observe
that it is not exponential tightness. But together with generator convergence
and the comparison principle (see further for the definition) exponential
containment will imply exponential tightness according to Theorem 6.14, [9].

###  5.3 Fleming semigroup generator

We now consider when $\mathcal{H}$, (5.4), is the generator of the Fleming
semigroup $\mathbf{V}_{t}$, (5.3). The proof is a rather cumbersome sequence
of arguments. One of the important conditions for $\mathcal{H}$ to be the
generator is the existence of solutions to the equation
$(I-\beta\mathcal{H})f=h$ for each $h$ and small positive $\beta$ (Chapter 6,
[9]). It is so called the viscosity equation. This condition is difficult to
verify. The comparison principle overcomes this difficulty.

Let us rewrite the equation using Hamiltoniam:

$f(\gamma)-\beta
H\left(\gamma,\tfrac{\mathrm{d}f}{\mathrm{d}\gamma}(\gamma)\right)-h(\gamma)=0.$
(5.7)

Let $u$ and $v$ be subsolution and supersolution of equation (5.7) (see
Definition 5.4 in Appendix 1). We say that the equation verifies the
comparison principle if for any subsolution $u$ and supersolution $v$, it is
verified that $u\leq v$.

The proof of the comparison principle for the case under study is given in
Appendix 1, see Proposition 5.5. When the comparison principle holds, if there
exists a viscosity solution, it is unique.

###  5.4 Rate function. Variational representation.

We have established that Hamiltonian $H$ is Fleming group generator. The next
our goal is to represent the rate function $I(\boldsymbol{\mathfrak{\gamma}})$
as Legendre transform of the Hamiltonian that is to obtain the variational
form of the rate function

$I(\boldsymbol{\mathfrak{\gamma}})=\int_{0}^{T}\sup_{\varkappa}\left\\{\varkappa\dot{\boldsymbol{\mathfrak{\gamma}}}(t)-H(\boldsymbol{\mathfrak{\gamma}}(t),\varkappa)\right\\}\mathrm{d}t.$
(5.8)

This expression is difficult to obtain directly from Fleming semigroup terms
(see (5.2), (5.3)) and require some results from control theory. The Fleming
semigroup in the large deviation is interpreted as Nisio semigroup in control
theory. The proof of (5.8) uses a method developed in a number of works. The
detail presentation of the method is represented in Capter 8 from [9], see
also [11], [4], for the application of the method.

The method is a series of transformations of the generator $g^{N}$ (4.1). The
first non-linear transformation provided the Fleming semigroup
$\mathbf{V}_{t}$ and its generator is
$H=\lambda\gamma\left(e^{\varkappa}+e^{-\varkappa}-2\right)$, see Section 5.1
and Section 5.3. The Fleming semigroup has the exponential form not easy to
investigate. The last transformation is the Nisio semigroup construction.

####  5.4.1 Nisio Semigroup

In this stage the method shows that the Fleming semigroup coincides with the
so-called Nisio semigroup $\mathbf{W}(t)$ from the control theory (see [9],
Chapter 8). The control theory provides the variational representation (5.8)
for the rate function.

Before starting let us observe that $H(\gamma,\varkappa)$ is a convex function
on the second coordinate $\varkappa$, and

$H(\gamma,\varkappa)=\sup_{\gamma}\left(\varkappa\gamma-L(\gamma,\varkappa)\right),$
(5.9)

where

$L(\gamma,\varkappa):=\sup_{\varkappa}\left(\varkappa\gamma-H(\gamma,\varkappa)\right).$
(5.10)

The Nisio semigroup has the following form

$\mathbf{W}(t)f(x)=\sup_{\boldsymbol{\mathfrak{\gamma}}\in\mathcal{AC}[0,T],\\\
\boldsymbol{\mathfrak{\gamma}}(0)=\gamma}f(\gamma(t))-\int_{0}^{t}L(\gamma(s),\dot{\gamma}(s))ds,$
(5.11)

where $\boldsymbol{\mathfrak{\gamma}}\in\mathcal{AC}[0,T]$, and
$\mathcal{AC}[0,T]$ is the set of all absolutely continuous paths on $[0,T]$.

A more easy approach is to use a relaxed presentation of Nisio semigroup
instead of (5.11). Let $U=\\{\dot{\boldsymbol{\mathfrak{\gamma}}}(\cdot)\\}$
and $\mathcal{M}_{m}(U)$ be the space of Borel measures $\lambdabar$ on
$U\times[0,T]$ satisfying $\lambdabar(U\times[0,t])=t$ for all $t\in[0,T]$.
The measure $\lambdabar$ is known as a relaxed control (see detailed
definition in Appendix 2).

The relaxed controls determine the Nisio semigroup as

$\mathbf{\bar{W}}(t)f(x)=\sup_{(\boldsymbol{\mathfrak{\gamma}},\lambdabar),\
\boldsymbol{\mathfrak{\gamma}}(0)=x}f(\boldsymbol{\mathfrak{\gamma}}(t))-\iint_{U\times[0,1]}L(\boldsymbol{\mathfrak{\gamma}}(s),u)\lambdabar(du\times
ds),$ (5.12)

####  5.4.2 Variational representation of rate function

Finally, the relation (5.8) follows from [4], Theorem 3.1, which states that

  1. 1.

$\mathbf{V}_{t}(f)=\bar{\mathbf{W}}(t)f$

  2. 2.

$I({\boldsymbol{\mathfrak{\gamma}}})=\inf_{\lambdabar:\
(\boldsymbol{\mathfrak{\gamma}},\lambdabar)}\iint_{U\times[0,T]}L(\boldsymbol{\mathfrak{\gamma}}(s),u)\lambdabar(du\times
ds)$;

  3. 3.

the rate function can be written as an action integral:

$I(\boldsymbol{\mathfrak{\gamma}})=\int_{0}^{T}L(\boldsymbol{\mathfrak{\gamma}}(s),\dot{\boldsymbol{\mathfrak{\gamma}}}(s))ds,$
(5.13)

if $\boldsymbol{\mathfrak{\gamma}}$ is absolutely continuous function.

We have omitted the required conditions.

###  5.5 Hamiltonian system

At that point, we have made four steps to obtain LDP and the rate function.

  1. Step 1. Section 5.1 established the convergence of the sequence of operators $\mathcal{H}^{N}$, where we obtained the limiting operator $\mathcal{H}$.

  2. Step 2. Section 5.2 checked the exponential compact containment condition.

  3. Step 3. Section 5.3 with Appendix 1 established the comparison principle. It proves that the limiting operator $\mathcal{H}$ generates the semigroup $\mathbf{V}_{t}$: $\frac{d}{dt}\mathbf{V}(f)=\mathcal{H}(\mathbf{V}_{t}(f))$.

According to Theorem 6.14 from [9] these three steps assure that the sequence
of our processes is exponentially tight and satisfies an LDP with rate
function expressed via the Fleming semigroup (see formulas for rate functions
(5.30) and (5.31) in [9]). Practically the exponetial tightness implies that
the rate function is a good rate function (see, for example, Theorem 3.7 [9]).

This result does not provide a useful presentation for the rate function. The
next fourth step of Feng & Kurtz theory provides the practical version of the
rate function based on control theory.

  1. Step 4. Section 5.4 with Appendix 2 state and checked conditions when the rate function $I$ can be written as an action integral (5.13). (Condition 8.9 and 8.11 from [9] are stated in Appendix 2. They are easily verified. See, for example, Propositions 3.6 and 3.8 from [4]).

Here, based on a variational representation of the rate function, fortunately,
we can obtain the rate functional (5.14) explicitly. Moreover, we can find the
optimal path for the following rare event: starting at the point
$\boldsymbol{\mathfrak{\gamma}}(0)$ the process will stay at time $T$ at the
point $\boldsymbol{\mathfrak{\gamma}}(T)$ far from their mean,
$\boldsymbol{\mathfrak{\gamma}}(0)\neq\boldsymbol{\mathfrak{\gamma}}(T)$ (see
example in Appendix 3.)

Let us now find paths between fixed boundary conditions. To this end, we
should investigate the following Hamiltonian system

$\left\\{\begin{aligned}
&\dot{\boldsymbol{\mathfrak{\gamma}}}=\lambda\boldsymbol{\mathfrak{\gamma}}\left(e^{\varkappa}-e^{-\varkappa}\right)\\\
&\dot{\varkappa}=-\lambda\left(e^{\varkappa}+e^{-\varkappa}-2\right)\end{aligned}\right.$

First equation of Hamiltonian system is the derivative on $\varkappa$,
$\dot{{\boldsymbol{\mathfrak{\gamma}}}}=\frac{\partial H}{\partial\varkappa}$,
to find the supremum. The second one is Euler–Lagrange equation.

Rate function: The first equation provides the explicit formula for the rate
function. Let $z=e^{\kappa}$. For given functions
$\boldsymbol{\mathfrak{\gamma}}(t),\dot{\boldsymbol{\mathfrak{\gamma}}}(t)$ we
obtain the quadratic equation on $z$

$z^{2}-\frac{\dot{\boldsymbol{\mathfrak{\gamma}}}}{\lambda\boldsymbol{\mathfrak{\gamma}}}z-1=0,$

with admissible solution

$z(t)=e^{\kappa(t)}=\frac{\dot{\boldsymbol{\mathfrak{\gamma}}}}{2\lambda\boldsymbol{\mathfrak{\gamma}}}+\sqrt{\left(\frac{\dot{\boldsymbol{\mathfrak{\gamma}}}}{2\lambda\boldsymbol{\mathfrak{\gamma}}}\right)^{2}+1}.$

Finally,

$I(\boldsymbol{\mathfrak{\gamma}})=\int_{0}^{T}\left[\dot{\boldsymbol{\mathfrak{\gamma}}}\ln\left(\frac{\dot{\boldsymbol{\mathfrak{\gamma}}}}{2\lambda\boldsymbol{\mathfrak{\gamma}}}+\sqrt{\left(\frac{\dot{\boldsymbol{\mathfrak{\gamma}}}}{2\lambda\boldsymbol{\mathfrak{\gamma}}}\right)^{2}+1}\right)-\dot{\boldsymbol{\mathfrak{\gamma}}}+2\lambda\boldsymbol{\mathfrak{\gamma}}-\frac{(2\lambda\boldsymbol{\mathfrak{\gamma}})^{2}}{\dot{\boldsymbol{\mathfrak{\gamma}}}+\sqrt{\dot{\boldsymbol{\mathfrak{\gamma}}}^{2}+(2\lambda\boldsymbol{\mathfrak{\gamma}})^{2}}}\right]dt$
(5.14)

Solution of the system: Let $z=e^{\varkappa}$ then
$\dot{\varkappa}=\frac{\dot{z}}{z}$. The second equation of the above system
is

$\frac{\dot{z}}{z}=-\lambda\left(z+\frac{1}{z}\right)+2\lambda,$

or

$\dot{z}=-\lambda(z-1)^{2}.$

The solution is

$z(t)=\left\\{\begin{aligned} \frac{1}{\lambda t-c_{1}}+1,&\mbox{ when
}\boldsymbol{\mathfrak{\gamma}}(t)\nequiv const;\\\ 1,&\mbox{ when
}\boldsymbol{\mathfrak{\gamma}}(t)\equiv const.\end{aligned}\right.$

where $c_{1}$ is some constant.

Now the first equation of the system (when
$\boldsymbol{\mathfrak{\gamma}}\nequiv const$) is

$\dot{\boldsymbol{\mathfrak{\gamma}}}=\lambda\boldsymbol{\mathfrak{\gamma}}\left(z(t)-\frac{1}{z(t)}\right)=\lambda\boldsymbol{\mathfrak{\gamma}}\left(\frac{1}{\lambda
t-c_{1}}+\frac{1}{\lambda t-c_{1}+1}\right),$

or

$\frac{\mathrm{d}\
\ln(\boldsymbol{\mathfrak{\gamma}})}{\mathrm{d}t}=\lambda\left(\frac{1}{\lambda
t-c_{1}}+\frac{1}{\lambda t-c_{1}+1}\right)$

Then

$\boldsymbol{\mathfrak{\gamma}}(t)=\left\\{\begin{array}[]{rl}(\lambda
t-c_{1})(\lambda t-c_{1}+1)c_{2},&\mbox{ if
}\boldsymbol{\mathfrak{\gamma}}(t)\nequiv const\\\ const,&\mbox{ if
}\boldsymbol{\mathfrak{\gamma}}(t)\equiv const\in[0,1].\end{array}\right.$

where $c_{2}$ is some positive constant. The parameters $c_{1},c_{2}$ are
uniquely determined by the boundary conditions
$\boldsymbol{\mathfrak{\gamma}}(0),\boldsymbol{\mathfrak{\gamma}}(T)$ and by
the fact that the solution $\boldsymbol{\mathfrak{\gamma}}(t)\in[0,1]$ when
$t\in[0,T]$.

The examples of the solutions are in Appendix 3.

## Acknowledgment

AY thanks A. Logachov for fruitful discussions. AY thanks FAPESP for the
financial support via the grant 2017/10555-0.

## Appendix 1: comparison principle

The following proposition proves the comparison principle. It is similar to
Proposition 3 in [11]. We adapt the proof in [11] for our case. Let us recall
some basic definitions from [11] slightly adapted for our notations. Recall,
see (5.5), that we are working with our Hamiltonian $H(x,\varkappa)=\lambda
x\left(e^{\varkappa}+e^{-\varkappa}-2\right)$.

######  Definition 5.4.

$u$ is called a (viscosity) subsolution if

  1. $u$ is bounded, upper semi-continuous;

  2. for every $\varphi\in\mathbf{C}_{1}[0,1]$ and $x_{0}:=\mbox{argmax}(u(x)-\varphi(x))$ we have $u(x_{0})-\beta H(x_{0},\varphi^{\prime}(x_{0}))-h(x_{0})\leq 0.$

We say that $v$ is a (viscosity) supersolution if

  1. $v$ is bounded, lower semi-continuous;

  2. for every $\varphi\in\mathbf{C}_{1}[0,1]$ and $x_{0}:=\mbox{argmin}(v(x)-\varphi(x))$ we have $v(x_{0})-\beta H(x_{0},\varphi^{\prime}(x_{0}))-h(x_{0})\geq 0.$

We say that $u$ is a (viscosity) solution if it is both a sub and super
solution.

######  Proposition 5.5.

Let $\beta>0$ and $h\in\mathbf{C}[0,1]$. Then the comparison principle holds
true for

$f(x)-\beta H(x,f^{\prime}(x))-h(x)=0.$ (5.15)

Proof. With small adaptation the proof follows [11], Section 4. See especially
[11], Section 4.1, where comparison principle verified for a one-dimensional
Erenfest model similar to our case.

For given $\beta>0$, $h\in\mathbf{C}[0,1]$ let $u,v$ be some sub- and super-
solutions of (5.15). Choose the following penalized function
$\Psi(x,y)=\frac{1}{2}(x-y)^{2}$. For a positive increasing parameter $\alpha$
(for example, we can suppose $\alpha$ as natural numbers) we define the
following points $x_{\alpha},y_{\alpha}\in[0,1]$

$u(x_{\alpha})-v(y_{\alpha})-\frac{\alpha}{2}(x_{\alpha}-y_{\alpha})^{2}=\sup_{x,y\in
E}\left\\{u(x)-v(y)-\frac{\alpha}{2}(x-y)^{2}\right\\}.$

Since the functions $u,v$ are bounded the limit
$\lim_{\alpha\to\infty}\alpha\Psi(x_{\alpha},y_{\alpha})$ should be zero. All
limit points of $(x_{\alpha},y_{\alpha})$ should take the form $(z,z)$ such
that

$u(z)-v(z)=\sup_{x\in E}u(x)-v(x).$

It is the general result, and we refer to Lemma 4 in [11].

For such chosen sequence of points $x_{\alpha},y_{\alpha}$ according to
Proposition 2 in [11] we need to prove that

$\liminf_{\alpha\to\infty}H(x_{\alpha},\alpha(x_{\alpha}-y_{\alpha}))-H(y_{\alpha},\alpha(x_{\alpha}-y_{\alpha}))\leq
0.$

In our case

$H(x_{\alpha},\alpha(x_{\alpha}-y_{\alpha}))-H(y_{\alpha},\alpha(x_{\alpha}-y_{\alpha}))=\lambda(x_{\alpha}-y_{\alpha})\left(e^{\alpha(x_{\alpha}-y_{\alpha})}+e^{-\alpha(x_{\alpha}-y_{\alpha})}-2\right).$
(5.16)

Lemma 5 in [11] provides more control: accoding this lemma we have

$\sup_{\alpha}H(y_{\alpha},\alpha(x_{\alpha}-y_{\alpha}))=\lambda
y_{\alpha}\left(e^{\alpha(x_{\alpha}-y_{\alpha})}+e^{-\alpha(x_{\alpha}-y_{\alpha})}-2\right)<\infty.$
(5.17)

Thus, suppose that $z\in(0,1]$ then, according to (5.17) the sequence
$\alpha(x_{\alpha}-y_{\alpha})$ is bounded, and it provides 5.16 (see some
details in Proposition 3, [11]). The case $z=0$ does not follow the proof in
[11].

If $z=0$, we prove directly that $v(0)>u(0)$. Let us suppose for simplicity
that $u$ and $v$ are continuous functions. Then for any small $\varepsilon>0$
we can find the probe functions
$\varphi_{\varepsilon}^{(1)},\varphi_{\varepsilon}^{(2)}\in\mathbf{C}_{1}[0,1]$
such that

$\displaystyle x_{0}^{(1)}$
$\displaystyle:=\mbox{argmax}(u(x)-\varphi_{\varepsilon}^{(1)}(x))<\varepsilon$
$\displaystyle x_{0}^{(2)}$
$\displaystyle:=\mbox{argmin}(v(x)-\varphi_{\varepsilon}^{(2)}(x))<\varepsilon$

and

$\displaystyle u(x_{0}^{(1)})-\beta
H\left(x_{0}^{(1)},\frac{\mathrm{d}}{\mathrm{d}x}\varphi_{\varepsilon}^{(1)}(x_{0}^{(1)})\right)-h(x_{0}^{(1)})\leq
0,$ $\displaystyle v(x_{0}^{(2)})-\beta
H\left(x_{0}^{(2)},\frac{\mathrm{d}}{\mathrm{d}x}\varphi_{\varepsilon}^{(2)}(x_{0}^{(2)})\right)-h(x_{0}^{(2)})\geq
0.$

Getting $\varepsilon\to 0$ we obtain that $v(0)>u(0)$. The case of
semicontinuous $u$ and $v$ is proved in a similar way adding some conditions
for probe functions. It finishes the proof of the proposition. $\Box$

## Appendix 2: control theory

We need to verify if this semigroup is well-defined: we need to check
Conditions 8.9 and 8.11 from [9]. To verify these conditions the following
definition of a control equation is required. Based on trivial relation
(deterministic control)

$f(\boldsymbol{\mathfrak{\gamma}}(t))-f(\boldsymbol{\mathfrak{\gamma}}(0))=\int_{0}^{t}f^{\prime}(\boldsymbol{\mathfrak{\gamma}}(s))\dot{\boldsymbol{\mathfrak{\gamma}}}(s)ds,$

the control theory introduce the notion of relaxed control equation, where the
deterministic velocity $\dot{\boldsymbol{\mathfrak{\gamma}}}$ is substituted
by a (control) measure $\lambdabar$, see definition below.

######  Definition 5.6.

([9], Definition 8.1, or [4], Definition 3.5) Let $U$ and $E$ be complete and
separable metric spaces. Let $A:\ Dom(A)\subset B(E)\to M(E\times U)$ be a
single valued linear operator. Let $\mathcal{M}_{m}(U)$ be the space of Borel
measures $\lambdabar$ on $U\times[0,1]$ satisfying
$\lambdabar(U\times[0,t])=t$ for all $t\in[0,1]$. The measure $\lambdabar$ is
known as a relaxed control. We say that the pair $(\mathbf{x},\lambdabar)\in
D_{E}[0,1]\times\mathcal{M}_{m}(U)$ satisfies the relaxed control equation for
$A$ if and only if:

  1. 1.

for any $f\in Dom(A)$ any $t\in[0,1]$

$\iint_{U\times[0,1]}\left|A(f)(\mathbf{x}(s),u)\right|\lambdabar(du\times
ds)<\infty,$

  2. 2.

for any $f\in Dom(A)$ any $t\in[0,1]$

$f(\mathbf{x}(t))-f(\mathbf{x}(0))=\iint_{U\times[0,1]}\left|A(f)(\mathbf{x}(s),u)\right|\lambdabar(du\times
ds).$

Observe that the deterministic control equation is a relaxed control equation
assuming

$A(f)(x,v)=f^{\prime}(x)v\mbox{ and }\lambdabar(dv,ds)=\delta_{v(s)}(dv)ds.$
(5.18)

Moreover, in our case the operator $H$ is $H(x,f^{\prime}(x))$ for each
$x\in[0,1]$, and (5.9) can be rewritten in terms of operator $A$ as

$H(f)(x)=H(x,f^{\prime}(x))=\sup_{u}\left(A(f)(x,u)-L(x,u)\right).$ (5.19)

Let $\mathcal{Y}$ be the set of all such pairs $(\mathbf{x},\lambdabar)$.
Instead of (5.11) the relaxed controls determine the Nisio semigroup as

$\mathbf{V}(t)f(x)=\sup_{(\mathbf{x},\lambdabar)\in\mathcal{Y},\
\mathbf{x}(0)=x}f(\mathbf{x}(t))-\iint_{U\times[0,1]}L(\mathbf{x}(s),u)\lambdabar(du\times
ds)$ (5.20)

Theorem 3.1 (section 4.3.2, see [4]) provides the variational and control
representation of the rate function.

We finish the section with Conditions 8.9 and 8.11 from [9].

Condition 8.9 [9]

  1. (1)

$A\subset C_{b}(E)\times C(E\times U)$ is single-valued and $Dom(A)$ separates
points.

  2. (2)

$\Gamma\subset E\times U$ is closed, and for each $x_{0}\in E$, there exists
$(\mathbf{x},\lambdabar)$ such that $\mathbf{x}(0)=x_{0}$ and

$\int_{U\times[0,t]}\mathbb{I}_{\Gamma}(\mathbf{x}(s),u)\lambdabar(du\times
ds)=t,$

where $\mathbb{I}_{B}(\cdot)$ is an indicator of a set $B$.

  3. (3)

$L(x,u):E\times U\to[0,\infty]$ is a lower semicontinuous function, and for
each $c\in[0,\infty)$ and compact $K\subset E$,

$\\{(x,u)\in\Gamma:\ L(x,u)\leq c\\}\cap(K\times U)$

is relatevely compact.

  4. (4)

For each compact $K\subset E,T>0$, and $0\leq M<\infty$, there exists a
compact $\hat{K}\equiv\hat{K}(K,T,M)\subset E$ such that $\mathbf{x}(0)\in K$,

$\int_{U\times[0,t]}\mathbb{I}_{\Gamma}(\mathbf{x}(s),u)\lambdabar(du\times
ds)=t,\ \mbox{ and }\ \int_{U\times[0,T]}L(\mathbf{x}(s),u)\lambdabar(du\times
ds)\leq M$

imply $\mathbf{x}(t)\in\hat{K},\ 0\leq t\leq T$.

  5. (5)

For each $f\in Dom(A)$ and compact set $K\subset E$, there exists a right
continuous, nondecreasing function $\psi_{f,K}:\ [0,\infty)\to[0,\infty)$ such
that

$|Af(x,u)|\leq\psi_{f,K}(L(x,u)),\ \ \forall\ (x,u)\in\Gamma\cap(K\times U),$

and

$\lim_{r\to\infty}\frac{\psi_{f,K}(r)}{r}=0.$

Condition 8.11 [9] For each $x_{0}\in E$ and each $f\in Dom(\mathcal{H})$,
there exists $(\mathbf{x},\lambdabar)$ with $\mathbf{x}(0)=x_{0}$ and

$\int_{U\times[0,t]}\mathbb{I}_{\Gamma}(\mathbf{x}(s),u)\lambdabar(du\times
ds)=t,$

such that for $0\leq t_{1}<t_{2}$

$\int_{t_{1}}^{t_{2}}\mathcal{H}f(\mathbf{x}(s))ds\leq\int_{U\times(t_{1},t_{2}]}\left(Af(\mathbf{x}(s),u)-L(\mathbf{x}(s),u)\right)\lambdabar(ds\times
du).$

These conditions are easily verified in our case. See also [4].

## Appendix 3

Consider some cases.

Case $\boldsymbol{\mathfrak{\gamma}}(0)=\boldsymbol{\mathfrak{\gamma}}(T)=y$.
In this case the solution is constant,
$\boldsymbol{\mathfrak{\gamma}}(t)\equiv\boldsymbol{\mathfrak{\gamma}}(0)$.

Case
$0\leq\boldsymbol{\mathfrak{\gamma}}(0)<\boldsymbol{\mathfrak{\gamma}}(T)$.
Consider first the case when $\boldsymbol{\mathfrak{\gamma}}(0)=0$ and
$\boldsymbol{\mathfrak{\gamma}}(T)\neq 0$. In this case the formulas are
simple

$\displaystyle c_{1}=0,\ \
c_{2}=\frac{\boldsymbol{\mathfrak{\gamma}}(T)}{\lambda T(\lambda T+1)}$
$\displaystyle x(t)=\lambda t(\lambda
t+1)\frac{\boldsymbol{\mathfrak{\gamma}}(T)}{\lambda T(\lambda T+1)}$

In Figure 2 we plot four the function $\boldsymbol{\mathfrak{\gamma}}(t)$ for
case when $T=2$ and
$\boldsymbol{\mathfrak{\gamma}}(2)=0.1,\boldsymbol{\mathfrak{\gamma}}(2)=0.3,\boldsymbol{\mathfrak{\gamma}}(2)=0.5$
and $\boldsymbol{\mathfrak{\gamma}}(2)=0.9$.

Figure 2: Here we plot the optimal paths for the case $T=2$ and boundaries
conditions $\boldsymbol{\mathfrak{\gamma}}(0)=0$ and
$\boldsymbol{\mathfrak{\gamma}}(0)<\boldsymbol{\mathfrak{\gamma}}(2)\in\\{01,0.3,0.5,0.9\\}$.
The upper small plot shows the whole parabolas – solutions of system of
Hamiltonian equations.  Figure 3: Here we plot the optimal paths for the case
$T=2$ and boundaries conditions $\boldsymbol{\mathfrak{\gamma}}(0)=0$ and
$\boldsymbol{\mathfrak{\gamma}}(2)\in\\{0.0,0.3,0.5,0.7,1.0\\}$. In order to
show whole parabolas, we add the upper small plot, which shows the same
parabolas in larger ”time” interval $t\in[-10,10]$. Rectangle outlined in red
is the area of optimal trajectories $\boldsymbol{\mathfrak{\gamma}}(t)$:
$[0,1]\times[0,T]$.

We should also note that when values $\boldsymbol{\mathfrak{\gamma}}(0)$ and
$\boldsymbol{\mathfrak{\gamma}}(T)$ approach each other, then the parabola
$\boldsymbol{\mathfrak{\gamma}}(t)$ transforms into horizontal line
$\boldsymbol{\mathfrak{\gamma}}(t)\equiv\boldsymbol{\mathfrak{\gamma}}(0)$.

If $\boldsymbol{\mathfrak{\gamma}}(0)>0$ and
$\boldsymbol{\mathfrak{\gamma}}(0)<\boldsymbol{\mathfrak{\gamma}}(T)$, then
$c_{1}$ we find as the solution of quadratic equation

$c_{1}^{2}+c_{1}\left(\frac{2\lambda
T\boldsymbol{\mathfrak{\gamma}}(0)}{\boldsymbol{\mathfrak{\gamma}}(T)-\boldsymbol{\mathfrak{\gamma}}(0)}-1\right)-\frac{\lambda
T(\lambda
T+1)\boldsymbol{\mathfrak{\gamma}}(0)}{\boldsymbol{\mathfrak{\gamma}}(T)-\boldsymbol{\mathfrak{\gamma}}(0)}=0,\
\ c_{2}=\frac{\boldsymbol{\mathfrak{\gamma}}(0)}{c_{1}(c_{1}-1)}$ (5.21)

The value of $c_{1}$ which provides the admissible function
$\boldsymbol{\mathfrak{\gamma}}(t)$, i.e.
$\boldsymbol{\mathfrak{\gamma}}(t)\in[0,1]$ for all $t\in[0,T]$ is

$c_{1}=\frac{1}{2}-\frac{\lambda
T\boldsymbol{\mathfrak{\gamma}}(0)}{\boldsymbol{\mathfrak{\gamma}}(T)-\boldsymbol{\mathfrak{\gamma}}(0)}-\sqrt{\left(\frac{1}{2}-\frac{\lambda
T\boldsymbol{\mathfrak{\gamma}}(0)}{x(T)-\boldsymbol{\mathfrak{\gamma}}(0)}\right)^{2}+\frac{\lambda
T(\lambda
T+1)\boldsymbol{\mathfrak{\gamma}}(0)}{\boldsymbol{\mathfrak{\gamma}}(T)-\boldsymbol{\mathfrak{\gamma}}(0)}},\
\ c_{2}=\frac{\boldsymbol{\mathfrak{\gamma}}(0)}{c_{1}(c_{1}-1)}$

For the time moment $T=2$ see Figure 3. The here considered case corrsponds to
the boundary conditions $x(0)=0.5,x(2)=0.7$ and $x(0)=0.5,x(2)=1$ in the plot.

Case
$0\leq\boldsymbol{\mathfrak{\gamma}}(T)<\boldsymbol{\mathfrak{\gamma}}(0)$.
Consider first the case when $\boldsymbol{\mathfrak{\gamma}}(T)=0$ and
$\boldsymbol{\mathfrak{\gamma}}(0)\neq 0$. In this case constants which
defines the admissible trajectories are

$c_{1}=\lambda T+1,\ \ c_{2}=\frac{\boldsymbol{\mathfrak{\gamma}}(0)}{\lambda
T(\lambda T+1)}\mbox{ and
}\boldsymbol{\mathfrak{\gamma}}(t)=\lambda(T-t)(\lambda(T-t)+1)\frac{x(0)}{\lambda
T(\lambda T+1)}$

Example can be found in Figure 3 for the case $T=2$ and boundary condition
$\boldsymbol{\mathfrak{\gamma}}(0)=1/2$ and
$\boldsymbol{\mathfrak{\gamma}}(2)=0$. The optimal trajectories are parabolas
again, and they are symmetric version of the Figure 2.

If $\boldsymbol{\mathfrak{\gamma}}(T)>0$ and
$\boldsymbol{\mathfrak{\gamma}}(0)>\boldsymbol{\mathfrak{\gamma}}(T)$, then
$c_{1}$ we find as the solution of the same quadratic equation (5.21). The
root of (5.21) which provides the admissible function
$\boldsymbol{\mathfrak{\gamma}}(t)$ is

$c_{1}=\frac{1}{2}-\frac{\lambda
Tx(0)}{\boldsymbol{\mathfrak{\gamma}}(T)-\boldsymbol{\mathfrak{\gamma}}(0)}+\sqrt{\left(\frac{1}{2}-\frac{\lambda
T\boldsymbol{\mathfrak{\gamma}}(0)}{\boldsymbol{\mathfrak{\gamma}}(T)-\boldsymbol{\mathfrak{\gamma}}(0)}\right)^{2}+\frac{\lambda
T(\lambda
T+1)\boldsymbol{\mathfrak{\gamma}}(0)}{\boldsymbol{\mathfrak{\gamma}}(T)-\boldsymbol{\mathfrak{\gamma}}(0)}},\
\ c_{2}=\frac{\boldsymbol{\mathfrak{\gamma}}(0)}{c_{1}(c_{1}-1)}$

As an example, see Figure:3 with $T=2$,
$\boldsymbol{\mathfrak{\gamma}}(0)=0.5,\boldsymbol{\mathfrak{\gamma}}(2)=0.3$
and
$\boldsymbol{\mathfrak{\gamma}}(0)=0.5,\boldsymbol{\mathfrak{\gamma}}(2)=0.0$.

Observe that extremal points of the corresponding parabolas are greater then
$T$, when these extremal points in the case
$\boldsymbol{\mathfrak{\gamma}}(0)<\boldsymbol{\mathfrak{\gamma}}(T)$ are
negative.

## References

  * [1] D. Gorbunov, V. Rubakov (2011) Introduction to the theory of the early Universe, World Scientific.
  * [2] V.A. Rubakov, Cosmology, arXiv:1804.11230v1
  * [3] E. Sather (1996) The Mystery of the matter asymmetry, SLAC Beam Line 26N1, 31-37 Report number: SLAC-REPRINT-1996-074; www.slac.stanford.edu pubs beamline
  * [4] Bermolen, P., Goicoechea, V., Jonckheere, M., and Mordecki, E. (2022). Large Deviation Principle for the Greedy Exploration Algorithm over Erdös-Rényi Graphs. ALEA, Lat. Am. J. Probab. Math. Stat. 19, 439-456.
  * [5] Billingsley, P. (2008). Probability and measure. John Wiley & Sons.
  * [6] A. A. Borovkov, E. A. Pechersky (1975) Convergence of the distributions of integral functionals, Siberian Math. J., 16:5, 685–698
  * [7] Dobrushin, R. L. and Pechersky, E. A. (1994). Large deviations for tandem queueing systems. Journal of Applied Mathematics and Stochastic Analysis, 7(3), 301-330.
  * [8] Ethier, S. N. and Kurtz, T. G. (2009). Markov processes: characterization and convergence. John Wiley & Sons.
  * [9] Feng, J., and Kurtz, T. G. (2006). Large deviations for stochastic processes (No. 131). American Mathematical Soc..
  * [10] Fleming, W. H. (1985). A stochastic control approach to some large deviations problems. In Recent Mathematical Methods in Dynamic Programming (pp. 52-66). Springer, Berlin, Heidelberg.
  * [11] Kraaij, R. (2016). Large deviations for finite state Markov jump processes with mean-field interaction via the comparison principle for an associated Hamilton–Jacobi equation. Journal of Statistical Physics, 164(2), 321-345.
  * [12] Logachov, A. V., Suhov, Y. M., Vvedenskaya, N. D., and Yambartsev, A. A. (2021). A principle of large deviations for birth-death processes with a linear rate of downward jumps. arXiv preprint arXiv:2112.05877.
  * [13] Logachov, A. V. E., Sukhov, Y. M., Vvedenskaya, N. D., and Yambartsev, A. A. (2020). A remark on normalizations in a local large deviations principle for inhomogeneous birth–and–death process. Siberian Electron. Math. Reports, 17(0), 1258-1269.
  * [14] Logachov, A. V. E., Suhov, Y. M., Vvedenskaya, N. D., and Yambartsev, A. A. (2018). A Local Large Deviation Principle for Inhomogeneous Birth–Death Processes. Problems of Information Transmission, 54(3), 263-280.
  * [15] Mogulskii, A., Pechersky, E., and Yambartsev, A. (2014). Large deviations for excursions of non-homogeneous Markov processes. Electronic Communications in Probability, 19, 1-8.
  * [16] Sakharov, A. D. (1991) Violation of CP invariance, C asymmetry, and baryon asymmetry of the Universe. Pis’ma Z. Eksp. Teor. Fiz. 5, 32–35 (1967); Sov. Phys. Usp. 34, 392–393.
  * [17] The $T2K$ Collaboration (2020) Constraint on the matter–antimatter symmetry-violating phase in neutrino oscillations. Nature, v.580, n.7803, p. 339–344. https://doi.org/10.1038/s41586-020-2177-0