# Large deviations for infectious diseases models

Peter Kratz and Etienne Pardoux111Aix-Marseille Université, CNRS, Centrale
Marseille, I2M, UMR 7373, 13453 Marseille, France, e-mail adress:
<EMAIL_ADDRESS>
##  1 Introduction

Consider a model of infectious disease dynamics where the total number of
individuals is constant over time, equal to $N$, and we denote by $Z^{N}(t)$
the vector of proportions of this population in each compartment (susceptible,
infectious, removed, etc.). Our probabilistic model takes into account each
event of infection, removal, etc. It takes the form

$Z^{N,x}(t):=Z^{N}(t):=x+\frac{1}{N}\sum_{j=1}^{k}h_{j}P_{j}\Big{(}\int_{0}^{t}N\beta_{j}(Z^{N}(s))ds\Big{)}.$
(1)

Here, $(P_{j})_{1\leq j\leq k}$ are i.i.d. standard Poisson processes. The
$h_{j}\in\mathds{Z}^{d}$ denote the $k$ respective jump directions with jump
rates $\beta_{j}(x)$ and $x\in A$ (where $A$ is the “domain” of the process).
The $d$ components of the process denote the “proportions” of individuals in
the various compartments. Usually $A$ is a compact or at least a bounded set.
For example, in the models we have in mind the compartment sizes are non-
negative, hence $A\subset\mathds{R}^{d}_{+}$.

As we shall prove below, $Z^{N,x}_{t}\to Y^{x}_{t}$ as $N\to\infty$, where
$Y^{x}_{t}$ is the solution of the ODE

$Y^{x}(t):=Y(t):=x+\int_{0}^{t}b(Y^{x}(s))ds,$ (2)

with

$b(z):=\sum_{j=1}^{k}\beta_{j}(z)h_{j}.$

This Law of Large Numbers result goes back to Kurtz (1978) (see the version in
Theorem 3.1 below, where a rate of convergence is given).

Most of the literature on mathematical models of disease dynamics treats
deterministic models of the type of (2). When an epidemics is established, and
each compartment of the model contains a significant proportion of the total
population, if $N$ is large enough, the ODE (2) is a good model for the
epidemics. The original stochastic model (1), which we believe to be more
realistic than the (2), can be considered as a stochastic perturbation of (2).
However, we know from the work of Freidlin and Wentzell (2012), that small
Brownian perturbations of an ODE will eventually produce a large deviation
from its law of large numbers limit. For instance, if the ODE starts in the
basin of attraction of an locally stable equilibrium, the solution of the ODE
converges to that equilibrium, and stays for ever close to that equilibrium.
The theory of Freidlin and Wentzell, based upon the theory of Large
Deviations, predicts that soon or later the solution of a random perturbation
of that ODE will exit the basin of attraction of the equilibrium. The aim of
this paper is to show that the Poissonian perturbation (1) of (2) behaves
similarly. This should allow us to predicts the time taken by an endemic
equilibrium to cease, and a disease–free equilibrium to replace it.

We shall apply at the end of this paper our results to the following example.

######  Example 1.1.

We consider a so-called $SIRS$ model without demography ($S(t)$ being the
number of susceptible individuals, $I(t)$ the number of infectious individuals
and $R(t)$ the number of removed/immune individuals at time $t$). We let
$\beta>0$ and assume that the average number of new infections per unit time
is $\beta S(t)I(t)/N$.222The reasoning behind this is the following. Assume
that an infectious individuals meets on average $\alpha>0$ other individuals
in unit time. If each contact of a susceptible and an infectious individual
yields a new infection with probability $p$, the average number of new
infections per unit time is $\beta S(t)I(t)/N$, where $\beta=p\alpha$ since
all individuals are contacted with the same probability (hence $S(t)/N$ is the
probability that a contacted individual is susceptible). For $\gamma,\nu>0$,
we assume that the average number of recoveries per unit time is $\gamma I(t)$
and the average number of individuals who lose immunity is $\nu R(t)$. As
population size is constant, we can reduce the dimension of the model by
solely considering the proportion of infectious and removed at time $t$. Using
the notation of equations (1) and (2), we have

$\displaystyle A=\\{x\in\mathds{R}^{2}_{+}|0\leq x_{1}+x_{2}\leq 1\\},\quad
h_{1}=(1,0)^{\top},\quad h_{2}=(-1,1)^{\top},\quad h_{3}=(0,-1)^{\top},$
$\displaystyle\beta_{1}(z)=\beta z_{1}(1-z_{1}-z_{2}),\quad\beta_{2}(z)=\gamma
z_{1},\quad\beta_{3}(z)=\nu z_{2}.$

It is easy to see that in this example the ODE (2) has a disease free
equilibrium $\bar{x}=(0,0)^{\top}$. This equilibrium is asymptotically stable
if $R_{0}=\beta/\gamma<1$. $R_{0}$ is the so-called _basic reproduction
number_. It denotes the average number of secondary cases infected by one
primary case during its infectious period at the start of the epidemic (while
essentially everybody is susceptible). If $R_{0}>1$, $\bar{x}$ is unstable and
there exists a second, endemic equilibrium
$x^{*}=(\frac{\nu(\beta-\gamma)}{\beta(\gamma+\nu)},\frac{\gamma(\beta-\gamma)}{\beta(\gamma+\nu})$
which is asymptotically stable. While in the deterministic model the
proportion of infectious and removed individuals converges to the endemic
equilibrium $x^{*}$, the disease will go extinct soon or later in the
stochastic model.

$S$$I$$R$$\beta SI/N$(infection rate)$\nu R$(loss of immunity)$\gamma
I$(recovery rate) Figure 1: Transmission diagram of the $SIRS$ model without
demography.

Our results also apply to other models like the $SIV$ model ($V$ like
_vaccinated_) and the $S_{0}IS_{1}$ (with two levels of susceptibility), see
Kratz et al. (2015) and the references therein. These two models have the
property that for certain values of their parameters, both the disease–free
equilibrium and one of the endemic equilibria are locally stable. Our results
predict the time taken by the solution of the stochastic equation to leave the
basin of attraction of the endemic equilibrium. We shall discuss those and
other applications elsewhere in the future.

There is already a vast literature on the theory of large deviations for
systems with Poissonian inputs, see Dupuis and Ellis (1997), Dupuis et al.
(1991), Feng and Kurtz (2006), Shwartz and Weiss (1995), among others.

However, the assumptions in those works are not satisfied in our case. The
difficulty is the following. For obvious reasons, the solution of our SDE (1)
must remain in $\mathds{R}^{d}_{+}$. This implies that some rates vanish when
one of the components of $Z^{N,x}(t)$ vanishes. However, the expression of the
large deviation fucntional (as well as the ratio of the probabilities in the
Girsanov theorem) involves the logarithm of those rates, which hence explodes
as the rate vanishes. The same happens with the computer network models which
was the motivation of the work of Shwartz and Weiss (1995), and this led them
to consider situations with vanishing rates in Shwartz and Weiss (2005).
However, even the assumptions in that paper are not fully satisfied in our
models (see our discussion below in section 2.2). For that reason, in order to
avoid the ackward situation where we would have to cite both Shwartz and Weiss
(1995) and Shwartz and Weiss (2005), and add some arguments to cope with our
specific situation, we prefered to rewrite the whole theory, so as to cover
the situation of the epidemiological models in a self–consistent way. We must
however recognize that the work of Shwartz and Weiss has been an importance
source of inspiration for this work.

Let us now discuss one subtlety of our models. In the models without
demography, i.e. the models where the total population remains constant, then
we choose $N$ as this total population, so that the various components of the
vector $Z^{N,x}_{t}$ are the proportions of the total population in the
various compartments of the model, that is each component of $Z^{N,x}_{t}$ at
any time is of the form $k/N$, where $k\in\mathds{Z}_{+}$, and also
$\sum_{i=1}^{d}Z^{N,x}_{i}(t)=1$, if $Z^{N,x}_{i}(t)$ denotes the $i$–th
component of the vector $Z^{N,x}(t)$, $1\leq i\leq d$. In this case, provided
we start our SDE from a point of the type $(k_{1}/N,\ldots,k_{d}/N)$, where
$k_{1},\ldots,k_{d}\in\mathds{Z}_{+}$, then the solution visits only such
points, and cannot escape the set $\mathds{R}_{+}^{d}$ without hitting first
its boundary, where the rates for exiting vanish. Consequently $Z^{N,x}_{t}$
remains in $\mathds{R}_{+}^{d}$ for all time a.s. However, if we start our
process outside the above grid, or if the total population size does not
remain constant, the components of the vector $Z^{N,x}_{t}$ multiplied by $N$
need not be integers. Then some of the components of $Z^{N,x}_{t}$ might
become negative, and one can still continue to define $Z^{N,x}_{t}$ provided
for any $1\leq i\leq d$ and $1\leq j\leq k$, the rule
$x_{i}=0\Rightarrow\beta_{j}(x)=0$ is extended to $x_{i}\leq
0\Rightarrow\beta_{j}(x)=0$. However, in order to make things simpler, we
restrict ourselves in this paper to the situation where all coordinates of the
vector $NZ^{N,x}_{0}$ are integers, and the same is true with $NZ^{N,x}_{t}$
for all $t>0$. In particular, we shall consider equation (1) only with a
starting point $x$ such that all coordinates of $Nx$ are integers. This will
be explicitly recalled in the main statements, and implicitly assumed
everywhere. We shall consider more general situations in a further
publication.

The paper is organized as follows Our set–up is made precise and the general
assumptions are formulated in section 2. Section 3 is devoted to the law of
large numbers. In section 4 we study the rate function. The Large Deviations
lower bound is established in section 5 and the Large Deviations upper bound
in section 6. Section 7 treats the exit time from a domain, including the case
of a characteristic boundary. Finally in section 8 we show how our results
apply to the SIRS model (which requires an additional argument), and a
Girsanov theorem is formulated in the Appendix.

##  2 Set-up

We consider a set $A\subset\mathds{R}^{d}$ (whose properties we specify below)
and define the grids

$\mathds{Z}^{d,N}:=\\{x\in\mathds{R}^{d}|x_{i}=j/N\text{ for some
}j\in\mathds{Z}\\},\quad A^{N}:=A\cap\mathds{Z}^{d,N}.$

We rewrite the process defined by Equation (1) as

$\displaystyle Z^{N,x}(t):=Z^{N}(t)$
$\displaystyle:=x+\frac{1}{N}\sum_{j=1}^{k}h_{j}P_{j}\Big{(}\int_{0}^{t}N\beta_{j}(Z^{N}(s))ds\Big{)}$
(3)
$\displaystyle=x+\int_{0}^{t}b(Z^{N}(s))ds+\frac{1}{N}\sum_{j}h_{j}M_{j}\Big{(}\int_{0}^{t}N\beta_{j}(Z^{N}(s))ds\Big{)},$

where the $M_{j}(t)=P_{j}(t)-t$ are the compensated Poisson processes
corresponding to the $P_{j}$ ($j=1,\dots,k$).

Let us shortly comment on this definition. In the models we have in mind, the
components of $Z^{N}$ usually denote the proportions of individuals in the
respective compartments. It is hence plausible to demand that the starting
point is in $\mathds{Z}^{d,N}$. However, it is not sufficient to simply
restrict our analysis to those starting points with $x\in A^{N}$ as this does
obviously not imply $x\in A^{M}$ for all $M>N$. Note that $U^{N}(t)=NZ^{N}(t)$
would solve the SDE

$U^{N}(t)=Nx+\sum_{j=1}^{k}h_{j}P_{j}\left(\int_{0}^{t}\beta_{j,N}(U^{N}(s))\right),$

where $N\beta_{j}(x)=\beta_{j,N}(Nx)$. Here the coefficients of the vector
$U^{N}(t)$ are the numbers of individuals from the population in each
compartment. The equation for $U^{N}(t)$ is really the original model, where
all events of infection, recovery, loss of immunity, etc. are modeled.
Dividing by $N$ leads to a process which has a law of large number limits as
$N\to\infty$. The crucial assumption for this procedure to make sense is that
$N^{-1}\beta_{j,N}(Nx)$ does not depend upon $N$, which is typically the case
in the epidemics models, see in particular Example 1.1.

We first introduce the following notations. For $x\in A$ and
$y\in\mathds{R}^{d}$, let

$\displaystyle V_{x}$
$\displaystyle:=\Big{\\{}\mu\in\mathds{R}^{k}_{+}|\mu_{j}>0\text{ only if
}\beta_{j}(x)>0\Big{\\}},$ $\displaystyle V_{x,y}$
$\displaystyle:=\Big{\\{}\mu\in V_{x}|y=\sum_{j}\mu_{j}h_{j}\Big{\\}}.$

As $V_{x,y}$ is sometimes independent of $x$ or $V_{x,y}=\emptyset$, we also
define for $y\in\mathds{R}^{d}$,

$\tilde{V}_{y}:=\Big{\\{}\mu\in\mathds{R}^{d}_{+}|y=\sum_{j}\mu_{j}h_{j}\Big{\\}}$

We define the cone spanned by a (finite) set of vectors $(v_{j})_{j}$
($v_{j}\in\mathds{R}^{d}$ by

$\mathcal{C}((v_{j})_{j}):=\Big{\\{}v=\sum_{j}\alpha_{j}v_{j}|\alpha_{j}\geq
0\Big{\\}}.$

Similarly, we define the cone generated by the jump directions $(h_{j})_{j}$
at $x\in A$ by

$\mathcal{C}_{x}:=\mathcal{C}((h_{j})_{j:\beta_{j}(x)>0})=\Big{\\{}v=\sum_{j:\beta_{j}(x)>0}\mu_{j}h_{j}|\mu_{j}\geq
0\Big{\\}}.$

Note that

$\mathcal{C}_{x}=\mathcal{C}=\\{v=\sum_{j=1}^{k}\mu_{j}h_{j}|\mu_{j}\geq 0\\}$
(4)

whenever $x\in\mathring{A}$, since $\beta_{j}(x)>0$ for all $1\leq j\leq k$ if
$x\in\mathring{A}$. Also, in part of this paper, we shall assume that the
$\log\beta_{j}$’s are bounded, which then means that (4) is true for all $x\in
A$.

We define the following upper and lower bounds of the rates. Let $\rho>0$.

$\displaystyle\bar{\beta}$ $\displaystyle:=\sup_{x\in
A,\,j=1,\dots,k}\beta_{j}(x)\in\bar{\mathds{R}}_{+},$
$\displaystyle\underline{\beta}$ $\displaystyle:=\inf_{x\in
A,\,j=1,\dots,k}\beta_{j}(x)\in\mathds{R}_{+},$
$\displaystyle\underline{\beta}(\rho)$
$\displaystyle:=\inf\big{\\{}\beta_{j}(x)|j=1,\dots,k,\,x\in A\text{ and
}|x-z|\geq\rho\forall z\in A\text{ with
}\beta_{j}(z)=0\big{\\}}\in\mathds{R}_{+},$ $\displaystyle\bar{h}$
$\displaystyle:=\sup_{j=1,\dots,k}|h_{j}|\in\mathds{R}_{+}.$

###  2.1 The Legendre-Fenchel transform and the rate function

We define the following transforms. For $x\in A$, $y\in\mathbb{R}^{d}$, let

$\displaystyle\ell(x,\mu)$
$\displaystyle:=\sum_{j}\big{\\{}\beta_{j}(x)-\mu_{j}+\mu_{j}\log\big{(}\tfrac{\mu_{j}}{\beta_{j}(x)}\big{)}\big{\\}},$

with the convention $0\log(0/\alpha)=0$ for all $\alpha\in\mathds{R}$, and

$\overline{L}(x,y):=\begin{cases}\inf_{\mu\in V_{x,y}}\ell(x,\mu)&\text{if
}V_{x,y}\not=\emptyset\\\ +\infty&\text{otherwise}.\end{cases}$ (5)

Now let, for $x,y$ as above and $\theta\in\mathbb{R}^{d}$,

$\tilde{\ell}(x,y,\theta)=\langle\theta,y\rangle-\sum_{j}\beta_{j}(x)\big{(}\operatorname{e}^{\langle\theta,h_{j}\rangle}-1\big{)},$

and define

$\displaystyle\underline{L}(x,y)$
$\displaystyle:=\sup_{\theta\in\mathds{R}^{d}}\tilde{\ell}(x,y,\theta),$ (6)

######  Remark 2.1.

For $\mu\in\tilde{V}_{y}\setminus V_{x,y}$, we have $\ell(x,\mu)=\infty$ and
hence

$L(x,y)=\inf_{\mu\in\tilde{V}_{y}}\ell(x,\mu).$

We first show

######  Lemma 2.2.

Let $x\in A$, $y\in\mathcal{C}_{x}$, $\theta\in\mathds{R}^{d}$ and $\mu\in
V_{x,y}$. Then

$\tilde{\ell}(x,y,\theta)\leq\ell(x,\mu),$

in particular

$\underline{L}(x,y)\leq\overline{L}(x,y).$

###### Proof.

The result is obvious if $V_{x,y}=\emptyset$. If not, for $\mu\in V_{x,y}$,
with

$f_{j}(z)=\mu_{j}z-\beta_{j}(x)(e^{z}-1),$
$\displaystyle\tilde{\ell}(x,y,\theta)$
$\displaystyle=\sum_{j}\mu_{j}\langle\theta,h_{j}\rangle-\beta_{j}(x)(\exp(\langle\theta,h_{j}\rangle)-1)$
$\displaystyle=\sum_{j}f_{j}(\langle\theta,h_{j}\rangle)$
$\displaystyle\leq\sum_{j}f_{j}(\log\mu_{j}/\beta_{j}(x))$
$\displaystyle=\ell(x,\mu),$

since $f_{j}$ achieves its maximum at $z=\log[\mu_{j}/\beta_{j}(x)]$. ∎

We will show below that under appropriate assumptions,
$\underline{L}(x,y)=\overline{L}(x,y)$, and we shall write $L(x,y)$ for the
common value of those two quantities.

For any $T>0$, we define

$\displaystyle C([0,T];A)$ $\displaystyle:=\\{\phi:[0,T]\rightarrow
A|\phi\text{ continuous}\\},$ $\displaystyle D([0,T];A)$
$\displaystyle:=\\{\phi:[0,T]\rightarrow A|\phi\text{ c\\`{a}dl\\`{a}g}\\}.$

On $C([0,T];A)$ (or $D([0,T];A)$), $\operatorname{d}_{C}$ denotes the metric
corresponding to the supremum-norm, denoted by $\|\cdot\|$. Whenever the
context is clear, we write $\operatorname{d}:=\operatorname{d}_{C}$. On
$D([0,T];A)$ we denote by $\operatorname{d}_{D}$ the metric given, e.g., in
Billingsley (1999), Sections 12.1 and 12.2 which defines the Skorohod topology
in such a way that the resulting space is Polish. The resulting metric spaces
are denoted by $C([0,T];A;\operatorname{d}_{C})$,
$D([0,T];A;\operatorname{d}_{C})$ and $D([0,T];A;\operatorname{d}_{D})$,
respectively (where the metrics are omitted, whenever they are clear from the
context).

We now introduce a candidate $I$ for the rate function. For
$\phi:[0,T]\rightarrow A$, let

$\displaystyle I_{T}(\phi)$
$\displaystyle:=\begin{cases}\int_{0}^{T}L(\phi(t),\phi^{\prime}(t))dt&\text{
if }\phi\text{ is absolutely continuous}\\\ \infty&\text{
otherwise.}\end{cases}$

For $x\in A$ and $\phi:[0,T]\rightarrow A$, let

$I_{T,x}(\phi):=\begin{cases}I_{T}(\phi)&\text{ if }\phi(0)=x\\\ \infty&\text{
otherwise.}\end{cases}$

###  2.2 Assumptions on the process $Z^{N}$

We do not assume that the rates are bounded away form zero (as in Shwartz and
Weiss (1995)) and allow for them to vanish near the boundary (cf. the
discussion in the introduction). Shwartz and Weiss (2005) allow for vanishing
rates. We generalize these assumptions as we outline below. The difference is
essentially Assumption 2.3 (C) below.

######  Assumption 2.3.

  1. (A)

Assumptions on the set $A$.

    1. (A1)

The set $A$ is compact and $A=\overline{\mathring{A}}$. Furthermore, there
exists a constant $\lambda_{0}>0$ such that for all $N\in\mathds{N}$, $z\in
A^{N}$ and $j=1,\dots,k$ with $\beta_{j}(z)>0$,

$z+\frac{h_{j}}{N}\in
A^{N}\quad\text{and}\quad|\tilde{z}-z|\geq\frac{\lambda_{0}}{N}\text{ for all
}\tilde{z}\text{ with }\beta_{j}(\tilde{z})=0.$

    2. (A2)

There exist open balls $B_{i}=B(x_{i},r_{i})$444Here (and later) $B(x,r)$
denotes the open ball around $x$ with radius $r$., $i=1,\dots,I_{1},\dots,I$
($0<I_{1}<I$) such that

$x_{i}\in\partial A\text{ for }i\leq I_{1}\quad\text{and}\quad
x_{i}\in\mathring{A}\text{ for }i>I_{1}$

and

$A\subset\bigcup_{i\leq I}B_{i},\quad\partial A\subset\bigcup_{i\leq
I_{1}}B_{i}\quad\text{and}\quad B_{i}\cap\partial A=\emptyset\text{ for
}i>I_{1}.$

    3. (A3)

There exist (universal) constants $\lambda_{1},\lambda_{2}>0$ and vectors
$v_{i}$ ($i\leq I_{1}$, w.l.o.g., we assume $0<|v_{i}|\leq 1$; for notational
reasons, we set $v_{i}=0$ for $i>I_{1}$) such that for all $x\in B_{i}\cap A$,

$B(x+tv_{i},\lambda_{1}t)\subset A\quad\text{for all }t\in(0,\lambda_{2}).$

and $\operatorname{dist}(x+tv_{i},\partial A)$ is increasing for
$t\in(0,\lambda_{2})$.

    4. (A4)

There exists a Lipschitz continuous mapping $\psi_{A}:\mathds{R}^{d}\to A$
such that $\psi_{A}(x)=x$ whenever $x\in A$.

  2. (B)

Assumptions on the rates $\beta_{j}$.

    1. (B1)

The rates $\beta_{j}:A\rightarrow\mathds{R}_{+}$ are Lipschitz continuous.

    2. (B2)

For $x\in\mathring{A}$, $j=1,\dots,k$, $\beta_{j}(x)>0$ and
$\mathcal{C}((h_{j})_{j})=\mathds{R}^{d}$.

    3. (B3)

For all $x\in\partial A$ there exists a constant
$\lambda_{3}=\lambda_{3}(x)>0$ such that

$y\in\mathcal{C}_{x},|y|\leq\lambda_{3}\Rightarrow x+y\in A.$

    4. (B4)

There exists a (universal) constant $\lambda_{4}>0$ such that for all $i\leq
I_{1}$, $x\in B_{i}\cap A$ and

$v\in\mathcal{C}_{1,i}:=\Big{\\{}\frac{\tilde{v}}{|\tilde{v}|}\Big{|}\tilde{v}=v_{i}+w\text{
for }w\in\mathds{R}^{d},|w|\leq\frac{\lambda_{1}}{3-\lambda_{1}}\Big{\\}},$
$\beta_{j}(x)<\lambda_{4}\Rightarrow\beta_{j}(x+\cdot v)\text{ is increasing
in }(0,\lambda_{2}).$

  3. (C)

There exists an $\eta_{0}>0$ such that for all $N\in\mathds{N}$, $\epsilon>0$
there exists a constant $\delta(N,\epsilon)>0$ (decreasing in $N$ and in
$\epsilon$) such that for all $i\leq I_{1}$, $x\in B_{i}$ there exists a
$\mu^{i}=\mu^{i}(x)\in\tilde{V}_{v_{i}}$ and

$\mathbb{P}\Big{[}\sup_{t\in[0,\eta_{0}]}|\tilde{Z}^{N,x}(t)-\phi^{x}(t)|\geq\epsilon\Big{]}\leq\delta(N,\epsilon),$
(7)

where $\tilde{Z}^{N,x}$ denotes the solution of (3) if the rates $\beta_{j}$
are replaced by the rates $\tilde{\mu}_{j}^{i}$ for

$\tilde{\mu}_{j}^{i}(z):=\begin{cases}\mu_{j}^{i}&\text{ if }z+\epsilon
h_{j}\in A\text{ for all }\epsilon\text{ small enough}\\\
0&\text{else}\end{cases}$

and $\phi^{x}=x+tv_{i}$ as before.555We do not necessarily have $\mu^{i}\in
V_{x,v_{i}}$ for all $x\in B_{i}$; this might not be the case if $x\in\partial
A$. In such a case $V_{x,v_{i}}=\emptyset$ is possible, cf. the discussion
about $x=(1,0)^{\top}$ for the SIRS model below.

Furthermore, there exists a constant $\alpha\in(0,1/2)$ and a sequence
$\epsilon_{N}$ such that

$\epsilon_{N}<\frac{1}{N^{\alpha}}\quad\text{and}\quad\frac{\delta(N,\epsilon_{N})}{\epsilon_{N}}\rightarrow
0\text{ as }N\rightarrow\infty.$ (8)

and

$\rho^{\alpha}\log\underline{\beta}(\rho)\rightarrow 0\quad\text{as
}\rho\rightarrow 0.$ (9)

Let us comment on Assumption 2.3. Assumption (A) is essentially Assumption 2.1
of Shwartz and Weiss (2005). We want to remark that Assumption 2.1 (iv) of
Shwartz and Weiss (2005) is not included here as it is redundant (see Lemma
3.5 of Shwartz and Weiss (2005); cf. also the discussion preceding Lemma 5.5).
In the epidemiological models we want to consider, $A$ is a compact, convex
$d$-polytope, i.e., $\partial A$ is composed by $d-1$-dimensional hyperplanes.
For example for the SIRS model in Example 1.1,

$A=\\{x\in\mathds{R}^{2}|0\leq x_{1}+x_{2}\leq 1\\}.$

In line with the Assumption (A1), let us note that we always want to choose
the starting point $x$ of equation (1) to belong to $A^{N}$. If that would not
be the case, then in our simplest models the solution $Z^{N,x}$ might exit the
domain $A$. Choosing the starting point arbitrarily in $A$ would force us to
let the rates $\beta_{j}$ depend upon $N$ (and vanish) near the boundary. Note
that the coordinates of the vector $Z^{N,x}_{t}$ are proportions of the
population in various compartments. The coordinates of the vector
$NZ^{N,x}_{t}$ are integers, while those of the vectors $h_{j}$ belong to the
set $\\{-1,0,1\\}$.

Note that in all situations we have in mind, both the set $A$ itself and its
boundary can be covered by a finite number of balls. These balls can
furthermore be chosen in such a way that those centered in the interior do not
intersect with the boundary. For the SIRS model, we can for instance define
the balls covering the boundary by $B(x,3/(4m))$ for large $m\in\mathds{N}$
and $x=(i/m,j/m)^{\top}$ for $i=0,\dots,m$, $j=0$ or $i=0$, $j=0,\dots,m$ or
$i+j=m$. The vectors $v_{i}$ can be defined to be the inside normal vectors
for those balls with $x\not\in\\{(0,0)^{\top},(1,0)^{\top},(0,1)^{\top}\\}$.
For the remaining three balls, we define $v_{i}$ by the normalizations of

$(1/2,1/2)^{\top},\quad(-1/2,1/4)^{\top}\quad\text{respectively}\quad(1/2,-1/2)^{\top}.$

In general, the constant $\lambda_{1}$ can be interpreted to be given via the
“angle” of the vector $v_{i}$ to the boundary. We have $\lambda_{1}\leq 1$. It
is straightforward that Assumption (A) is satisfied for the SIRS model. We
also note that Assumption (A) is not very restrictive, see Shwartz and Weiss
(2005) Lemma 2.1. In particular, every convex, compact set with non-empty
interior satisfies the assumption.

Most of Assumption (B) is taken from Assumption 2.2 of Shwartz and Weiss
(2005). We outline the difference below. Assumption (B1) is quite standard and
ensures in particular that the ODE (2) admits a unique solution. For the
compartmental epidemiological models we consider, the rates are usually
polynomials and hence this assumption is satisfied. Assumption (B2) implies
that within $\mathring{A}$ it is possible to move into all directions. Only by
approaching the boundary the rates are allowed to vanish. (B3) implies that at
least locally the convex cone $x+\mathcal{C}_{x}$ is included in $A$. In
particular, it is not possible to exit the set $A$ from its boundary.
Assumption (B4) differs slightly from the corresponding assumption in Shwartz
and Weiss (2005). While in Shwartz and Weiss (2005), it is implied that close
to the boundary, “small” rates are increasing while following the vector
$v_{i}$, we assume this for a set of vector in a “cone” around $v_{i}$. We
note that for $i\leq I_{1}$, $x\in B_{i}$, $v\in\mathcal{C}_{1,i}$, we have
(cf. Assumption (A3))

$\operatorname{dist}(x+tv,\partial A)\geq\operatorname{dist}(x+tv_{i},\partial
A)-t\frac{\lambda_{1}}{3-\lambda_{1}}\ \\\ \geq
t\lambda_{1}\Big{(}\frac{2-\lambda_{1}}{3-\lambda_{1}}\Big{)}.$

It is easily seen that this assumption is satisfied for the SIRS model. In
addition to this, Shwartz and Weiss (2005) also require that (cf. the meaning
of $\lambda_{4}$ in Assumption (B4))

$v_{i}\in\mathcal{C}\big{(}\\{h_{j}|\inf_{x\in
B_{i}}\beta_{j}(x)>\lambda_{4}\\}\big{)}.$ (10)

In order to apply the theory to epidemiological models, we have to remove this
assumption. To see this, consider the SIRS model and the point
$x=(1,0)^{\top}$ with corresponding ball $B$ containing it. We readily observe
that a vector $v$ pointing “inside” $A$ (as required by Assumption (A3)) which
is generated by only those $h_{j}$ whose corresponding rates are bounded away
from zero in $B$ does not exist. We hence replace this assumption by
Assumption (C), which follows from (10). Indeed, if Assumption (10) holds, the
$\mu^{i}$ representing $v_{i}$ can be chosen in such a way that the directions
corresponding to components $\mu^{i}_{j}>0$ do not point outside $A$ in
$B_{i}$. Hence, $\tilde{\mu}^{i}\equiv\mu^{i}$ (as long as the process is in
$B_{i}$) and the LLN Theorem 3.1 can be applied. In general, Theorem 3.1
cannot be applied as the rates $\tilde{\mu}^{i}$ can be discontinuous. Note
that the assumption can only fail if $x\in\partial A$. Else, the process is
equal to the process with constant rates $\mu^{i}$ on the set

$\Big{\\{}\sup_{t\in[0,\eta_{0}]}|\tilde{Z}^{N,x}(t)-\phi^{x}(t)|<\epsilon\Big{\\}}$

for all small enough $\epsilon>0$ and Theorem 3.1 is applicable. We note that
Assumption (C) implies that

$\delta(N,\epsilon)\rightarrow 0\text{ as }N\rightarrow\infty\text{ for all
}\epsilon>0.$

Moreover, as $\delta(N,\cdot)$ is decreasing, we can choose $\epsilon_{N}$ in
such a way that

$\epsilon_{N}=\frac{1}{N^{\alpha}}\quad\text{for some }\alpha\in(0,1).$ (11)

We remark here (and further discuss this important issue below) that
Assumption 2.3 (C) may well fail to be satisfied. To this end, we consider the
SIRS model an $x\in A$ with $x_{1}=0$. We hence have $\beta_{2}(x)=0$ and
hence the process $Z^{N,x}$ cannot enter the interior of $A$. Therefore, we
have

$\mathbb{P}\Big{[}\sup_{t\in[0,\eta_{0}]}|Z^{N,x}(t)-\phi^{x}(t)|\geq\epsilon\Big{]}=1$

for $\epsilon$ small enough. Assumption (C) can hence be considered as a means
to ensure that the process can enter the interior of $A$ from every point on
the boundary. (9) implies that

$\int_{0}^{\eta}|\log\underline{\beta}(\rho)|d\rho\rightarrow 0\quad\text{as
}\eta\rightarrow 0,$ (12)

since $\rho^{\alpha/2}|\log\underline{\beta}(\rho)|\leq C$ for appropriate $C$
and hence $|\log\underline{\beta}(\rho)|\leq C/\rho^{\alpha/2}$ is integrable,
and hence in particular that the rate $I(\phi)$ of linear functions $\phi$ is
finite, as it is shown in Lemma 5.7 below.

It remains to show that Assumption 2.3 (C) is satisfied for the SIRS model.
This is accomplished in section 8.

Exploiting (9), it is easy to prove

######  Lemma 2.4.

Under the Assumption 2.3 (C), for all $i\leq I$, $x\in A\cap B_{i}$, let
$\phi^{x}(t):=x+tv_{i}$. For all $\epsilon>0$, there exists an $\eta>0$
(independent of $i$, $x$) such that for all $i\leq I$ and all $x\in A\cap
B_{i}$,

$I_{\eta,x}(\phi^{x})<\epsilon.$

Note that for $i>I_{1}$, we have $\phi^{x}(t)=x$ for all $t$.

##  3 Law of large numbers

We first prove the law of large numbers by Kurtz (1978) with the rate of
convergence given as in Shwartz and Weiss (1995) Theorem 5.3.

######  Theorem 3.1.

Let $Z^{N.x}$ and $Y^{x}$ be given as in Equation (3) and (2) respectively,
and assume that the rates $\beta_{j}$ are bounded and Lipschitz continuous.
Then there exist constants $\tilde{C}_{1}=\tilde{C}_{1}(T)>0$ (independent of
$\epsilon$) and $\tilde{C}_{2}(\epsilon)=\tilde{C}_{2}(T,\epsilon)>0$ with
$\tilde{C}_{2}(\epsilon)=O(\epsilon^{2})$ as $\epsilon\rightarrow 0$ such that

$\mathbb{P}\Big{[}\sup_{t\in[0,T]}|Z^{N,x}(t)-Y^{x}(t)|\geq\epsilon\Big{]}\leq\tilde{C}_{1}\exp\big{(}-N\tilde{C}_{2}(\epsilon)\big{)}.$

$C_{1}$ and $C_{2}$ can be chosen independently of $x$.

Before we prove Theorem 3.1, we require some auxiliary results. We first have

######  Lemma 3.2.

Let $T>0$. Suppose that $f:D([0,T];A)\times\mathds{R}\rightarrow\mathds{R}$
and $G:D([0,T];A)\times\mathds{R}\times\mathds{R}\rightarrow\mathds{R}$ are
such that for all $\rho>0$,

$M(t):=\exp\big{(}\rho f(Z^{N,x},t)-G(Z^{N,x},t,\rho)\big{)}$

is a right-continuous martingale with mean one. Suppose furthermore that
$R:\mathds{R}\times\mathds{R}\rightarrow\mathds{R}$ is increasing in the first
argument and

$G(\phi,t,\rho)\leq R(t,\rho)$

for all $\phi\in D([0,T];A)$ and $\rho>0$. Then for all $\epsilon>0$

$\mathbb{P}\Big{[}\sup_{t\in[0,T]}f(Z^{N,x},t)\geq\epsilon\Big{]}\leq\inf_{\rho>0}\exp\big{(}R(T,\rho)-\rho\epsilon).$

###### Proof.

Fix $\rho>0$. Then by the assumptions of the lemma,

$\displaystyle\mathbb{P}\Big{[}\sup_{t\in[0,T]}f(Z^{N,x},t)\geq\epsilon\Big{]}$
$\displaystyle=\mathbb{P}\Big{[}\sup_{t\in[0,T]}\exp\big{(}\rho
f(Z^{N,x},t)\big{)}\geq\exp\big{(}\rho\epsilon\big{)}\Big{]}$
$\displaystyle\leq\mathbb{P}\Big{[}\sup_{t\in[0,T]}\exp\big{(}\rho
f(Z^{N,x},t)-G(Z^{N,x},t,\rho)\big{)}\geq\exp\big{(}\rho\epsilon-R(T,\rho)\big{)}\Big{]}$
$\displaystyle\leq\exp\big{(}R(T,\rho)-\rho\epsilon\big{)}$

where the last inequality is Doob’s martingale inequality, see, e.g. Theorem
II.1.7 in Revuz and Yor (2005). ∎

The next result is an easy exercise which we leave to the reader.

######  Lemma 3.3.

Let $Y$ be a $d$–dimensional random vector. Suppose that there exist numbers
$a>0$ and $\delta>0$ such that for all $\theta\in\mathds{R}^{d}$ with
$|\theta|=1$

$\mathbb{P}\big{[}\langle\theta,Y\rangle\geq a\big{]}\leq\delta.$

Then

$\mathbb{P}\big{[}|Y|\geq a\sqrt{d}\big{]}\leq 2d\delta.$

The main step towards the proof of Theorem 3.1 is the following Lemma

######  Lemma 3.4.

Assume that $\beta_{j}$ ($j=1,\dots,k$) is bounded and that $Y^{x}$ is a
solution of (2). Then for all $\theta\in\mathds{R}^{d}$ with $|\theta|=1$ and
all $T>0$, there is a function
$\tilde{C}:\mathds{R}_{+}\rightarrow\mathds{R}_{+}$ (independent of $x$) such
that

$\mathbb{P}\Big{[}\sup_{t\in[0,T]}\Big{\\{}\langle
Z^{N,x}(t)-Y^{x}(t),\theta\rangle-\int_{0}^{t}\sum_{j=1}^{k}\big{(}\beta_{j}(Z^{N,x}(s))-\beta_{j}(Y^{x}(s))\big{)}\langle
h_{j},\theta\rangle
ds\Big{\\}}\geq\epsilon\Big{]}\leq\exp\big{(}-N\tilde{C}(\epsilon)\big{)},$

and moreover

$0<\lim_{\epsilon\rightarrow
0}\tilde{C}(\epsilon)/\epsilon^{2}<\infty,\quad\text{and}\quad\lim_{\epsilon\rightarrow\infty}\tilde{C}(\epsilon)/\epsilon=\infty.$

###### Proof.

Let

$\displaystyle\mathcal{N}^{\theta}_{t}$ $\displaystyle=\langle
Z^{N,x}(t)-Y^{x}(t),\theta\rangle-\int_{0}^{t}\sum_{j=1}^{k}\big{(}\beta_{j}(Z^{N,x}(s))-\beta_{j}(Y^{x}(s))\big{)}\langle\theta,h_{j}\rangle
ds$ $\displaystyle=\frac{1}{N}\sum_{j=1}^{k}\langle h_{j},\theta\rangle
M_{j}\left(N\int_{0}^{t}\beta_{j}(Z^{N,x}_{s})ds\right).$

We want to use Lemma 3.2, with $f(Z^{N,x},t)=\mathcal{N}^{\theta}_{t}$. It is
not hard to check that if we define

$G(Z^{N,x},t,\rho)=N\sum_{j=1}^{k}\left(e^{\frac{\rho}{N}\langle
h_{j},\theta\rangle}-1-\frac{\rho}{N}\langle
h_{j},\theta\rangle\right)\int_{0}^{t}\beta_{j}(Z^{N,x}(s))ds,$

we have that

$M(t)=\exp\left(\rho f(Z^{N,x},t)-G(Z^{N,x},t,\rho)\right)$

is a martingale. Hence from Lemma 3.2, with $a=\rho/N$,

$\^{E}\mathbb{P}\left(\sup_{0\leq t\leq
T}\mathcal{N}^{\theta}_{t}>\epsilon\right)\leq\min_{a>0}\exp\left(N\overline{\beta}T\left[\sum_{j=1}^{k}\left\\{e^{a\langle
h_{j},\theta\rangle}-1-a\langle
h_{j},\theta\rangle\right\\}-a\epsilon\right]\right).$

The main inequality of the Lemma is established, with

$\tilde{C}(\epsilon)=\overline{\beta}T\max_{a>0}\left[a\epsilon-\sum_{j=1}^{k}\left\\{e^{a\langle
h_{j},\theta\rangle}-1-a\langle h_{j},\theta\rangle\right\\}\right].$

It is not hard to show that as $\epsilon\to 0$,

$\frac{\tilde{C}(\epsilon)}{\epsilon^{2}}\to\frac{\overline{\beta}T}{2\sum_{j=1}^{k}\langle
h_{j},\theta\rangle^{2}}.$

Consider now the case where $\epsilon$ is large. If $\langle
h_{j},\theta\rangle\leq 0$ for $1\leq j\leq k$, then for
$\epsilon>-\sum_{j}\langle h_{j},\theta\rangle$,
$\tilde{C}(\epsilon)=+\infty$, which means that a certain event has
probability zero. Now consider the more interesting case where $\langle
h_{j},\theta\rangle>0$ for at least one $1\leq j\leq k$. If we choose
$a_{\epsilon}$ such that

$\sum_{j=1}^{k}\left\\{e^{a_{\epsilon}\langle
h_{j},\theta\rangle}-1-a_{\epsilon}\langle
h_{j},\theta\rangle\right\\}=\epsilon,$

then $a_{\epsilon}\to\infty$ as $\epsilon\to\infty$, while
$\tilde{C}(\epsilon)\geq\epsilon(a_{\epsilon}-1)$, which completes the proof
of the Lemma. ∎

###### Proof of Theorem 3.1.

We deduce from Lemma 3.4 and a variant of Lemma 3.3 that

$\^{E}\mathbb{P}\left(\sup_{0\leq t\leq
T}\frac{1}{N}\left|\sum_{j=1}^{k}h_{j}M_{j}\left(N\int_{0}^{t}\beta_{j}(Z^{N,x}_{s})ds\right)\right|>\epsilon\right)\leq
2de^{-N\tilde{C}^{\prime}(\epsilon)},$ (13)

where $\tilde{C}^{\prime}(\epsilon)=\tilde{C}(\epsilon/\sqrt{d})$. In view of
the Lipschitz property of $b$, we have

$\displaystyle Z^{N,x}_{t}-Y^{x}_{t}$
$\displaystyle=\int_{0}^{t}\left[b(Z^{N,x}_{s})-b(Y^{x}_{s})\right]ds+\frac{1}{N}\sum_{j=1}^{k}h_{j}M_{j}\left(N\int_{0}^{t}\beta_{j}(Z^{N,x}_{s})ds\right),$
$\displaystyle\sup_{0\leq s\leq t}\left|Z^{N,x}_{s}-Y^{x}_{s}\right|$
$\displaystyle\leq K\int_{0}^{t}\sup_{0\leq r\leq
s}\left|Z^{N,x}_{r}-Y^{x}_{r}\right|ds+\sup_{0\leq s\leq
t}\frac{1}{N}\left|\sum_{j=1}^{k}h_{j}M_{j}\left(N\int_{0}^{s}\beta_{j}(Z^{N,x}_{r})dr\right)\right|.$

The result now follows from (13) and Gronwall’s Lemma. ∎

We can deduce from Theorem 3.1.

######  Corollary 3.5.

Let $M$ be a compensated standard Poisson process. Then there exist constants
$C_{1}=C_{1}(T)>0$ (independent of $\epsilon$) and
$C_{2}(\epsilon)=C_{2}(T,\epsilon)>0$ with $C_{2}(\epsilon)=O(\epsilon^{2})$
as $\epsilon\rightarrow 0$ such that

$\mathbb{P}\Big{[}\sup_{t\in[0,T]}\frac{|M(tN)|}{N}\geq\epsilon\Big{]}\leq
C_{1}\exp\big{(}-NC_{2}(\epsilon)\big{)}.$

$C_{1}$ and $C_{2}$ can be chosen independently of $x$.

###### Proof.

We apply Theorem 3.1 to $d=k=1$, $\beta_{1}(x)\equiv 1$ and $h_{1}=1$. Hence,

$|Z^{N}(t)-Y(t)|=\frac{|M(tN)|}{N}.$

The result follows directly. ∎

We shall need below the

######  Lemma 3.6.

Let $\beta_{j}$ ($j=1,\dots,k$) be bounded. Then there exist positive
constants $\tilde{C}_{1}$ and $\tilde{C}_{2}$ independent of $x$ such that for
all $0\leq s<t\leq T$ and for all $\epsilon>0$,

$\mathbb{P}\Big{[}\sup_{r\in[s,t]}|Z^{N,x}(r)-Z^{N,x}(s)|\geq\epsilon\Big{]}\leq\exp\Big{(}-N\epsilon\tilde{C}_{1}\log\Big{(}\frac{\epsilon\tilde{C}_{2}}{t-s}\Big{)}\Big{)}.$

###### Proof.

Let $\xi^{N}_{s,t}$ denote the number of jumps of the process $Z^{N,x}$ on the
time interval $[s,t]$. It is plain that

$\\{\sup_{r\in[s,t]}|Z^{N,x}(r)-Z^{N,x}(s)|\geq\epsilon\\}\subset\\{\xi^{N}_{s,t}\geq
CN\epsilon\\},$

for some universal constant $C>0$. Now $\xi^{N}_{s,t}$ is stochastically
dominated by a Poisson random variable with parameter $C^{\prime}N(t-s)$, for
some other constant $C^{\prime}>0$. Now let $\Theta$ be a Poisson r.v. with
parameter $\lambda$. For any $a,b>0$,

$\displaystyle\mathbb{P}(\Theta>b)$
$\displaystyle=\mathbb{P}\left(e^{a\Theta}>e^{ab}\right)$
$\displaystyle\leq\exp\left(\lambda(e^{a}-1)-ab\right),$

which, with the optimal choice $a=\log(b/\lambda)$, reads

$\mathbb{P}(\Theta>b)\leq\exp\left(b-\lambda-b\log(b/\lambda)\right)\leq\exp\left(-b\log\left(\frac{b}{e\lambda}\right)\right).$

The result follows by applying this inequality with
$\lambda=C^{\prime}N(t-s)$, and $b=CN\epsilon$. ∎

##  4 Properties of the rate function

###  4.1 Properties of the Legendre Fenchel transform

In this subsection we assume that the $\beta_{j}$’s are bounded and
continuous. We recall that $\ell$, $\overline{L}$, $\tilde{\ell}$ and
$\underline{L}$ have been defined in section 2.1, and start with

######  Lemma 4.1.

  1. 1.

For all $x\in A$,
$\underline{L}(x,\cdot):\mathcal{C}_{x}\rightarrow\mathds{R}_{+}$ is convex
and lower semicontinuous.

  2. 2.

For all $y\in\mathds{R}^{d}$,

$\underline{L}(x,y)\geq\underline{L}\Big{(}x,\sum_{j}\beta_{j}(x)h_{j}\Big{)}=0$

with strict inequality if $y\not=\sum_{j}\beta_{j}(x)h_{j}$.

###### Proof.

  1. 1.

We readily observe that $\tilde{\ell}(x,\cdot,\theta)$ is linear and hence
convex. As the supremum of these functions, the function
$\underline{L}(x,\cdot)$ is convex.

Lower semicontinuity follows as $\underline{L}(x,\cdot)$ is the supremum of a
family of continuous functions.

  2. 2.

Let first $y=\sum_{j}\beta_{j}(x)h_{j}$. We have

$\displaystyle\underline{L}(x,y)$
$\displaystyle=\sup_{\theta}\Big{\\{}\sum_{j}\beta_{j}(x)\langle
h_{j},\theta\rangle-\sum_{j}\beta_{j}(x)\big{(}\exp\langle
h_{j},\theta\rangle-1\big{)}\Big{\\}}$
$\displaystyle=\sup_{\theta}\Big{\\{}\sum_{j}\beta_{j}(x)\big{(}\langle
h_{j},\theta\rangle-\exp\langle h_{j},\theta\rangle+1\big{)}\Big{\\}}$
$\displaystyle=0$

as $\beta_{j}(x)\geq 0$ and $e^{z}\geq z+1$ for all $z\in\mathds{R}$ with
equality for $z=0$.

Let now $y$ be such that $\underline{L}(x,y)=0$. This implies

$\langle y,\theta\rangle-\sum_{j}\beta_{j}(x)\big{(}\exp\langle
h_{j},\theta\rangle-1\big{)}\leq 0\quad\text{for all
}\theta\in\mathds{R}^{d},$

in particular for $\theta=\epsilon e_{i}$ (where $e_{i}$ is the
$i^{\text{th}}$ unit-vector and $\epsilon>0$; in the following $h_{j}^{i}$ is
the $i^{\text{th}}$ component of $h_{j}$),

$\epsilon y_{i}\leq\sum_{j}\beta_{j}(x)\big{(}\exp(\epsilon
h_{j}^{i})-1\big{)}.$

Dividing by $\epsilon$ and letting $\epsilon\rightarrow 0$, we deduce that

$y_{i}\leq\sum_{j}\beta_{j}(x)h_{j}^{i}.$

For $\theta=-\epsilon e_{i}$ the converse inequality follows accordingly.

∎

######  Remark 4.2.

The function $\underline{L}(x,\cdot)$ is even strictly convex, see Corollary
4.10 below.

######  Lemma 4.3.

Assume that $\beta_{j}$ ($j=1,\dots,k$) is bounded. Then, there exist
constants $C_{1}$ and $B_{1}$ such that for all $|y|\geq B_{1}$,
$x\in\mathbb{R}^{d}$,

$\underline{L}(x,y)\geq C_{1}|y|\log(|y|).$

###### Proof.

Let

$\theta:=y\frac{\log|y|}{\bar{h}|y|},$

hence provided $|y|\geq 1$,

$\underline{L}(x,y)\geq\frac{|y|\log|y|}{\bar{h}}-k\bar{\beta}|y|$

which grows like $|y|\log|y|$ as $|y|\rightarrow\infty$. ∎

We now have

######  Lemma 4.4.

There exists a constant $C_{2}$ such that for all $x\in A$,
$y\in\mathcal{C}_{x}$, there exists a $\mu\in V_{x,y}$ with

$|\mu|\leq C_{2}|y|.$

###### Proof.

We first note that there are only finitely many convex cones $\mathcal{C}_{x}$
and we can hence restrict our attention to a fixed $x\in A$.

We proceed by contradiction. Assume that for all $n$ there exists a
$y^{n}\in\mathcal{C}_{x}$ such that for all $\mu\in V_{x,y^{n}}$,

$|\mu|\geq n|y^{n}|.$

We note that for any $y\in\mathcal{C}_{x}$, there exists a minimal
representation $\mu\in V_{x,y}$ (in the sense that $\tilde{\mu}\in
V_{x,y}\Rightarrow\max_{j}\tilde{\mu}_{j}\geq\max_{j}\mu_{j}$). Indeed, let
$\\{\mu^{n},:n\geq 1\\}\subset V_{x,y}$ be such that, as $n\to\infty$,

$\max_{j}\mu^{n}_{j}\downarrow\inf_{\mu\in
V_{x,y}}\left(\max_{j}\mu_{j}\right).$

There exists a subsequence along which
$\mu^{n}\rightarrow\mu\in\mathds{R}^{d}_{+}$ as $n\rightarrow\infty$. If
$\mu_{j}>0$, we have $\mu^{n}_{j}>0$ for $n$ large enough and hence
$\beta_{j}(x)>0$. Hence $\mu\in V_{x,y}$ since moreover

$\sum_{j}\mu_{j}h_{j}=\lim_{n}\sum_{j}\mu^{n}_{j}h_{j}=y.$

Given $y^{n}$, we denote this minimal representation by $\bar{\mu}^{n}$. We
now define

$\tilde{y}^{n}:=\frac{y^{n}}{|\bar{\mu}^{n}|},\quad\text{hence}\quad|\tilde{y}^{n}|\leq\frac{1}{n}.$

Furthermore, it is easy to see that minimal representations for the
$\tilde{y}^{n}$ are given by

$\tilde{\mu}^{n}:=\frac{\bar{\mu}^{n}}{|\bar{\mu}^{n}|},\quad\text{hence}\quad|\tilde{\mu}^{n}|=1.$

Boundedness implies (after possibly the extraction of a subsequence)
$\tilde{\mu}^{n}\rightarrow\tilde{\mu}$ with $|\tilde{\mu}|=1$. We let $n$
large enough such that for all $j$

$\tilde{\mu}^{n}_{j}>0\Rightarrow\tilde{\mu}^{n}_{j}>\frac{\tilde{\mu}_{j}}{2}$

(note that for at least one $j$, $\tilde{\mu}_{j}>0$). We have

$0=\lim_{n}\tilde{y}^{n}=\lim_{n}\sum_{j}\tilde{\mu}^{n}_{j}h_{j}=\sum_{j}\tilde{\mu}_{j}h_{j}$

and therefore

$\tilde{y}^{n}=\sum_{j}\tilde{\mu}_{j}^{n}h_{j}=\sum_{j;\tilde{\mu}_{j}>0}\underbrace{\Big{(}\tilde{\mu}_{j}^{n}-\frac{\tilde{\mu}_{j}}{2}\Big{)}}_{=:\hat{\mu}^{n}_{j}>0}h_{j}+\sum_{j;\tilde{\mu}_{j}=0}\underbrace{\tilde{\mu}_{j}^{n}}_{=:\hat{\mu}^{n}_{j}}h_{j},$

a contradiction to the minimality of the $\tilde{\mu}^{n}_{j}$. ∎

We require the following result

######  Lemma 4.5.

Let $x\in A$.

  1. 1.

$\ell(x,\mu)\geq 0$ for $\mu\in V_{x}$ and
$\ell(x,\cdot):V_{x}\longrightarrow\mathds{R}_{+}$ is strictly convex and has
compact level sets $\\{\mu\in V_{x}|\ell(x,\mu)\leq\alpha\\}$.

  2. 2.

Let $y\in\mathcal{C}_{x}$. Then there exists a unique $\mu^{*}=\mu^{*}(y)$
such that

$\ell(x,\mu^{*})=\inf_{\mu\in V_{x,y}}\ell(x,\mu).$

  3. 3.

There exist constants $C_{3},C_{4},C_{5},B_{2}>0$ (which depend only upon
$\sup_{x\in A}\max_{i\leq j\leq k}\beta_{j}(x)$), such that

$\displaystyle|\mu^{*}(y)|$ $\displaystyle\leq C_{3}|y|\quad\text{if
}|y|>B_{2},$ (14) $\displaystyle|\mu^{*}(y)|$ $\displaystyle\leq
C_{4}\quad\text{if }|y|\leq B_{2},$ (15) $\displaystyle|\mu^{*}(y)|$
$\displaystyle\geq C_{5}|y|\quad\text{for all }y.$ (16)

  4. 4.

$\overline{L}(x,\cdot),\mu^{*}:\mathcal{C}_{x}\rightarrow\mathds{R}_{+}$ are
continuous.

###### Proof.

  1. 1.

We define the function $f(z)=1-z+z\log z$ for $z\geq 0$ and note that for
$\mu\in V_{x}$,

$\ell(x,\mu)=\sum_{j,\beta_{j}(x)>0}\beta_{j}(x)f\Big{(}\tfrac{\mu_{j}}{\beta_{j}(x)}\Big{)}.$

We readily observe (by differentiation) that $f\geq 0$ and that $f$ is
strictly convex. Thus the first two assertions follow.

As $V_{x}$ is closed and $\ell(x,\cdot)$ is continuous, the level sets are
closed. Compactness follows form the fact that
$\lim_{x\rightarrow\infty}f(x)=\infty$.

  2. 2.

Existence of a minimizer follows from the fact that $V_{x,y}$ is closed.
Uniqueness follows from the strict convexity of $\ell(x,\cdot)$.

  3. 3.

By the definition of $\ell$, there exists a $B_{2}=B_{2}(\bar{\beta}(x))>0$
and $C=C(\bar{\beta}(x))>0$ such that for $y\in\mathcal{C}_{x}$ with $|y|\geq
B_{2}$ (and appropriate $\mu\in V_{x,y}$ according to Lemma 4.4),

$\ell(x,\mu^{*}(y))\leq\ell(x,\mu)\leq C|y|\log|y|.$

On the other hand, assume that for all $n$ there exists an
$y^{n}\in\mathcal{C}_{x}$ with $|y^{n}|\geq B_{2}$ such that

$|\mu^{*}(y^{n})|\geq n|y^{n}|.$

This implies for an appropriate constant $\tilde{C}$ and $n$ large enough

$\ell(x,\mu^{*}(y^{n}))\geq n\tilde{C}|y|\log|y|,$

a contradiction. Hence Inequality (14) follows.

Assume now that for all $n$ there exists an $y^{n}\in\mathcal{C}_{x}$ with
$|y^{n}|\leq B_{2}$,

$|\mu^{*}(y^{n})|\geq n,\quad\text{hence
}\lim_{n\rightarrow\infty}\ell(x,\mu^{*}(y^{n}))\rightarrow\infty.$

However, Lemma 4.4 implies that there exists an $\mu^{n}\in V_{x,y^{n}}$ and a
constant $C=C(\bar{\beta}(x),B_{2})$ independent of $n$ with

$\ell(x,\mu^{n})\leq C,$

a contradiction. Hence Inequality (15) follows.

Finally, Inequality (16) follows from the definition of $V_{x,y}$.

  4. 4.

Let $y,y^{n}\in\mathcal{C}_{x}$ with $y^{n}\rightarrow y$. By 3., the sequence
$(\mu^{*}(y^{n}))_{n}$ is bounded and hence there exists a convergent
subsequence, say (by abuse of notation)

$\mu^{*}(y^{n})\rightarrow\mu^{*}\quad\text{with }\mu_{j}^{*}\geq 0\text{ for
all }j.$

In particular, we have

$\sum_{j}\mu^{*}_{j}h_{j}=y.$ (17)

We have

$\displaystyle y^{n}$ $\displaystyle=\sum_{j}\mu_{j}^{*}(y^{n})h_{j}$
$\displaystyle=(1-\epsilon^{n})\sum_{j}\mu_{j}^{*}h_{j}+\sum_{j}\big{(}\mu_{j}^{*}(y^{n})-\mu_{j}^{*}+\epsilon^{n}\mu_{j}^{*}\big{)}h_{j}$
$\displaystyle=(1-\epsilon^{n})\sum_{j}\mu_{j}^{*}(y)h_{j}+\sum_{j}\tilde{\mu}_{j}^{n}h_{j},$
(18)

where we have used (17),
$\mu^{\ast}(y)=\operatorname{argmax}_{\mu}\ell(x,\mu)$,
$\tilde{\mu}_{j}^{n}=\mu_{j}^{*}(y^{n})-\mu_{j}^{*}+\epsilon^{n}\mu_{j}^{*}$,
and

$\epsilon^{n}=\begin{cases}\frac{2\max_{j}|\mu_{j}^{*}(y^{n})-\mu_{j}^{*}|}{\min_{j;\mu_{j}^{*}>0}\mu_{j}^{*}},&\text{if
$\min_{j;\mu_{j}^{*}>0}\mu_{j}^{*}>0$};\\\ 1/n,&\text{otherwise}.\end{cases}$

In particular, we have $0\leq\tilde{\mu}_{j}^{n}\rightarrow 0$ as
$n\rightarrow\infty$. By Equation (18), 2. and the continuity of $\ell$, we
have

$\displaystyle\ell(x,\mu^{*}(y^{n})))$
$\displaystyle\leq\ell(x,(1-\epsilon^{n})\mu^{*}(y)+\tilde{\mu}^{n})$
$\displaystyle\leq\ell(x,\mu^{*}(y))+\delta(n)$ (19)

with $\delta(n)\rightarrow\infty$ as $n\rightarrow\infty$. This implies (again
by the continuity of $\ell$)

$\ell(x,\mu^{*})\leq\ell(x,\mu^{*}(y))$

and hence $\mu^{*}=\mu^{*}(y)$ by 2. As this holds true for all convergent
subsequences of $(\mu^{*}(y^{n}))_{n}$, this establishes the continuity of
$\mu^{*}(\cdot)$.

The continuity of $\overline{L}(x,\cdot)$ follows directly from this and the
continuity of $\ell$.

∎

######  Remark 4.6.

Assume that for $x\in A$, $\mathcal{C}_{x}=\mathcal{C}_{\tilde{x}}$ for all
$\tilde{x}$ in some neighborhood $U$ of $x$. Then the function $\ell:U\times
V_{x,y}\rightarrow\mathds{R}_{+}$ is continuous and hence we have that
$\mu^{*}(y)=\mu^{*}(x,y)$ as given in Lemma 4.5 is also continuous in $x$ (as
the $\operatorname{argmin}$ of a continuous function).

We have moreover

######  Lemma 4.7.

  1. 1.

Let $x\in A$. For all $B>0$, there exists a constant $C_{6}=C_{6}(x,B)>0$ such
that for all $y\in\mathcal{C}_{x}$ with $|y|\leq B$ and
$\theta\in\mathds{R}^{d}$ with $\tilde{\ell}(x,y,\theta)\geq-1$,666The
constant $-1$ can be replaced by any other constant $-C$ ($C>0$). Note that
$C_{6}$ then depends on $C$ with $C_{6}$ increasing in $C$.

$\langle\theta,h_{j}\rangle\leq C_{6}\quad\text{for all }j\text{ with
}\beta_{j}(x)>0.$

If $\log\beta_{j}(\cdot)$ ($j=1\dots,k$) is bounded, $C_{6}$ can be chosen
independently of $x$.

  2. 2.

Let $x\in A$ and $y\in\mathcal{C}_{x}$. If $(\theta_{n})_{n}$ is a maximizing
sequence of $\tilde{\ell}(x,y,\cdot)$ and for some $j=1,\dots,k$,

$\liminf_{n\rightarrow\infty}\langle\theta_{n},h_{j}\rangle=-\infty,$

then

$\mu_{j}=0\quad\text{for all }\mu\in V_{x,y}.$

Conversely, there exists a constant $\tilde{C}_{6}=\tilde{C}_{6}(B)>0$ such
that if $|y|\leq B$ and $\mu_{j}>0$ for some $\mu\in V_{x,y}$, then

$\liminf_{n\rightarrow\infty}\langle\theta_{n},h_{j}\rangle>-\tilde{C}_{6}.$

###### Proof.

  1. 1.

Let $|y|\leq B$, $C_{2}$ and $\mu\in V_{x,y}$ be according to Lemma 4.4.
Define the functions from $\mathbb{R}$ into itself

$f_{j}(z):=\mu_{j}z-\beta_{j}(x)(\operatorname{e}^{z}-1).$

Note that $f_{j}(z)=0$ if $\beta_{j}(x)=0$, and
$\operatorname{argmax}_{z}f_{j}(z)=\log\mu_{j}/\beta_{j}(x)$ if
$\beta_{j}(x)>0$. Let

$\^{E}\tilde{C}(x,B)=\sup_{j;\ \beta_{j}(x)>0}\sup_{|\mu|\leq
C_{2}B}f_{j}\left(\log\frac{\mu_{j}}{\beta_{j}(x)}\right).$

If $x$, $y$ and $\theta$ are as in the statement, and $1\leq j\leq k$ is such
that $\beta_{j}(x)>0$ and $\langle\theta,h_{j}\rangle>0$, then

$\sum_{j^{\prime}\not=j}f_{j^{\prime}}(\langle\theta,h_{j^{\prime}}\rangle)=\tilde{\ell}(x,y,\theta)-f_{j}(\langle\theta,h_{j}\rangle),$

hence in view of the assumption,

$f_{j}(\langle\theta,h_{j}\rangle)\geq-1-(k-1)\tilde{C}(x,B),$

As $f_{j}(z)\rightarrow-\infty$ as $z\rightarrow\infty$, the assertion
follows.

  2. 2.

If $\liminf_{n\rightarrow\infty}\langle\theta_{n},h_{j}\rangle=-\infty$ and
$\mu\in V_{x,y}$ with $\mu_{j}>0$, then 1. implies that
$\tilde{\ell}(x,y,\theta_{n})\rightarrow-\infty$, a contradiction.

The second assertion follows accordingly.

∎

We now prove

######  Lemma 4.8.

  1. 1.

Let $x\in A$ and $y\in\mathcal{C}_{x}$. Then there exists a maximizing
sequence $(\theta_{n})_{n}$ of $\tilde{\ell}(x,y,\cdot)$ and constants
$\tilde{s}_{j}$ (for all $j=1,\dots,k$ for which there exists a $\mu\in
V_{x,y}$ with $\mu_{j}>0$) such that

$\lim_{n\rightarrow\infty}\langle\theta_{n},h_{j}\rangle=\tilde{s}_{j}\in\mathds{R}.$

The constants $\tilde{s}_{j}$ are bounded uniformly over bounded sets of
$y\in\mathcal{C}_{x}$.

In particular, there exists a maximizing sequence $(\theta_{n})_{n}$ such that
for all $j=1,\dots,k$ with $\beta_{j}(x)>0$,

$\lim_{n\rightarrow\infty}\exp(\langle\theta_{n},h_{j}\rangle)=s_{j}\in\mathds{R}.$

  2. 2.

Let $x\in A$ and $y\not\in\mathcal{C}_{x}$. Then $\underline{L}(x,y)=\infty$.

###### Proof.

  1. 1.

By Lemma 4.7,

$-\tilde{C}_{6}=-\tilde{C}_{6}(|y|)<\langle\theta_{n},h_{j}\rangle\leq
C_{6}=C_{6}(|y|)$

for all $n$ and for all $j$ with $\mu_{j}>0$ for some $\mu\in V_{x,y}$. The
first assertion follows by taking appropriate subsequences.

For the second assertion, we have to consider those $j$ with $\mu_{j}=0$ for
all $\mu\in V_{x,y}$ although $\beta_{j}(x)>0$. If
$\liminf_{n\rightarrow\infty}\langle\theta_{n},h_{j}\rangle=-\infty$, we take
further subsequences and obtain (with a slight abuse of notation)

$\lim_{n\rightarrow\infty}\exp(\langle\theta_{n},h_{j}\rangle)=0.$

  2. 2.

Let $y\not\in\mathcal{C}_{x}$ and $v$ be the projection of $y$ on
$\mathcal{C}_{x}$. Hence, $0=\langle y-v,v\rangle\geq\langle
y-v,\tilde{v}\rangle$ for all $\tilde{v}\in\mathcal{C}_{x}$. For $z=y-v$
($\not=0$ as $y\not\in\mathcal{C}_{x}$), we have $\langle z,y\rangle=\langle
z,z\rangle+\langle z,v\rangle>0$ and $\langle z,h_{j}\rangle\leq 0$ for all
$j$ with $\beta_{j}(x)>0$. If we set $\theta_{n}=nz$, we obtain
$\ell(x,y,\theta_{n})\rightarrow\infty$.

∎

###  4.2 Equality of $\underline{L}$ and $\overline{L}$

We can now finally establish

######  Theorem 4.9.

For all $x\in A$, $y\in\mathds{R}^{d}$,

$\underline{L}(x,y)=\overline{L}(x,y).$

###### Proof.

In view of Lemma 2.2, it suffices to prove that
$\overline{L}(x,y\leq\underline{L}(x,y)$. We first note that we have
$\underline{L}(x,y)<\infty$ if and only if $y\in\mathcal{C}_{x}$ by Lemma 4.8
2\. and Lemma 4.11. As the same is true for $\overline{L}(x,y)$ by definition,
we can restrict our attention to the case $y\in\mathcal{C}_{x}$.

We choose a maximizing sequence $(\theta_{n})_{n}$ according to Lemma 4.7 and
obtain

$\lim_{n}\langle\theta_{n},y\rangle=\underline{L}(x,y)+\sum_{j}\beta_{j}(x)(s_{j}-1);$
(20)

here we set $s_{j}=0$ if $\beta_{j}(x)=0$. We now differentiate with respect
to $\theta$ and obtain for all $n$

$\nabla_{\theta}\tilde{\ell}(x,y,\theta_{n})=y-\sum_{j;\beta_{j}(x)>0}\beta_{j}(x)h_{j}\exp(\langle\theta_{n},h_{j}\rangle);$

hence (by the fact that $(\theta_{n})_{n}$ is a maximizing sequence and the
limit of $\nabla_{\theta}\tilde{\ell}(x,y,\theta_{n})$ exists),

$\lim_{n}\nabla_{\theta}\tilde{\ell}(x,y,\theta_{n})=y-\sum_{j;\beta_{j}(x)>0}\beta_{j}(x)s_{j}h_{j}=0.$

We set,

$\mu_{j}^{*}:=\beta_{j}(x)s_{j},$ (21)

in particular

$y=\sum_{j}\mu_{j}^{*}h_{j}\quad\text{and}\quad\mu^{*}\in V_{x,y}.$

Therefore,

$\displaystyle\overline{L}(x,y)$ $\displaystyle\leq\ell(x,\mu^{*})$
$\displaystyle=\sum_{j}\beta_{j}(x)-\mu^{*}_{j}+\mu_{j}^{*}\log\big{(}\tfrac{\mu_{j}^{*}}{\beta_{j}(x)}\big{)}$
$\displaystyle=\sum_{j}\beta_{j}(x)(1-s_{j})+\mu_{j}^{*}\log s_{j}$
$\displaystyle=\underline{L}(x,y),$

where we have used (20) and (21) for the last identity. The assertion follows.
∎

From now on, we shall write $L(x,y)$ for the quantity
$\underline{L}(x,y)=\overline{L}(x,y)$.

We now prove the strict convexity of $L(x,\cdot)$.

######  Corollary 4.10.

For all $x\in A$, $L(x,\cdot):\mathcal{C}_{x}\rightarrow\mathds{R}_{+}$ is
strictly convex.

###### Proof.

For strict convexity, we exclude the case that $\beta_{j}(x)=0$ for all $j$
(as then $L(x,y)=\infty$ for all $y\not=0$ and the assertion is trivial).

Convexity was proven in Lemma 4.1. Assume now that for
$y,\tilde{y}\in\mathcal{C}_{x}$ and $\lambda\in(0,1)$,

$L(x,\lambda y+(1-\lambda)\tilde{y})=\lambda
L(x,y)+(1-\lambda)L(x,\tilde{y}).$

In other words,

$\displaystyle\sup_{\theta}\Big{\\{}\langle\theta,\lambda
y+(1-\lambda)\tilde{y}\rangle-\sum_{j}\beta_{j}(x)(\operatorname{e}^{\langle\theta,h_{j}\rangle}-1)\Big{\\}}$
$\displaystyle\qquad=\sup_{\theta}\Big{\\{}\lambda\big{[}\langle\theta,y\rangle-\sum_{j}\beta_{j}(x)(\operatorname{e}^{\langle\theta,h_{j}\rangle}-1)\big{]}+(1-\lambda)\big{[}\langle\theta,\tilde{y}\rangle-\sum_{j}\beta_{j}(x)(\operatorname{e}^{\langle\theta,h_{j}\rangle}-1)\big{]}\Big{\\}}$
$\displaystyle\qquad=\lambda\sup_{\theta}\Big{\\{}\langle\theta,y\rangle-\sum_{j}\beta_{j}(x)(\operatorname{e}^{\langle\theta,h_{j}\rangle}-1)\Big{\\}}+(1-\lambda)\sup_{\theta}\Big{\\{}\langle\theta,\tilde{y}\rangle-\sum_{j}\beta_{j}(x)(\operatorname{e}^{\langle\theta,h_{j}\rangle}-1)\Big{\\}}.$

Hence, if $(\theta_{n})_{n}$ is a maximizing sequence for
$\tilde{\ell}(x,\lambda y+(1-\lambda)\tilde{y},\cdot)$, it is also a
maximizing sequence for $\tilde{\ell}(x,y,\cdot)$ and
$\tilde{\ell}(x,\tilde{y},\cdot)$. As in the proof of Theorem 4.9, this
implies

$\displaystyle\lim_{n\rightarrow\infty}\nabla_{\theta}\tilde{\ell}(x,y,\theta_{n})$
$\displaystyle=y-\lim_{n\rightarrow\infty}\sum_{j}\beta_{j}(x)(\operatorname{e}^{\langle\theta_{n},h_{j}\rangle}-1)=0,$
$\displaystyle\lim_{n\rightarrow\infty}\nabla_{\theta}\tilde{\ell}(x,\tilde{y},\theta_{n})$
$\displaystyle=\tilde{y}-\lim_{n\rightarrow\infty}\sum_{j}\beta_{j}(x)(\operatorname{e}^{\langle\theta_{n},h_{j}\rangle}-1)=0.$

Hence $y=\tilde{y}$ as required. ∎

###  4.3 Further properties of the Legendre Fenchel transform

In this subjection, we assume that the $\log\beta_{j}$’s are bounded. In this
case $\mathcal{C}_{x}=\mathcal{C}=\mathbb{R}^{d}$ for all $x$.

We have

######  Lemma 4.11.

  1. Assume that $\log\beta_{j}$ ($j=1,\dots,k$) is bounded.

  2. 1.

For all $B>0$ exists a constant $C_{7}=C_{7}(B)>0$ such that for all $x\in A$,
$y\in\mathcal{C}$ with $|y|\leq B$,

$L(x,y)\leq C_{7}.$

  3. 2.

For all $x\in A$, $L(x,\cdot):\mathcal{C}\rightarrow\mathds{R}_{+}$ is
continuous.

###### Proof.

  1. 1.

Let $x\in A$, $y\in\mathcal{C}$. By Lemma 4.4 and Theorem 4.9 below888note
that this result is not used for the proof of Theorem 4.9., we obtain

$\displaystyle L(x,y)$
$\displaystyle\leq\sum_{j,\beta_{j}(x)>0}\beta_{j}(x)-\mu_{j}+\mu_{j}\log\mu_{j}-\mu_{j}\log\beta_{j}(x)$
$\displaystyle\leq k(\overline{\beta}+C|y|\log
C+C|y|\log|y|+C|y||\log\underline{\beta}|).$

The assertion follows.

  2. 2.

The assertion follows directly from 1., Lemma 4.1 1.

∎

We have moreover

######  Lemma 4.12.

Assume that $\log\beta_{j}$ ($j=1,\dots,k$) is bounded. For all $\rho>0$,
$\epsilon>0$, $C_{8}>0$, there exists a constant $B_{3}=B(C_{8},\epsilon)$
such that for all $x\in A$, $y\in\mathcal{C}$ with $|y|\leq C_{8}$,

$\sup_{|\theta|\leq
B}\tilde{\ell}(x,y,\theta)\geq\sup_{\theta\in\mathds{R}^{d}}\tilde{\ell}(x,y,\theta)-\epsilon=L(x,y)-\epsilon.$

###### Proof.

We first fix $x\in A$ and define the compact set

$\tilde{\mathcal{C}}:=\\{y\in\mathcal{C}||y|\leq C_{8}\\}.$

We fix $\delta>0$ and define for $y\in\tilde{\mathcal{C}}$,

$\displaystyle z(y,\delta)$ $\displaystyle:=y+\sum_{j}\delta h_{j},$
$\displaystyle N^{y,\delta}$
$\displaystyle:=\big{\\{}y+\sum_{j}\alpha_{j}h_{j}|\alpha_{j}\in(-\delta,\delta)\big{\\}}.$

For all $y\in\tilde{\mathcal{C}}$, $N^{y,\delta}$ is relatively open (with
respect to $\mathcal{C}$) and $y\in N^{y,\delta}$. Hence there exists a finite
cover $N_{1},\dots,N_{n}$ of $\tilde{\mathcal{C}}$, where
$N_{i}:=N^{y_{i},\delta}$ for appropriate $y_{i}\in\tilde{\mathcal{C}}$; we
define $z_{i}:=z(y_{i},\delta)$.

We use the continuity of $L(x,\cdot):\mathcal{C}\rightarrow\mathds{R}_{+}$
(cf. Lemma 4.11 2.) and the fact that $\tilde{\mathcal{C}}$ is compact, we
obtain for $\delta$ small enough that for all $y\in\tilde{\mathcal{C}}$, $v\in
N^{y,\delta}$,

$|L(x,v)-L(x,z(y,\delta))|<\frac{\epsilon}{4}.$ (22)

We let $\theta_{i}$ be almost optimal for $z_{i}$ in the sense that

$\tilde{\ell}(x,z_{i},\theta_{i})\geq L(x,z_{i})-\frac{\epsilon}{4}.$ (23)

We now set $B^{x}:=\max_{i}|\theta_{i}|$ and let $y\in\tilde{\mathcal{C}}$,
say $y\in N_{i}$. Then, making use successively of (22) and (23), we obtain

$\displaystyle L(x,y)$ $\displaystyle\leq L(x,z_{i})+\frac{\epsilon}{4}$
$\displaystyle\leq\tilde{\ell}(x,z_{i},\theta_{i})+\frac{\epsilon}{2}$
$\displaystyle=\tilde{\ell}(x,y,\theta_{i})+\frac{\epsilon}{2}+\langle\theta_{i},z_{i}-y\rangle.$
(24)

We have $z_{i}-y=\sum_{j}\mu_{j}h_{j}$ for appropriate
$\mu_{j}=\alpha_{j}+\delta\in(0,2\delta)$ and by Lemma 4.7 (cf. also
Inequality (22)), $\langle\theta_{i},h_{j}\rangle\leq C_{6}$. Hence

$\langle z_{i}-y,\theta_{i}\rangle=\sum_{j}\mu_{j}\langle
h_{j},\theta_{i}\rangle\leq 2kC_{6}\delta\leq\frac{\epsilon}{4},$ (25)

provided we choose $\delta$ such that $8kC_{6}\delta\leq\epsilon$. Therefore
by Inequalities (24) and (25) for all $y\in\tilde{\mathcal{C}}$,

$L(x,y)\leq\tilde{\ell}(x,y,\theta_{i})+\epsilon\quad\text{(recall that
$|\theta_{i}|\leq B^{x}$)}.$ (26)

Let now for all $x\in A$, $B^{x}$ be the bound obtained above belonging to
$\frac{\epsilon}{4}$.999Note that $B^{x}$ depends on $x$ only through
$\beta(x)$. Let furthermore $x,\tilde{x}\in A$ with
$|\beta(x)-\beta(\tilde{x})|<\delta$ for some $\delta>0$,
$y\in\tilde{\mathcal{C}}$, $|y|\leq C_{8}$ and $\theta\in\mathds{R}^{d}$ such
that $\tilde{\ell}(x,y,\theta)\geq-1$ (which implies
$\langle\theta,h_{j}\rangle\leq C_{6}$ by Lemma 4.7). This implies

$|\tilde{\ell}(x,y,\theta)-\tilde{\ell}(\tilde{x},y,\theta)|\leq\sum_{j}|\beta_{j}(x)-\beta_{j}(\tilde{x})|\operatorname{e}^{C_{6}}<\frac{\epsilon}{4}$
(27)

for $\delta$ small enough (and independent of $x,\tilde{x},y,\theta$). Let now
be $\tilde{\theta}$ be almost optimal for $\tilde{x},y$. Using twice (27) and
once (26), we obtain

$\displaystyle L(\tilde{x},y)$
$\displaystyle\leq\tilde{\ell}(\tilde{x},y,\tilde{\theta})+\frac{\epsilon}{4}$
$\displaystyle\leq\tilde{\ell}(x,y,\tilde{\theta})+\frac{\epsilon}{2}$
$\displaystyle\leq\sup_{|\theta|\leq
B^{x}}\tilde{\ell}(x,y,\theta)+\frac{3\epsilon}{4}$
$\displaystyle\leq\sup_{|\theta|\leq
B^{x}}\tilde{\ell}(\tilde{x},y,\theta)+\epsilon$

We can cover the compact interval $[\underline{\beta},\bar{\beta}]$ by
finitely many $\tilde{\delta}$-neighborhoods of $\beta^{i}$. The assertion
follows by taking the maximum of the corresponding $B^{i}$ (cf. Footnote 9). ∎

######  Lemma 4.13.

Assume that $\log\beta_{j}$ ($j=1,\dots,k$) is bounded. There exist constants
$B_{4}$ and $C_{9}$ such that for all, $x\in A$ and $y\in\mathcal{C}$,

$L(x,y)\leq\begin{cases}C_{9}&\text{if }|y|\leq B_{4}\\\
C_{9}|y|\log|y|&\text{if }|y|>B_{4}.\end{cases}$

###### Proof.

From the formula for $\overline{L}(x,y)$ and Lemma 4.4, we have

$\displaystyle L(x,y)$
$\displaystyle\leq\sum_{j}\bar{\beta}+C|y|\log|y|+C|y||\log\underline{\beta}|$
$\displaystyle\leq
k\cdot\big{(}\bar{\beta}+C|y|\log|y|+C|y||\log\underline{\beta}|\big{)}.$

∎

We also obtain the continuity of $L$ in $x$

######  Lemma 4.14.

Assume that $\log\beta_{j}$ ($j=1,\dots,k$) is bounded and continuous. For all
$y\in\mathcal{C}$,

$L(\cdot,y):A\rightarrow\mathds{R}_{+}$

is continuous. The continuity is uniform over bounded $y$.

###### Proof.

We let $y\in\mathcal{C}$ with $|y|\leq B$, $0<\epsilon<1$. and $x,\tilde{x}\in
A$. Let $\theta$ such that

$L(x,y)\leq\tilde{\ell}(x,y,\theta)+\frac{\epsilon}{2}.$

We have by the continuity of the $\beta_{j}$ and Lemma 4.7,

$|\tilde{\ell}(x,y,\theta)-\tilde{\ell}(\tilde{x},y,\theta)|\leq\sum_{j}|\beta_{j}(x)-\beta_{j}(\tilde{x})|e^{C_{6}}<\frac{\epsilon}{2}$

if $|x-\tilde{x}|<\delta$ for appropriate $\delta>0$ (independent of
$x,\tilde{x}\in A$ and $y$ with $|y|\leq B$). Thus,

$L(x,y)\leq\tilde{\ell}(\tilde{x},y,\theta)+\epsilon\leq
L(\tilde{x},y)+\epsilon.$

Reversing the roles of $x$ and $\tilde{x}$ proves the assertion. ∎

Combinig Lemma 4.14 and Lemma 4.11, we deduce the

######  Corollary 4.15.

Assume that $\log\beta_{j}$ ($j=1,\dots,k$) is bounded and continuous. Then
$L:A\times\mathbb{R}^{d}\to\mathbb{R}_{+}$ is continuous.

###  4.4 The rate function

Recall that for $\phi:[0,T]\rightarrow A$, we let

$\displaystyle I_{T}(\phi)$
$\displaystyle:=\begin{cases}\int_{0}^{T}L(\phi(t),\phi^{\prime}(t))dt&\text{
if }\phi\text{ is absolutely continuous}\\\ \infty&\text{
otherwise.}\end{cases}$

For $x\in A$ and $\phi:[0,T]\rightarrow A$, let

$I_{T,x}(\phi):=\begin{cases}I_{T}(\phi)&\text{ if }\phi(0)=x\\\ \infty&\text{
otherwise.}\end{cases}$

We first have the following statement, which follows readily from point 2 in
Lemma 4.1.

######  Lemma 4.16.

Assume that $x\in A$. If $\phi$ solves the ODE (2), then $I_{T,x}(\phi)=0$.
Conversely, if the ODE (2) admits a unique solution $Y^{x}$ and
$I_{T,x}(\phi)=0$, then $\phi(t)=Y^{x}(t)$ for all $t\in[0,T]$.

In the next statement, $B_{1}$ refers to the constant appearing in Lemma 4.3.

######  Lemma 4.17.

Assume that $\beta_{j}$ ($j=1,\dots,k$) is bounded.

  1. 1.

Let $K,\epsilon>0$. There exits $\delta>0$ such that for all $\phi$ with
$I_{T,x}(\phi)\leq K$ and for all finite collections of non-overlapping
subintervals of $[0,T]$, $[s_{1},t_{1}],\dots,[s_{J},t_{J}]$, with
$\sum_{i}(t_{i}-s_{i})=\delta$,

$\sum_{i}|\phi(t_{i})-\phi(s_{i})|<\epsilon.$

  2. 2.

Let $K>0$. Then, for all constants $B\geq B_{1}$ and for all $\phi$ with
$I_{T,x}(\phi)\leq K$,

$\int_{0}^{T}\mathds{1}_{\\{|\phi^{\prime}(t)|\geq B\\}}dt<\frac{K}{C_{1}B\log
B}.$

###### Proof.

  1. 1.

Note first that

$f(\alpha):=\inf_{x,y}\Big{\\{}\frac{L(x,y)}{|y|}\Big{|}|y|\geq\alpha\Big{\\}}\rightarrow\infty$

as $\alpha\rightarrow\infty$ by Lemma 4.3. For
$g(t):=\mathds{1}_{\cup_{j}[s_{j},t_{j}]}$ and $\alpha=1/\sqrt{\delta}$, we
obtain by the fact that $\phi$ is absolutely continuous

$\displaystyle\sum_{j}|\phi(t_{j})-\phi(s_{j})|$
$\displaystyle\leq\int_{0}^{T}|\phi^{\prime}(t)|g(t)dt$
$\displaystyle\leq\int_{0}^{T}\alpha\mathds{1}_{\\{|\phi^{\prime}(t)|\leq\alpha\\}}g(t)dt+\int_{0}^{T}\mathds{1}_{\\{|\phi^{\prime}(t)|>\alpha\\}}\frac{L(\phi(t),\phi^{\prime}(t))}{L(\phi(t),\phi^{\prime}(t))/|\phi^{\prime}(t)|}g(t)dt$
$\displaystyle\leq\alpha\delta+\frac{I_{T,x}(\phi)}{f(\alpha)}$
$\displaystyle\leq\sqrt{\delta}+\frac{K}{f(1/\sqrt{\delta})}\rightarrow 0,$

as $\delta\rightarrow 0$.

  2. 2.

For $B>0$, we define the function

$f(B):=\inf_{x,y}\Big{\\{}\frac{L(x,y)}{|y|}\,\Big{|}\,|y|\geq B\Big{\\}}.$

By Lemma 4.3, $f(B)\geq C_{1}\log B$ for $B\geq B_{1}$.

$\displaystyle\int_{0}^{T}\mathds{1}_{\\{|\phi^{\prime}(t)|\geq B\\}}dt$
$\displaystyle\leq\frac{1}{B}\int_{0}^{T}|\phi^{\prime}(t)|\mathds{1}_{\\{|\phi^{\prime}(t)|\geq
B\\}}dt$
$\displaystyle=\frac{1}{B}\int_{0}^{T}\frac{L(\phi(t),\phi^{\prime}(t))|\phi^{\prime}(t)|}{L(\phi(t),\phi^{\prime}(t))}\mathds{1}_{\\{|\phi^{\prime}(t)|\geq
B\\}}dt$ $\displaystyle\leq\frac{K}{Bf(B)}\leq\frac{K}{C_{1}B\log B}.$

∎

######  Theorem 4.18.

Assume that $\log\beta_{j}$ ($j=1,\dots,k$) is bounded and continuous. Let
$\phi\in D([0,T];A)$ with $I_{T,x}(\phi)<\infty$. For all $\epsilon>0$, there
exists a $\delta>0$ such that for

$\tilde{\phi}:[0,T]\rightarrow A\quad\text{with}\quad\sup_{0\leq t\leq
T}|\tilde{\phi}(t)-\phi(t)|<\delta,$
$\Big{|}\int_{0}^{T}\big{(}L(\tilde{\phi}(t),\phi^{\prime}(t))-L(\phi(t),\phi^{\prime}(t))\big{)}dt\Big{|}<\epsilon.$

###### Proof.

We choose $B\geq B_{1}\vee B_{4}$ large enough such that for $x\in A$,
$y\in\mathcal{C}_{x}=\mathcal{C}$ (independent of $x$) with $|y|\geq B$ (cf.
Lemmas 4.3 and 4.13),

$C_{1}|y|\log|y|\leq L(x,y)\leq C_{9}|y|\log|y|.$

As $I_{T,x}(\phi)<\infty$, the set
$\\{t|\phi^{\prime}(t)\not\in\mathcal{C}\\}$ is a Lebesgue null-set and we
assume w.l.o.g. that for all $t$, $\phi^{\prime}(t)\in\mathcal{C}$. We hence
obtain that

$\displaystyle\int_{0}^{T}L(\tilde{\phi}(t),\phi^{\prime}(t))\mathds{1}_{\\{|\phi^{\prime}(t)|\geq
B\\}}dt$
$\displaystyle\leq\int_{0}^{T}C_{9}|\phi^{\prime}(t)|\log|\phi^{\prime}(t)|\mathds{1}_{\\{|\phi^{\prime}(t)|\geq
B\\}}dt$
$\displaystyle\leq\frac{C_{9}}{C_{1}}\int_{0}^{T}L(\phi(t),\phi^{\prime}(t))\mathds{1}_{\\{|\phi^{\prime}(t)|\geq
B\\}}dt.$

From this and Lemma 4.17, we can choose $B$ large enough such that

$\displaystyle\sup\left(\int_{0}^{T}L(\tilde{\phi}(t),\phi^{\prime}(t))\mathds{1}_{\\{|\phi^{\prime}(t)|\geq
B\\}}dt,\int_{0}^{T}L(\phi(t),\phi^{\prime}(t))\mathds{1}_{\\{|\phi^{\prime}(t)|\geq
B\\}}dt\right)<\frac{\epsilon}{4}.$

By Lemma 4.14, there exists an $\delta>0$ such that for all $x,\tilde{x}\in A$
with $|x-\tilde{x}|<\delta$ and $y\in\mathcal{C}$ with $|y|\leq B$,

$|L(x,y)-L(\tilde{x},y)|<\frac{\epsilon}{2T}.$

We obtain for $\sup_{0\leq t\leq T}|\tilde{\phi}(t)-\phi(t)|<\delta$,

$\displaystyle\Big{|}\int_{0}^{T}\big{(}L(\tilde{\phi}(t),\phi^{\prime}(t))-L(\phi(t),\phi^{\prime}(t))\big{)}dt\Big{|}$
$\displaystyle\qquad\leq\Big{|}\int_{0}^{T}L(\tilde{\phi}(t),\phi^{\prime}(t))\mathds{1}_{\\{|\phi^{\prime}(t)|\geq
B\\}}dt\Big{|}+\Big{|}\int_{0}^{T}L(\phi(t),\phi^{\prime}(t))\mathds{1}_{\\{|\phi^{\prime}(t)|\geq
B\\}}dt\Big{|}$
$\displaystyle\qquad\qquad+\int_{0}^{T}\big{|}L(\tilde{\phi}(t),\phi^{\prime}(t))-L(\phi(t),\phi^{\prime}(t))\big{|}\mathds{1}_{\\{|\phi^{\prime}(t)|<B\\}}dt$
$\displaystyle\qquad<\epsilon.$

∎

###  4.5 $I$ is a good rate function

We first have

######  Lemma 4.19.

For $\delta>0$, $x\in A$ and $y\in\mathds{R}^{d}$, we define

$L_{\delta}(x,y):=\sup_{\theta\in\mathds{R}^{d}}\tilde{\ell}_{\delta}(x,y,\theta),$

where

$\tilde{\ell}_{\delta}(x,y,\theta):=\langle\theta,y\rangle-\sup_{z\in
A;|z-x|\leq\delta}\sum_{j}\beta_{j}(z)\big{(}\operatorname{e}^{\langle\theta,h_{j}\rangle}-1\big{)}.$

Since the $\beta_{j}$ are bounded and continuous, then

$L_{\delta}(x,y)\uparrow L_{0}(x,y)=L(x,y)$

and $L_{\delta}(x,y)$ is lower semicontinuous in $(\delta,x,y)$.

###### Proof.

It is easy to see that $\tilde{\ell}_{\delta}(x,y,\theta)$ is continuous is
$(x,y,\delta)$, hence the first assertion follows. The second assertion
follows from the fact that the supremum of a family of lower semicontinuous
functions is lower semicontinuous. ∎

We next establish (recall the metric $d_{D}$ introduced in subsection 2.1)

######  Lemma 4.20.

Let the $\beta_{j}$ be bounded and continuous. Then, $I_{T}$ is lower
semicontinuous with respect to the metric $d_{D}$ on $D([0,T];A)$.

###### Proof.

As $I_{T}(\phi)=\infty$ if $\phi$ is not absolutely continuous, we can
restrict our attention to sequences of absolutely continuous functions. As the
Skorohod topology relativized to $C([0,T];A)$ coincides with the uniform
topology (see, e.g., Billingsley (1999), Section 12, p.124), we can consider a
sequence of functions $\phi_{n}\in C([0,T];A)$ converging to a function $\phi$
under the uniform topology. We can furthermore assume that
$I_{T}(\phi_{n})\leq K$ for some $K$ and all $n\geq 1$. By Lemma 4.17, the
functions $\phi_{n}$ are hence uniformly absolutely continuous and therefore
the limit $\phi$ is absolutely continuous.

Therefore, for any given $\delta>0$, there exists a $\Delta>0$ such that

$|s-t|\leq\Delta\Rightarrow|\phi_{n}(s)-\phi_{n}(t)|\leq\delta\text{ for all
}n.$

We choose $\Delta$ smaller if necessary such that $T/\Delta=:J\in\mathds{N}$
and divide $[0,T]$ into subintervals $[t_{j},t_{j+1}]$, $j=1,\dots,J$ of
length $\leq\Delta$. We note that for $|x^{\prime}-x|\leq\delta$, we have
$L_{\delta}(x^{\prime},y)\leq L(x,y)$. Furthermore, we observe that
$L_{\delta}(x,\cdot)$ is convex as a supremum of linear functions and hence by
Jensen’s Inequality

$\displaystyle\int_{0}^{T}L(\phi_{n}(t),\phi_{n}^{\prime}(t))dt$
$\displaystyle\geq\sum_{j=1}^{J}\int_{t_{j}}^{t_{j+1}}L_{\delta}(\phi_{n}(t_{j}),\phi_{n}^{\prime}(t))dt$
$\displaystyle\geq\sum_{j=1}^{J}\Delta
L_{\delta}\Big{(}\phi_{n}(t_{j}),\frac{\phi_{n}(t_{j+1})-\phi_{n}(t_{j})}{\Delta}\Big{)}.$
(28)

We now further divide the interval $[0,T]$ into subintervals of length
$\Delta_{k}:=\Delta 2^{-k}$, $k\in\mathds{N}$, $[t_{j}^{k},t_{j+1}^{k}]$,
$j=1,\dots,J_{k}:=2^{k}J$ and define the functions

$\underline{\phi}^{k}(t):=\phi(t_{j}^{k})\quad\text{if
}t\in[t_{j}^{k},t_{j+1}^{k}],\quad\overline{\phi}^{k}(t):=\underline{\phi}^{k}(t+\Delta_{k}).$

Note that there exits a sequence $\delta_{k}\downarrow 0$ such that

$|s-t|<\Delta_{k}\Rightarrow|\phi_{n}(s)-\phi_{n}(t)|<\delta_{k}\text{ for all
}n.$

Hence by Inequality (28) and Lemma 4.19 for all $k\in\mathds{N}$,

$\displaystyle\liminf_{n\rightarrow\infty}\int_{0}^{T}L(\phi_{n}(t),\phi_{n}^{\prime}(t))dt$
$\displaystyle\geq\sum_{j=1}^{J_{k}}\Delta_{k}\liminf_{n\rightarrow\infty}L_{\delta_{k}}\Big{(}\phi_{n}(t_{j}^{k}),\frac{\phi_{n}(t_{j+1}^{k})-\phi_{n}(t_{j}^{k})}{\Delta_{k}}\Big{)}$
$\displaystyle\geq\int_{0}^{T-\Delta_{k}}L_{\delta_{k}}\Big{(}\underline{\phi}^{k}(t),\frac{\overline{\phi}^{k}(t)-\underline{\phi}^{k}(t)}{\Delta_{k}}\Big{)}dt.$
(29)

As $\phi$ is absolutely continuous, we have that for almost all $t\in[0,T]$,

$\frac{\overline{\phi}^{k}(t)-\underline{\phi}^{k}(t)}{\Delta_{k}}\rightarrow\phi^{\prime}(t)\quad\text{as
}k\rightarrow\infty.$

We conclude by using Inequality (29), Fatou’s Lemma and Lemma 4.19 again:

$\displaystyle\liminf_{n\rightarrow\infty}\int_{0}^{T}L(\phi_{n}(t),\phi_{n}^{\prime}(t))dt$
$\displaystyle\geq\liminf_{k\rightarrow\infty}\int_{0}^{T-\Delta_{k}}L_{\delta_{k}}\Big{(}\underline{\phi}^{k}(t),\frac{\overline{\phi}^{k}(t)-\underline{\phi}^{k}(t)}{\Delta_{k}}\Big{)}dt$
$\displaystyle\geq\int_{0}^{T}\liminf_{k\rightarrow\infty}\Big{(}\mathds{1}_{[0,T-\Delta_{k}]}(t)L_{\delta_{k}}\Big{(}\underline{\phi}^{k}(t),\frac{\overline{\phi}^{k}(t)-\underline{\phi}^{k}(t)}{\Delta_{k}}\Big{)}\Big{)}dt$
$\displaystyle\geq\int_{0}^{T}L(\phi(t),\phi^{\prime}(t))dt$

as required. ∎

We define for $K>0$, $x\in A$,

$\displaystyle\Phi(K)$ $\displaystyle=\big{\\{}\phi\in
D([0,T];A)|I_{T}(\phi)\leq K\big{\\}},$ $\displaystyle\Phi_{x}(K)$
$\displaystyle=\big{\\{}\phi\in D([0,T];A)|I_{T,x}(\phi)\leq K\big{\\}}.$

We have moreover

######  Proposition 4.21.

Assume that $\beta_{j}$ ($j=1,\dots,k$) are bounded and continuous. Let
furthermore $K>0$ and $\tilde{A}\subset A$ be compact.Then, the sets

$\bigcup_{x\in\tilde{A}}\Phi_{x}(K)$

are compact in $C([0,T];A)$.

###### Proof.

By Lemma 4.17, the functions in $\bigcup_{x\in\tilde{A}}\Phi_{x}(K)$ are
equicontinuous. As $\tilde{A}$ is compact, the Theorem of Arzelà-Ascoli hence
implies that $\bigcup_{x\in\tilde{A}}\Phi_{x}(K)$ has compact closure. Now,
the semicontinuity of $I$ (cf. Lemma 4.20) implies that
$\bigcup_{x\in\tilde{A}}\Phi_{x}(K)$ is closed which finishes the proof. ∎

We define for $S\subset D([0,T];A)$,

$I_{x}(S):=\inf_{\phi\in S}I_{T,x}(\phi).$

######  Lemma 4.22.

Assume that $\beta_{j}$ ($j=1,\dots,k$) are bounded and continuous. Let
$F\subset C([0,T];A)$ be closed. Then $I_{x}(F)$ is lower semicontinuous in
$x$.

###### Proof.

We let $x_{n}\rightarrow x$ with
$\liminf_{n\rightarrow\infty}I_{x_{n}}(F)=:K<\infty$. For simplicity, we
assume that $I_{x_{n}}(F)\leq K+\epsilon$ for some fixed $\epsilon>0$ and for
all $n$. By Proposition 4.21, we have that for all $\epsilon,\delta>0$,

$F\cap\Phi_{x_{n}}(K+\epsilon)\quad\text{and}\quad
F\cap\bigcup_{|x-y|\leq\delta}\Phi_{y}(K+\epsilon)$

are compact. By the semicontinuity of $I_{T}(\cdot)$ (cf. Lemma 4.20) and the
fact that a l.s.c. function attains its minimum on a compact set, there exist
$\phi_{n}\in F$ such that $I_{x_{n}}(F)=I_{T,x_{n}}(\phi_{n})$ (for $n$ large
enough). As the $\phi_{n}$ are in a compact set, there exists a convergent
subsequence with limit $\phi$, in particular $\phi(0)=x$. As $F$ is closed, we
have $\phi\in F$. We use Lemma 4.20 again and obtain

$I_{x}(F)\leq
I_{T}(\phi)\leq\liminf_{n\rightarrow\infty}I_{T}(\phi_{n})=\liminf_{n\rightarrow\infty}I_{x_{n}}(F)=K$

as required. ∎

The following result is a direct consequence of Lemma 4.22.

######  Lemma 4.23.

Assume that $\beta_{j}$ ($j=1,\dots,k$) is bounded and continuous. For
$F\subset D([0,T];A)$ closed and $x\in A$, we have

$\lim_{\epsilon\downarrow 0}\inf_{y\in A,\,|x-y|<\epsilon}I_{y}(F)=I_{x}(F)$

We can now establish the main result of this subsection.

######  Proposition 4.24.

Let the $\beta_{j}$ be bounded and continuous. For all $x$, $I_{x}$ is a good
rate function on $C([0,T];A)\cap\\{\phi|\phi(0)=x\\}$.

###### Proof.

It is clear that $I_{T}$ is non-negative as $L$ is non-negative. Furthermore,
it is lower semicontinuous by Lemma 4.20. By Proposition 4.21 its level sets
are compact. ∎

We have moreover

######  Corollary 4.25.

Let the $\beta_{j}$ be bounded and continuous. For all $x\in A$, $I_{T,x}$ is
a good rate function on $D([0,T];A)\cap\\{\phi|\phi(0)=x\\}$ under both
metrics $d_{C}$ and $d_{D}$.

###### Proof.

Since $I_{t,x}$ is finite only for absolutely continuous functions, it
suffices to consider sequences in $C([0,T];A)\cap\\{\phi|\phi(0)=x\\}$. Limits
of such sequences (under either metric) are continuous and convergence is
equivalent for both metrics (see, e.g., Billingsley (1999)). Lower
semicontinuity follows. Compactness of the level sets follows by Proposition
4.24 and the fact that the identity maps from $(C([0,T];A),d_{C})$ into
$(D([0,T];A),d_{C})$ and $(D([0,T];A),d_{D})$ are continuous . ∎

###  4.6 A property of non–exponential equivalence

It is worth wondering whether or not $\\{Z^{N,x_{N}}_{t},\ 0\leq t\leq T\\}$
and $\\{Z^{N,x}_{t},\ 0\leq t\leq T\\}$ are exponentially equivalent, whenever
$x_{N}\to x$ as $N\to\infty$. Indeed, Dembo and Zeitouni (2009) prove that
property for diffusions with small noise and Lipschitz coefficients, and use
it to establish certain results, of which we shall prove analogs below, but
without that exponential equivalence, which fails to hold in our Poissonian
case.

Let $x,y\in A$, and consider the processes

$\displaystyle Z^{N,x}_{t}$
$\displaystyle=x+\sum_{j=1}^{k}\frac{h_{j}}{N}P_{j}\left(N\int_{0}^{t}\beta_{j}(Z^{N,x}_{s})ds\right),$
$\displaystyle Z^{N,y}_{t}$
$\displaystyle=y+\sum_{j=1}^{k}\frac{h_{j}}{N}P_{j}\left(N\int_{0}^{t}\beta_{j}(Z^{N,y}_{s})ds\right).$

For any $\delta>0$, as $|x-y|\to 0$, we ask what is the limit, as $|x-y|\to
0$, of

$\limsup_{N\to\infty}\frac{1}{N}\log\mathbb{P}\left(\sup_{0\leq t\leq
T}|Z^{N,x}_{t}-Z^{N,y}_{t}|>\delta\right)\ \text{?}$ (30)

If that limit is $-\infty$, then we would have the above exponential
equivalence. We now show on a particularly simple example that this is not the
case. It is easy to infer that it in fact fails in the above generality,
assuming that the $\beta_{j}$’s are Lipschitz continuous and bounded. We
consider the case $d=1$, $A=\mathbb{R}_{+}$, $k=1$, $\beta(x)=x$, $h=1$. We
could truncate $\beta(x)$ to make it bounded, in order to comply with our
standing assumptions. The modifications below would be minor, but we prefer to
keep the simplest possible notations. Assume $0<x<y$ and consider the two
processes

$\displaystyle Z^{N,x}_{t}$
$\displaystyle=x+\frac{1}{N}P\left(N\int_{0}^{t}Z^{N,x}_{s}ds\right),$
$\displaystyle Z^{N,y}_{t}$
$\displaystyle=y+\frac{1}{N}P\left(N\int_{0}^{t}Z^{N,y}_{s}ds\right).$

It is plain that $0<Z^{N,x}_{t}<Z^{N,y}_{t}$ for all $N\geq 1$ and $t>0$. Let
$\Delta^{N,x,y}_{t}=Z^{N,y}_{t}-Z^{N,x}_{t}$. The law of
$\\{\Delta^{N,x,y}_{t},\ 0\leq t\leq T\\}$ is the same as that of the solution
of

$\Delta^{N,x,y}_{t}=y-x+\frac{1}{N}P\left(N\int_{0}^{t}\Delta^{N,x,y}_{s}ds\right).$

We deduce from Theorem 5.10 below (which is established in case of a bounded
coefficient $\beta(x)$, but it makes no difference here) that

$\displaystyle\liminf_{N\to\infty}\frac{1}{N}\log\mathbb{P}[\Delta^{N,x,y}_{1}>1]$
$\displaystyle\geq-\inf_{\phi(0)=y-x,\ \phi(1)>1}I_{1,y-x}(\phi)$
$\displaystyle\geq-I_{1,y-x}(\psi),$

with $\psi(t)=y-x+t$, hence

$\displaystyle I_{1,y-x}(\psi)$ $\displaystyle=\int_{0}^{1}L(y-x+t,1)dt$
$\displaystyle=\int_{0}^{1}[y-x+t-1-\log(y-x+t)]dt$
$\displaystyle=y-x+1/2-(y-x+1)\log(y-x+1)+(y-x)\log(y-x)$ $\displaystyle\to
1/2,$

as $y-x\to 0$. This clearly contradicts the exponential equivalence.

We note that the above process $Z^{N,x}_{t}$ can be shown to be “close” (in a
sense which is made very precise in Kurtz (1978)) to its diffusion
approximation

$X^{N,x}_{t}=x+\int_{0}^{t}X^{N,x}_{s}ds+\frac{1}{\sqrt{N}}\int_{0}^{t}\sqrt{X^{N,x}_{s}}dB_{s},$

where $\\{B_{t},\ t\geq 0\\}$ is standard Brownian motion. One can study large
deviations of this diffusion process from its Law of Large Numbers limit
(which is the same as that of $Z^{N,x}_{t}$). The rate function on the time
interval $[0,1]$ is now

$I(\phi)=\int_{0}^{1}\frac{(\phi^{\prime}(t)-\phi(t))^{2}}{\phi(t)}dt.$

Let again $\psi(t)=y-x+t$, now with $0=x<y$.
$I(\psi)=\log(1+y)-\log(y)-3/2+y\to+\infty$, as $y\to 0$. We see here that the
large deviations behaviour of the solution of the Poissonian SDE and of its
diffusion approximation differ dramatically, as was already noted by Pakdaman
et al. (2010) (see also the references in this paper).

##  5 Lower bound

We first establish the LDP lower bound under the assumption that the rates are
bounded away from zero, or in other words the $\log\beta_{j}$’s are bounded.
From this, we will derive later the general result.

###  5.1 LDP lower bound if the rates are bounded away from zero

We first note that if the $\beta_{j}$ are bounded away from zero, then the
convex cone $\mathcal{C}_{x}$ is dependent of $x$,
$\mathcal{C}_{x}=\mathcal{C}$ for all $x$. Note that this implies that the
“domain” $A$ of the process cannot be bounded.

We require a LDP for linear functions. This follows from the LLN (Theorem
3.1).

######  Proposition 5.1.

Assume that $\log\beta_{j}$ ($j=1,\dots,k$) is bounded and continuous. For any
$\epsilon>0$, $\delta>0$ there exists an $\tilde{\epsilon}>0$ such that for
$x\in A$, $y\in\mathcal{C}$ and $\mu\in V_{x,y}=\tilde{V}_{y}$,

$\liminf_{N\rightarrow\infty}\frac{1}{N}\log\Big{(}\inf_{z\in
A;|z-x|<\tilde{\epsilon}}\mathbb{P}\Big{[}\sup_{t\in[0,T]}|Z^{N,z}(t)-\phi^{x}(t)|<\epsilon\Big{]}\Big{)}\geq-\int_{0}^{T}\ell(\phi^{x}(t),\mu)dt-\delta,$

where

$\phi^{x}(t):=x+ty=x+t\sum_{j}\mu_{j}h_{j}.$

###### Proof.

We define

$F^{N,z}:=\Big{\\{}\sup_{t\in[0,T]}|Z^{N,z}(t)-\phi^{z}(t)|<\frac{\epsilon}{2}\Big{\\}}$

and let $\tilde{\epsilon}<\epsilon_{1}=\epsilon/2$. Let now
$\xi_{T}=\xi^{N,z}_{T}=\frac{d\tilde{\mathbb{P}}}{d\mathbb{P}}|_{\mathcal{F}_{T}}$
be given as in Theorem A.1 for initial value $z$ and constant rates
$\tilde{\beta}_{j}=\mu_{j}$. Then, with the notation
$\tilde{\mathbb{E}}_{F^{N,z}}[X]:=\tilde{\mathbb{E}}[X|F^{N,z}]$ and (recall
that $\xi_{T}\not=0$ $\tilde{\mathbb{P}}$-almost surely)

$X_{T}^{N,z}:=X_{T}:=\log\xi_{T}^{-1}=\sum_{\tau\leq
T}\Big{[}\log\beta_{j(\tau)}(Z^{N,z}(\tau-))-\log\mu_{j(\tau)}\Big{]}-N\sum_{j}\int_{0}^{T}\big{(}\beta_{j}(Z^{N,z}(t))-\mu_{j}\big{)}dt,$

$\displaystyle\liminf_{N\rightarrow\infty}\frac{1}{N}\log\Big{(}\inf_{z\in
A,\,|x-z|<\tilde{\epsilon}}\mathbb{P}\big{[}\sup_{t\in[0,T]}|Z^{N,z}(t)-\phi^{x}(t)|<\epsilon\big{]}\Big{)}$
$\displaystyle\qquad\geq\liminf_{N\rightarrow\infty}\frac{1}{N}\log\inf_{z\in
A,\,|x-z|<\tilde{\epsilon}}\mathbb{P}[F^{N,z}]$
$\displaystyle\qquad=\liminf_{N\rightarrow\infty}\frac{1}{N}\inf_{z\in
A,\,|x-z|<\tilde{\epsilon}}\log\mathbb{P}[F^{N,z}]$
$\displaystyle\qquad\geq\liminf_{N\rightarrow\infty}\frac{1}{N}\inf_{z\in
A,\,|x-z|<\tilde{\epsilon}}\log\tilde{\mathbb{E}}\big{[}\xi_{T}^{-1}\mathds{1}_{F^{N,z}}\big{]}$
$\displaystyle\qquad=\liminf_{N\rightarrow\infty}\frac{1}{N}\inf_{z\in
A,\,|x-z|<\tilde{\epsilon}}\log\big{(}\tilde{\mathbb{P}}[F^{N,z}]\tilde{\mathbb{E}}_{F^{N,z}}[\exp(X_{T})]\big{)}$
$\displaystyle\qquad\geq\liminf_{N\rightarrow\infty}\frac{1}{N}\inf_{z\in
A,\,|x-z|<\tilde{\epsilon}}\log\tilde{\mathbb{P}}[F^{N,z}]+\liminf_{N\rightarrow\infty}\frac{1}{N}\inf_{z\in
A,\,|x-z|<\tilde{\epsilon}}\log\tilde{\mathbb{E}}_{F^{N,z}}[\exp(X_{T})]$
$\displaystyle\qquad\geq\liminf_{N\rightarrow\infty}\inf_{z\in
A,\,|x-z|<\tilde{\epsilon}}\tilde{\mathbb{E}}_{F^{N,z}}\Big{[}\frac{X_{T}}{N}\Big{]},$
(31)

where we have used Corollary A.2 for the second inequality, Theorem 3.1 and
Jensen’s inequality on the last line. Note the independence of the constants
$\tilde{C}_{1},\tilde{C}_{2}$ of $z$ in Theorem 3.1 and hence

$\tilde{\mathbb{P}}[F^{N,z}]\rightarrow 1\quad\text{as
}N\rightarrow\infty\text{ independently of }z.$

We have

$\frac{1}{\tilde{\mathbb{P}}[F^{N,z}]}\tilde{\mathbb{E}}\big{[}\mathds{1}_{F^{N,z}}T\sum_{j}\mu_{j}\big{]}=T\sum_{j}\mu_{j}.$
(32)

By the fact that the $\beta_{j}$’s are bounded and continuous and by Theorem
3.1, we have for $j=1,\dots,k$,

$\sup_{t\in[0,T]}|\beta_{j}(Z^{N,z}(t))-\beta_{j}(\phi^{z}(t))|\rightarrow
0\quad\tilde{\mathbb{P}}-\text{a.s.}$

as $N\rightarrow\infty$ uniformly in $z$. This implies

$\frac{1}{\tilde{\mathbb{P}}[F^{N,z}]}\tilde{\mathbb{E}}\Big{[}\mathds{1}_{F^{N,z}}\int_{0}^{T}\sum_{j}\beta_{j}(Z^{N,z}(t))dt\Big{]}\longrightarrow\sum_{j}\int_{0}^{T}\beta_{j}(\phi^{z}(t))dt$
(33)

as $N\rightarrow\infty$ uniformly in $z$.

Let us now define the following processes. For $z\in A$, $j=1,\dots,k$ and
$0\leq t_{1}<t_{2}\leq T$ let

$Y_{j}^{N,z,t_{1},t_{2}}:=\frac{1}{N}\cdot\\#\text{jumps of }Z^{N,z}\text{ in
direction }h_{j}\text{ in }[t_{1},t_{2}].$

Let furthermore $\tau_{j}\in[0,T]$ denote the jump times of $Z^{N}$ in
direction $h_{j}$; we obtain

$\displaystyle\frac{1}{\tilde{\mathbb{P}}[F^{N,z}]}\sum_{j;\mu_{j}>0}\tilde{\mathbb{E}}\Big{[}\frac{1}{N}\mathds{1}_{F^{N,z}}\sum_{\tau_{j}}\log\mu_{j}\Big{]}$
$\displaystyle\qquad=\frac{1}{\tilde{\mathbb{P}}[F^{N,z}]}\sum_{j;\mu_{j}>0}\log\mu_{j}\Big{\\{}\tilde{\mathbb{E}}\big{[}Y_{j}^{N,z,0,T}\big{]}\tilde{\mathbb{P}}\big{[}F^{N,z}\big{]}+\widetilde{\operatorname{Cov}}(\mathds{1}_{F^{N,z}},Y_{j}^{N,z,0,T})\Big{\\}}$
$\displaystyle\qquad\rightarrow T\sum_{j}\mu_{j}\log\mu_{j},$ (34)

since, for a given set $F$,

$\displaystyle\tilde{\mathbb{E}}[Y_{j}^{N,z,t_{1},t_{2}}]$
$\displaystyle=(t_{2}-t_{1})\mu_{j}$
$\displaystyle\widetilde{\operatorname{Var}}[Y_{j}^{N,z,t_{1},t_{2}}]$
$\displaystyle=(t_{2}-t_{1})\mu_{j}$
$\displaystyle|\widetilde{\operatorname{Cov}}(\mathds{1}_{F},Y_{j}^{N,z,t_{1},t_{2}})|$
$\displaystyle\leq\sqrt{\widetilde{\operatorname{Var}}[\mathds{1}_{F}]}\sqrt{\widetilde{\operatorname{Var}}[Y_{j}^{N,z,t_{1},t_{2}}]}=\sqrt{\tilde{\mathbb{P}}[F]-\tilde{\mathbb{P}}[F]^{2}}\sqrt{(t_{2}-t_{1})\mu_{j}}.$

We now define the set

$\tilde{F}^{N,z}:=\Big{\\{}\sup_{t\in[0,T]}|Z^{N,z}(t)-\phi^{z}(t)|<\epsilon_{N}\Big{\\}}\quad\text{for
}\epsilon_{N}:=\epsilon\wedge\frac{1}{N^{1/3}};$

we have (for $N$ large enough)

$\tilde{\mathbb{P}}[\tilde{F}^{N,z}]\geq
1-\tilde{C}_{1}\exp\big{(}-N\tilde{C}_{2}(\epsilon_{N})\big{)}\longrightarrow
1$

as $N\rightarrow\infty$ uniformly in $z$ by Theorem 3.1. We furthermore let
$\bar{A}\subset A$ be compact such that for all $z$, $|z-x|<\tilde{\epsilon}$
and all $t\in[0,T]$, $\phi^{z}(t)\in\bar{A}$ and $Z^{N,z}(t)\in\bar{A}$ on
$F^{N,z}$. As the $\log\beta_{j}$ are bounded and uniformly continuous, there
exit constants $\tilde{\delta}_{N}>0$ with $\tilde{\delta}_{N}\downarrow 0$
such that

$\tilde{x},\bar{x}\in\bar{A},\,|\tilde{x}-\bar{x}|<\frac{2}{N^{1/3}}\Rightarrow|\log\beta_{j}(\tilde{x})-\log\beta_{j}(\bar{x})|<\tilde{\delta}_{N}.$

We define $\bar{\mu}=\max_{j}\mu_{j}$,

$M=M(N):=\lfloor TN^{{1}/{3}}k\bar{h}\bar{\mu}+1\rfloor$

and divide the interval $[0,T]$ into $M$ subintervals $[t_{r},t_{r+1}]$
($r=0,\dots,M-1$, $t_{r}=t_{r}(N)$) of length $\Delta=\Delta(N)$, i.e. for
$N\geq N_{0}$ independent of $z$ large enough,

$\Delta<\frac{1}{N^{1/3}k\bar{\mu}\bar{h}}.$

For $j$, $r=0,\dots,M-1$ and $\tau_{j},t\in[t_{r},t_{r+1}]$, since for
$|t-s|<\frac{1}{N^{1/3}k\bar{\mu}\bar{h}}$,
$|\phi^{z}(t)-\phi^{z}(s)|<\frac{1}{N^{1/3}}$, we have on $\tilde{F}^{N,z}$

$|Z^{N,z}(\tau_{j}-)-\phi^{z}(t)|\leq|Z^{N,z}(\tau_{j}-)-\phi^{z}(\tau_{j})|+|\phi^{z}(\tau_{j})-\phi^{z}(t)|\leq\frac{2}{N^{1/3}},$

and hence

$\inf_{t\in[t_{r},t_{r+1}]}\log\beta_{j}(\phi^{z}(t))-\tilde{\delta}_{N}\leq\log\beta_{j}(Z^{N,z}(\tau_{j}-))\leq\sup_{t\in[t_{r},t_{r+1}]}\log\beta_{j}(\phi^{z}(t))+\tilde{\delta}_{N}.$

We compute

$\displaystyle\frac{1}{\tilde{\mathbb{P}}[F^{N,z}]}\tilde{\mathbb{E}}\Big{[}\frac{1}{N}\mathds{1}_{F^{N,z}}\sum_{\tau}\log\beta_{j(\tau)}(Z^{N,z}(\tau-))\Big{]}$
$\displaystyle\qquad=\frac{1}{\tilde{\mathbb{P}}[F^{N,z}]}\sum_{j,\,\mu_{j}>0}\sum_{r=0}^{M-1}\tilde{\mathbb{E}}\Big{[}\frac{1}{N}\mathds{1}_{\tilde{F}^{N,z}}\sum_{\tau_{j}\in[t_{r},t_{r+1})}\log\beta_{j}(Z^{N,z}(\tau_{j}-))\Big{]}$
$\displaystyle\qquad\qquad+\frac{1}{\tilde{\mathbb{P}}[F^{N,z}]}\sum_{j,\,\mu_{j}>0}\sum_{r=0}^{M-1}\tilde{\mathbb{E}}\Big{[}\frac{1}{N}\mathds{1}_{F^{N,z}\setminus\tilde{F}^{N,z}}\sum_{\tau_{j}\in[t_{r},t_{r+1})}\log\beta_{j}(Z^{N,z}(\tau_{j}-))\Big{]}$
$\displaystyle\qquad\leq\frac{1}{\tilde{\mathbb{P}}[F^{N,z}]}\sum_{j,\,\mu_{j}>0}\sum_{r=0}^{M-1}\Big{(}\sup_{t\in[t_{r},t_{r+1})}\log\beta_{j}(\phi^{z}(t))+\tilde{\delta}_{N}\Big{)}\tilde{\mathbb{E}}\Big{[}\mathds{1}_{\tilde{F}^{N,z}}Y_{j}^{N,z,t_{r},t_{r+1}}\Big{]}$
$\displaystyle\qquad\qquad+\frac{1}{\tilde{\mathbb{P}}[F^{N,z}]}\sum_{j,\,\mu_{j}>0}\sum_{r=0}^{M-1}\log\bar{\beta}\tilde{\mathbb{E}}\Big{[}\mathds{1}_{F^{N,z}\setminus\tilde{F}^{N,z}}Y_{j}^{N,z,t_{r},t_{r+1}}\Big{]}$
$\displaystyle\qquad\leq\frac{1}{\tilde{\mathbb{P}}[F^{N,z}]}\sum_{j,\,\mu_{j}>0}\sum_{r=0}^{M-1}\sup_{t\in[t_{r},t_{r+1})}\log\beta_{j}(\phi^{z}(t))\Big{\\{}\tilde{\mathbb{E}}\big{[}\mathds{1}_{\tilde{F}^{N,z}}\big{]}\tilde{\mathbb{E}}\big{[}Y_{j}^{N,z,t_{r},t_{r+1}}\big{]}+\widetilde{\operatorname{Cov}}(\mathds{1}_{\tilde{F}^{N,z}},Y_{j}^{N,z,t_{r},t_{r+1}})\Big{\\}}$
$\displaystyle\qquad\qquad+\frac{1}{\tilde{\mathbb{P}}[F^{N,z}]}\sum_{j,\,\mu_{j}>0}\sum_{r=0}^{M-1}\log\bar{\beta}\Big{\\{}\tilde{\mathbb{E}}\big{[}\mathds{1}_{F^{N,z}\setminus\tilde{F}^{N,z}}\big{]}\tilde{\mathbb{E}}\big{[}Y_{j}^{N,z,t_{r},t_{r+1}}\big{]}+\widetilde{\operatorname{Cov}}(\mathds{1}_{F^{N,z}\setminus\tilde{F}^{N,z}},Y_{j}^{N,z,t_{r},t_{r+1}})\Big{\\}}$
$\displaystyle\qquad\qquad+\frac{1}{\tilde{\mathbb{P}}[F^{N,z}]}\tilde{\delta}_{N}k\bar{\mu}T$
$\displaystyle\qquad\leq\frac{\tilde{\mathbb{P}}[\tilde{F}^{N,z}]}{\tilde{\mathbb{P}}[F^{N,z}]}\sum_{j,\,\mu_{j}>0}\mu_{j}\sum_{r=0}^{M-1}\Delta\sup_{t\in[t_{r},t_{r+1})}\log\beta_{j}(\phi^{z}(t))$
$\displaystyle\qquad\qquad+\frac{1}{\tilde{\mathbb{P}}[F^{N,z}]}\Big{\\{}\log\bar{\beta}\sum_{j,\,\mu_{j}>0}\sum_{r=0}^{M-1}\Big{(}|\widetilde{\operatorname{Cov}}(\mathds{1}_{\tilde{F}^{N,z}},Y_{j}^{N,z,t_{r},t_{r+1}})|+|\widetilde{\operatorname{Cov}}(\mathds{1}_{F^{N,z}\setminus\tilde{F}^{N,z}},Y_{j}^{N,z,t_{r},t_{r+1}})|\Big{)}$
$\displaystyle\qquad\qquad+\tilde{\delta}_{N}k\bar{\mu}T+\tilde{\mathbb{P}}[F^{N,z}\setminus\tilde{F}^{N,z}]k\bar{\mu}T\log\bar{\beta}\Big{\\}}$
(35) $\displaystyle\qquad=:\overline{S}^{N,z}+\overline{U}^{N,z},$

where $\overline{S}^{N,z}$ and $\overline{U}^{N,z}$ are the first respectively
the second term in Inequality (35). In a similar fashion we obtain

$\displaystyle\frac{1}{\tilde{\mathbb{P}}[F^{N,z}]}\tilde{\mathbb{E}}\Big{[}\frac{1}{N}\mathds{1}_{F^{N,z}}\sum_{\tau}\log\beta_{j(\tau)}(Z^{N,z}(\tau-))\Big{]}$
$\displaystyle\qquad\geq\frac{\tilde{\mathbb{P}}[\tilde{F}^{N,z}]}{\tilde{\mathbb{P}}[F^{N,z}]}\sum_{j,\,\mu_{j}>0}\mu_{j}\sum_{r=0}^{M-1}\Delta\inf_{t\in[t_{r},t_{r+1})}\log\beta_{j}(\phi^{z}(t))$
$\displaystyle\qquad\qquad+\frac{1}{\tilde{\mathbb{P}}[F^{N,z}]}\Big{\\{}\log\underline{\beta}\sum_{j,\,\mu_{j}>0}\sum_{r=0}^{M-1}\Big{(}|\widetilde{\operatorname{Cov}}(\mathds{1}_{\tilde{F}^{N,z}},Y_{j}^{N,z,t_{r},t_{r+1}})|+|\widetilde{\operatorname{Cov}}(\mathds{1}_{F^{N,z}\setminus\tilde{F}^{N,z}},Y_{j}^{N,z,t_{r},t_{r+1}})|\Big{)}$
$\displaystyle\qquad\qquad-\tilde{\delta}_{N}k\bar{\mu}T+\tilde{\mathbb{P}}[F^{N,z}\setminus\tilde{F}^{N,z}]k\bar{\mu}T\log\underline{\beta}\Big{\\}}$
$\displaystyle\qquad=:\underline{S}^{N,z}+\underline{U}^{N,z};$

we first note that $\overline{U}^{N,z},\underline{U}^{N,z}\rightarrow 0$ as
$N\rightarrow\infty$ uniformly in $z$, since $\tilde{\delta}_{N}\rightarrow 0$
and $\tilde{\mathbb{P}}[F^{N,z}\setminus\tilde{F}^{N,z}]\rightarrow 0$ as
$N\rightarrow 0$ uniformly in $z$. Furthermore, as (up to a factor which
converges to $1$ uniformly in $z$) $\overline{S}^{N,z}$ and
$\underline{S}^{N,z}$ are upper respectively lower Riemann sums, we obtain

$\overline{S}^{N,z},\underline{S}^{N,z}\rightarrow\sum_{j}\mu_{j}\int_{0}^{T}\log\beta_{j}(\phi^{z}(t))dt$
(36)

as $N\rightarrow\infty$; since

$|\overline{S}^{N,z}-\underline{S}^{N,z}|\leq
2\frac{\tilde{\mathbb{P}}[\tilde{F}^{N,z}]}{\tilde{\mathbb{P}}[F^{N,z}]}k\bar{\mu}\tilde{\delta}_{N}\rightarrow
0$

uniformly in $z$, the convergence in (36) is likewise uniform in $z$.

The uniform convergence implies (cf. (31) - (34) and the preceding discussion)
that

$\liminf_{N\rightarrow\infty}\frac{1}{N}\log\Big{(}\inf_{z\in
A,\,|x-z|<\tilde{\epsilon}}\mathbb{P}\big{[}\sup_{t\in[0,T]}|Z^{N,z}(t)-\phi^{x}(t)|<\epsilon\big{]}\Big{)}\geq-\sup_{z\in
A,\,|x-z|<\tilde{\epsilon}}\int_{0}^{T}\ell(\phi^{z}(t),\mu)dt$

In combination with the uniform continuity of $\ell(\cdot,\mu)$ (recall the
boundedness of the $\log\beta_{j}$) this proves the assertion. ∎

The main building block for the lower bound is the following result.

######  Theorem 5.2.

Assume that $\log\beta_{j}$ ($j=1,\dots,k$) is bounded and continuous. Let
$\phi\in D([0,T];A)$ with $\phi(0)=x$ and $\epsilon>0$. Then,

$\liminf_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{P}\Big{[}\sup_{t\in[0,T]}|Z^{N,x}(t)-\phi(t)|<\epsilon\Big{]}\geq-
I_{T,x}(\phi).$

The convergence is uniform in $x\in A$.

###### Proof.

We can w.l.o.g. assume that $I_{T,x}(\phi)<\infty$ (and hence $\phi$ is
absolutely continuous) as else the assertion is trivial. We approximate the
function $\phi$ by a continuous piecewise linear function and then apply the
LDP for linear functions to each of these linear functions. To this end, we
let $\delta>0$ and divide the interval $[0,T]$ into $J$ subintervals of length
$\Delta=T/J$, $[t_{r-1},t_{r}]$ ($r=1,\dots,J$) such that the resulting
piecewise linear approximation

$\tilde{\phi}(t)=\phi(t_{r-1})+\frac{t-t_{r-1}}{\Delta}(\phi(t_{r})-\phi(t_{r-1}))$

satisfies

$\sup_{t\in[0,T]}|\phi(t)-\tilde{\phi}(t)|<\frac{\epsilon}{2}$

(recall that $\phi$ is continuous).

We now apply Theorem 4.18 twice (in Inequalities (37) and (39)) and choose $J$
large enough in order to assure

$\displaystyle\int_{0}^{T}L(\phi(t),\phi^{\prime}(t))dt$
$\displaystyle=\sum_{r=1}^{J}\int_{t_{r-1}}^{t_{r}}L(\phi(t),\phi^{\prime}(t))dt$
$\displaystyle\geq\sum_{r=1}^{J}\int_{t_{r-1}}^{t_{r}}L(\phi(t_{r-1}),\phi^{\prime}(t))dt-\frac{\delta}{4}$
(37)
$\displaystyle\geq\Delta\sum_{r=1}^{J}L\Big{(}\phi(t_{r-1}),\frac{\Delta\phi(t_{r})}{\Delta}\Big{)}-\frac{\delta}{4}$
(38)
$\displaystyle\geq\sum_{r=1}^{J}\int_{t_{r-1}}^{t_{r}}L\big{(}\tilde{\phi}(t),\tilde{\phi}^{\prime}(t)\big{)}dt-\frac{\delta}{2},$
(39)

where

$\Delta\phi(t_{r}):=\phi(t_{r})-\phi(t_{r-1}).$

Note that for Inequality (38), we have applied Jensen’s inequality and the
fact that $L$ is convex in its second argument (cf. Corollary 4.10). As
$I_{T,x}(\tilde{\phi})<\infty$, this implies

$\Delta\phi(t_{r})\in\mathcal{C}\quad\text{for all }r.$

We note that by the continuity of $L(\cdot,y)$, $\mu^{*}(x,y)$ (the minimizing
$\mu\in V_{x,y}=\tilde{V}_{y}$ for $\ell(x,\cdot)$) is “almost optimal” for
all $\tilde{x}$ sufficiently close to $x$ (in the sense that
$\ell(\tilde{x},\mu^{*}(x,y))$ is close to $L(\tilde{x},y)$). By dividing each
interval $[t_{r-1},t_{r}]$ into further subintervals $[s_{j-1},s_{j}]$ if
necessary, we can hence represent the directions $\Delta\phi(t_{k})/\Delta$ by

$\mu^{j}\in
V_{\tilde{\phi}(t),\Delta\phi(t_{r})/\Delta}=\tilde{V}_{\Delta\phi(t_{r})/\Delta}$

in such a way that

$L\Big{(}\tilde{\phi}(t),\frac{\Delta\phi(t_{r})}{\Delta}\Big{)}\geq\ell(\tilde{\phi}(t),\mu^{j})-\frac{\delta}{4T}\quad\text{for
all }t\in[s_{j-1},s_{j}].$

For simplicity of exposition, we assume that this further subdivision of the
intervals $[t_{r-1},t_{r}]$ is not required and denote the “almost optimal”
$\mu$’s by $\mu^{r}$ ($r=1,\dots,J$). Hence

$\int_{0}^{T}L(\phi(t),\phi^{\prime}(t))dt\geq\sum_{r=1}^{J}\int_{t_{r-1}}^{t_{r}}\ell(\tilde{\phi}(t),\mu^{r})-\frac{3\delta}{4}.$
(40)

Choose now $\tilde{\epsilon}=\tilde{\epsilon}_{J-1}$ according to Proposition
5.1 corresponding to $\epsilon/2$, $\delta/(4J)$, initial value
$\tilde{\phi}(t_{J-1})$ and time-horizon $\Delta$. Using the Markov property
of $Z^{N}$, we compute

$\displaystyle\mathbb{P}\Big{[}\sup_{t\in[0,T]}|Z^{N,x}(t)-\phi(t)|<\epsilon\Big{]}$
$\displaystyle\geq\mathbb{P}\Big{[}\sup_{t\in[0,t_{J-1}]}|Z^{N,x}(t)-\tilde{\phi}(t)|<\tilde{\epsilon}\Big{]}$
$\displaystyle\quad\cdot\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\inf_{z\in
A;|z-\tilde{\phi}(t_{J-1})|<\tilde{\epsilon}}\mathbb{P}\Big{[}\sup_{t\in[t_{J-1},T]}|Z^{N,z}(t)-\tilde{\phi}(t)|<\frac{\epsilon}{2}\Big{]};$

here, we denote (by a slight abuse of notation) the process starting at $z$ at
time $t_{J-1}$ by $Z^{N,z}$. Proposition 5.1 implies

$\displaystyle\liminf_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{P}\Big{[}\sup_{t\in[0,T]}|Z^{N,x}(t)-\phi(t)|<\epsilon\Big{]}$
$\displaystyle\qquad\geq\liminf_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{P}\Big{[}\sup_{t\in[0,t_{J-1}]}|Z^{N,x}(t)-\tilde{\phi}(t)|<\tilde{\epsilon}_{J-1}\Big{]}$
$\displaystyle\qquad\qquad+\liminf_{N\rightarrow\infty}\frac{1}{N}\log\Big{(}\inf_{z\in
A;|z-\tilde{\phi}(t_{J-1})|<\tilde{\epsilon}}\mathbb{P}\Big{[}\sup_{t\in[t_{J-1},T]}|Z^{N,z}(t)-\tilde{\phi}(t)|<\frac{\epsilon}{2}\Big{]}\Big{)}$
$\displaystyle\qquad\geq\liminf_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{P}\Big{[}\sup_{t\in[0,t_{J-1}]}|Z^{N,x}(t)-\tilde{\phi}(t)|<\tilde{\epsilon}_{J-1}\Big{]}-\int_{t_{J-1}}^{T}\ell(\tilde{\phi}(t),\mu^{J})dt-\frac{\delta}{4J}.$

Iterating this procedure, we obtain

$\liminf_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{P}\Big{[}\sup_{t\in[0,T]}|Z^{N,x}(t)-\phi(t)|<\epsilon\Big{]}\geq-\sum_{r=1}^{J}\int_{t_{r-1}}^{t_{r}}\ell(\tilde{\phi}(t),\mu^{r})dt-\frac{\delta}{4}$

and the assertion follows from Inequality (40) if we let $\delta\rightarrow
0$.

We note that the convergence is uniform in $x$ by the uniformity in
Proposition 5.1. ∎

######  Theorem 5.3.

Assume that $\log\beta_{j}$ ($j=1,\dots,k$) is bounded and continuous. Let
$G\subset D([0,T];A)$ be open and $x\in A$. Then,

$\liminf_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{P}[Z^{N,x}\in
G]\geq-\inf_{\phi\in G,}I_{T,x}(\phi).$

The convergence is uniform in $x\in A$.

###### Proof.

Let $\inf_{\phi\in G}I_{T,x}(\phi)=:I^{*}<\infty$; hence, for $\delta>0$,
there exists a $\phi^{\delta}\in G$ ($\phi(0)=x$) with
$I_{T,x}(\phi^{\delta})\leq I^{*}+\delta$. For small enough
$\epsilon=\epsilon(\phi^{\delta})>0$, we have

$\Big{\\{}\phi\in
D([0,T];A)|\sup_{t\in[0,T]};|\phi^{\delta}(t)-\phi(t)|<\epsilon\Big{\\}}\subset
G$

and therefore

$\mathbb{P}\Big{[}\sup_{t\in[0,T]}|Z^{N,x}(t)-\phi^{\delta}(t)|<\epsilon\Big{]}\leq\mathbb{P}[Z^{N,x}\in
G].$

This implies by Theorem 5.2 that for all $\delta>0$,

$\displaystyle\liminf_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{P}[Z^{N,x}\in
G]$
$\displaystyle\geq\liminf_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{P}\Big{[}\sup_{t\in[0,T]}|Z^{N,x}(t)-\phi^{\delta}(t)|<\epsilon\Big{]}$
$\displaystyle\geq-I_{T,x}(\phi^{\delta})$ $\displaystyle\geq-I^{*}-\delta.$

This implies

$\liminf_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{P}[Z^{N,x}\in
G]\geq-I^{*}$

as desired. ∎

We obtain the following result.

######  Corollary 5.4.

Assume that $\log\beta_{j}$ ($j=1,\dots,k$) is bounded and continuous. Then
for all $\phi\in D([0,T];A)$ with $\phi(0)=x$ and $\epsilon,\delta>0$, there
exists an $\tilde{\epsilon}>0$ such that

$\liminf_{N\rightarrow\infty}\frac{1}{N}\log\Big{(}\inf_{z\in
A;|x-z|<\tilde{\epsilon}}\mathbb{P}\Big{[}\sup_{t\in[0,T]}|Z^{N,z}-\phi(t)|<\epsilon\Big{]}\Big{)}\geq-
I_{T,x}(\phi)-\delta.$

###### Proof.

We assume w.l.o.g. that $I_{T,x}(\phi)<\infty$. By Theorem 5.3, there exists
an $N_{0}$ and $\tilde{\epsilon}$ such that for $N\geq N_{0}$ and $z$ with
$|z-x|<\tilde{\epsilon}$,

$\frac{1}{N}\log\mathbb{P}\Big{[}\sup_{t\in[0,T]}|Z^{N,z}-\phi(t)|<\epsilon\Big{]}\geq-\inf_{\tilde{\phi}:\|\phi-\tilde{\phi}\|<\epsilon}I_{T,x}(\tilde{\phi})-\delta\geq-
I_{T,x}(\phi)-\delta.$

The assertion follows. ∎

###  5.2 LDP lower bound with vanishing rates

In the following, we drop the assumption that the log-rates are bounded.
Instead, we rather consider situations, where Assumption 2.3 is satisfied.

We start by some preliminary considerations and assume that Assumption 2.3
(A1) and (A2) are satisfied. We note that there exists a constant $\alpha>0$
such that for all $x\in A$ there exists a $i\leq I$ such that
$B(x,\alpha)\subset B_{i}$. Indeed, assume that this is incorrect and consider
a sequence of points $x_{n}\in A$ such that $B(x_{n},1/n)$ is not contained in
any $B_{i}$. W.l.o.g., we can assume that $x_{n}\rightarrow x\in A$ (recall
that $A$ is compact). As $x\in B_{i_{0}}$ for some $i_{0}$, we have
$B(x_{n},1/n)\subset B_{i_{0}}$ for $n$ large enough, a contradiction.

######  Lemma 5.5.

Assume that $\beta_{j}$ ($j=1,\dots,k$) is bounded and that Assumption 2.3
(A1) and (A2) are satisfied. Then, for $T>0$, $K>0$, there exists a
$J=J(T,K)\in\mathds{N}$ such that for all $\phi\in D([0,T];A)$ with
$I_{T}(\phi)\leq K$, there exist

$0=t_{0}<t_{1}<\cdots<t_{J}=T\text{ and }i_{1},\dots,i_{J}\text{ such that
}\phi(t)\in B_{i_{r}}\text{ for }t\in[t_{r-1},t_{r}].$

Furthermore, for $r=1,\dots,J$,

$\operatorname{dist}(\phi(t_{r-1}),\partial
B_{i_{r}})\geq\alpha\quad\text{and}\quad\operatorname{dist}(\phi(t),\partial
B_{i_{r}})\geq\alpha/2\quad\text{for }t\in[t_{r-1},t_{r}]$

for $\alpha$ as before.

###### Proof.

By the considerations above, we have $B(x,\alpha)\subset B_{i_{1}}$ for an
appropriate $i_{1}$. We define

$\tilde{t}_{1}:=\inf\\{t\geq 0|B(\phi(t),\alpha/2)\not\subset
B_{i_{1}}\\}\wedge T>0.$

Now, there exists an $i_{2}$ such that $B(\phi(t_{1}),\alpha)\subset
B_{i_{2}}$. If $\tilde{t}_{1}<T$, we define

$\tilde{t}_{2}:=\inf\\{t\geq t_{1}|B(\phi(t),\alpha/2)\not\subset
B_{i_{2}}\\}\wedge T>\tilde{t}_{1}.$

In the same way, we proceed. By the uniform absolute continuity (Lemma 4.17)
of all $\phi$ with $I_{T}(\phi)\leq K$, we have

$\tilde{t}_{r}-\tilde{t}_{r-1}\geq\delta\quad\text{for a constant
}\delta>0\text{ independent of }\phi.$

The assertion hence follows for $J:=\lfloor\frac{T}{\delta}\rfloor+1$ and
$t_{r}:=r\delta$ ($r=1,\dots,J-1$), $t_{J}:=T$. ∎

We now define a function $\phi^{\eta}$ which is close to a given function
$\phi$ with $I_{T}(\phi)<\infty$. We assume that Assumption 2.3 (A) holds.
Hence, for $x\in B_{i}\cap A$ and $t\in(0,\lambda_{2})$,

$\operatorname{d}(x+tv_{i},\partial A)>\lambda_{1}t.$

Note that $\lambda_{1}\leq 1$. Let $\eta>0$ be small. We define for
$r=1,\dots,J$, with the notation $\sum_{j=1}^{0}...=0$,

$\eta_{r}:=\eta\sum_{j=1}^{r}\big{(}\frac{3}{\lambda_{1}}\big{)}^{j-1}.$

  * •

For $r=1,\dots,J$, $t\in\big{[}t_{r-1}+\eta_{r-1},t_{r-1}+\eta_{r}\big{]}$,

$\phi^{\eta}(t):=\phi(t_{r-1})+\eta\sum_{j=1}^{r-1}\Big{(}\frac{3}{\lambda_{1}}\Big{)}^{j-1}v_{i_{j}}+\big{(}t-t_{r-1}-\eta_{r-1}\big{)}v_{i_{r}}.$

  * •

For $r=1,\dots,J$, $t\in\big{[}t_{r-1}+\eta_{r},t_{r}+\eta_{r}\big{]}$,

$\phi^{\eta}(t):=\phi\big{(}t-\eta_{r}\big{)}+\eta\sum_{j=1}^{r}\Big{(}\frac{3}{\lambda_{1}}\Big{)}^{j-1}v_{i_{j}}.$

We make the following assumptions on $\eta$:

$\eta_{J}=\eta\sum_{r=1}^{J}\Big{(}\frac{3}{\lambda_{1}}\Big{)}^{r-1}\leq\frac{\alpha}{4}\wedge\min_{r=1,\dots,J}|t_{r}-t_{r-1}|\wedge\lambda_{2}.$

Therefore, we have the following properties for $\phi^{\eta}$:

  * •

For $r=1,\dots,J$, $t\in\big{[}t_{r-1}+\eta_{r-1},t_{r-1}+\eta_{r}\big{]}$,

$|\phi(t)-\phi^{\eta}(t)|\leq|\phi(t)-\phi(t_{r-1})|+\eta_{r-1}+(\eta_{r}-\eta_{r-1})=V_{t_{r}-t_{r-1}}(\phi)+\eta_{r}\rightarrow
0$ (41)

as $\eta\rightarrow 0$, where $V_{\cdot}(\phi)$ is the modulus of continuity
of $\phi$. Similarly, for $r=1,\dots,J$,
$t\in\big{[}t_{r-1}+\eta_{r},t_{r}+\eta_{r}\big{]}$,

$|\phi(t)-\phi^{\eta}(t)|\leq|\phi(t)-\phi(t-\eta_{r})|+\eta_{r}=V_{\eta_{r}}(\phi)+\eta_{r}\rightarrow
0$

as $\eta\rightarrow 0$.

  * •

For $r=1,\dots,J$, $t\in\big{[}t_{r-1}+\eta_{r-1},t_{r-1}+\eta_{r}\big{]}$,

$\operatorname{dist}(\phi^{\eta}(t),\partial
B_{i_{r}})\geq\operatorname{dist}(\phi(t_{r-1}),\partial
B_{i_{r}})-\eta_{r}\geq\alpha-\frac{\alpha}{4}.$

Similarly, for $r=1,\dots,J$,
$t\in\big{[}t_{r-1}+\eta_{r},t_{r}+\eta_{r}\big{]}$, hence
$t-\eta_{r}\in[t_{r-1},t_{r}]$,

$\operatorname{dist}(\phi^{\eta}(t),\partial
B_{i_{r}})\geq\operatorname{dist}(\phi(t-\eta_{r}),\partial
B_{i_{r}})-\eta_{r}\geq\frac{\alpha}{2}-\frac{\alpha}{4}.$

Hence, for $r=1,\dots,J$,
$t\in\big{[}t_{r-1}+\eta_{r-1},t_{r}+\eta_{r}\big{]}$,

$\operatorname{dist}(\phi^{\eta}(t),\partial B_{i_{r}})\geq\frac{\alpha}{4}.$

  * •

For $t\in[0,\eta]$,

$\operatorname{dist}(\phi^{\eta}(t),\partial A)\geq t\lambda_{1}.$ (42)

For $t\in\big{[}\eta,T+\eta_{J}\big{]}$,

$\operatorname{dist}(\phi^{\eta}(t),\partial A)\geq\lambda_{1}\eta.$ (43)

This can be seen by induction on $r=1,\dots,J$ (the induction hypothesis is
clear, cf. Inequality (42)). For $r=1,\dots,J$, we have (by induction
hypothesis and the assumptions on $\eta$)

$\phi^{\eta}\Big{(}t_{r-1}+\eta_{r-1}\Big{)}\in B_{i_{r}},\text{ and for
}r\geq 2,\
\operatorname{dist}\Big{(}\phi^{\eta}\Big{(}t_{r-1}+\eta_{r-1}\Big{)},\partial
A\Big{)}\geq\eta\lambda_{1}.$

From Assumption 2.3 (A3), the distance of $\phi^{\eta}(t)$ to the boundary is
increasing for
$t\in\big{[}t_{r-1}+\eta\sum_{j=1}^{r-1}\big{(}\frac{3}{\lambda_{1}}\big{)}^{j-1},t_{r-1}+\eta\sum_{j=1}^{r}\big{(}\frac{3}{\lambda_{1}}\big{)}^{j-1}\big{]}$,
and is at least

$\Big{(}t-\Big{(}t_{r-1}+\eta\sum_{j=1}^{r-1}\Big{(}\frac{3}{\lambda_{1}}\Big{)}^{j-1}\Big{)}\Big{)}\cdot\lambda_{1}\vee\eta\lambda_{1}.$

In particular,

$\operatorname{dist}\Big{(}\phi^{\eta}\Big{(}t_{r-1}+\eta\sum_{j=1}^{r}\Big{(}\frac{3}{\lambda_{1}}\Big{)}^{j-1}\Big{)},\partial
A\Big{)}\geq\eta\lambda_{1}\Big{(}\frac{3}{\lambda_{1}}\Big{)}^{r-1}.$

For
$t\in\big{[}t_{r-1}+\eta\sum_{j=1}^{r}\big{(}\frac{3}{\lambda_{1}}\big{)}^{j-1},t_{r}+\eta\sum_{j=1}^{r}\big{(}\frac{3}{\lambda_{1}}\big{)}^{j-1}\big{]}$,
we have

$\displaystyle\phi^{\eta}(t)$
$\displaystyle=\phi\Big{(}t-\eta\sum_{j=1}^{r}\Big{(}\frac{3}{\lambda_{1}}\Big{)}^{j-1}\Big{)}+\eta\Big{(}\frac{3}{\lambda_{1}}\Big{)}^{r-1}v_{i_{r}}+\eta\sum_{j=1}^{r-1}\Big{(}\frac{3}{\lambda_{1}}\Big{)}^{j-1}v_{i_{j}}$
$\displaystyle=\bar{\phi}(t)+\eta\sum_{j=1}^{r-1}\Big{(}\frac{3}{\lambda_{1}}\Big{)}^{j-1}v_{i_{j}}$

and therefore (by elementary calculus and the fact that $|v_{i}|\leq 1$)

$\displaystyle\operatorname{dist}(\phi^{\eta}(t),\partial A)$
$\displaystyle\geq\operatorname{dist}(\bar{\phi}(t),\partial
A)-\Big{|}\eta\sum_{j=1}^{r-1}\Big{(}\frac{3}{\lambda_{1}}\Big{)}^{j-1}v_{i_{j}}\Big{|}$
$\displaystyle\geq\eta\left(\frac{3}{\lambda_{1}}\right)^{r-1}\lambda_{1}\left(1-\frac{{\bf
1}_{r\geq 2}}{2}\right)$ $\displaystyle\geq\eta\lambda_{1}.$

We now have

######  Lemma 5.6.

Assume that Assumption 2.3 holds. Let $K>0$ and $\epsilon>0$. Then there
exists an $\eta_{0}=\eta_{0}(T,K,\epsilon)>0$ such that for all $\phi\in
D([0,T];A)$ with $I_{T}(\phi)\leq K$ and all $\eta<\eta_{0}$,

$I_{T}(\phi^{\eta})\leq I_{T}(\phi)+\epsilon,$

where $\phi^{\eta}(t)$ is defined as above.

###### Proof.

We first use Lemma 2.4 and chose $\eta<\eta_{1}$ small enough (independent of
$i$, $\phi$) such that

$\sum_{r=1}^{J}\int_{t_{r-1}+\eta_{r-1}}^{t_{r-1}+\eta_{r}}L(\phi^{\eta}(t),(\phi^{\eta})^{\prime}(t))dt=\sum_{r=1}^{J}I_{\eta_{r}-\eta_{r-1}}(\phi)<\frac{\epsilon}{2}.$

We now denote by $\mu^{*}(t)$ the optimal $\mu$ corresponding to
$(\phi(t),\phi^{\prime}(t))$ (cf. Lemma 4.5). We let $r=1,\dots,J$ and
$t\in[t_{r-1}+\eta_{r},t_{r}+\eta_{r}]$ and note that
$(\phi^{\eta})^{\prime}(t)=\phi^{\prime}(t-\eta_{r})$. By Theorem 4.9, we have

$L(\phi^{\eta}(t),\phi^{\prime}(t-\eta_{r}))\leq\ell(\phi^{\eta}(t),\mu^{*}(t-\eta_{r})).$
(44)

By the Lipschitz continuity of the $\beta_{j}$, we have

$|\beta_{j}(\phi^{\eta}(t))-\beta_{j}(\phi(t-\eta_{r}))|\leq\delta_{K}(\eta)$
(45)

where $\delta_{K}(\eta)$ is independent of $\phi$ and
$\delta_{K}(\eta)\rightarrow 0$ as $\eta\rightarrow 0$. We deduce from (44)
and (45)

$\displaystyle
L(\phi^{\eta}(t),\phi^{\prime}(t-\eta_{r}))-L(\phi(t-\eta_{r}),\phi^{\prime}(t-\eta_{r}))$
$\displaystyle\leq
k\delta_{K}(\eta)+\sum_{j}\mu_{j}^{*}(t-\eta_{r})\log\frac{\beta_{j}(\phi(t-\eta_{r}))}{\beta_{j}(\phi^{\eta}(t))}.$
(46)

Let

$\tilde{v}_{i_{r}}=\Big{(}\frac{3}{\lambda_{1}}\Big{)}^{r-1}v_{i_{r}}+\sum_{j=1}^{r-1}\Big{(}\frac{3}{\lambda_{1}}\Big{)}^{j-1}v_{i_{j}},\
\text{ and
}\hat{v}_{i_{r}}=\frac{\tilde{v}_{i_{r}}}{|\tilde{v}_{i_{r}}|}\in\mathcal{C}_{1,i_{r}}.$

By Assumption 2.3 (B4), there exists a constant $\lambda_{4}>0$ such that for
$z\in B_{i_{r}}$ and $\eta<\eta_{2}\leq\eta_{1}$ small enough (note that
$\eta_{2}$ depends on $\lambda_{1}$ and $\lambda_{2}$ but not directly on
$\phi$, except through $K$),

$\beta_{j}(z)<\lambda_{4}\Rightarrow\beta_{j}(z+\eta\tilde{v}_{i_{r}})\geq\beta_{j}(z),$

hence

$\log\frac{\beta_{j}(\phi(t-\eta_{r}))}{\beta_{j}(\phi^{\eta}(t))}<0\quad\text{if
}\beta_{j}(\phi(t-\eta_{r}))<\lambda_{4}.$ (47)

If $\beta_{j}(\phi(t-\eta_{r}))\geq\lambda_{4}$ (recall the definition of
$\delta_{K}(\eta)$ and choose $\eta<\eta_{3}<\eta_{2}$ small enough such that
$\delta_{K}(\eta)<\lambda_{4}/2$)

$\displaystyle\log\frac{\beta_{j}(\phi(t-\eta_{r}))}{\beta_{j}(\phi^{\eta}(t))}$
$\displaystyle\leq\log\frac{\beta_{j}(\phi(t-\eta_{r}))}{\beta_{j}(\phi(t-\eta_{r}))-\delta_{K}(\eta)}$
$\displaystyle\leq\log\frac{\lambda_{4}}{\lambda_{4}-\delta_{K}(\eta)}$
$\displaystyle=\log\frac{1}{1-\delta_{K}(\eta)/\lambda_{4}}$
$\displaystyle\leq\frac{2\delta_{K}(\eta)}{\lambda_{4}},$ (48)

since $\log(1/(1-x))<2x$ for $0<x\leq 1/2$.

From Lemma 4.3 and Lemma 4.5, there exist (universal, i.e., independent of
$x$) constants $B\geq B_{1}\vee B_{2}$, $B>1$, $C_{1}$, $C_{3}$ such that for
$|y|\geq B$, and $x\in A$,

$L(x,y)\geq C_{1}|y|\log|y|,$ (49) $|\mu^{*}|=|\mu^{*}(x,y)|\leq C_{3}|y|.$
(50)

Hence if $|\phi^{\prime}(t-\eta_{r})|\geq B$, using (46), (47), (48) and (50)
for the first inequality and (49) for the second, we get

$\displaystyle
L(\phi^{\eta}(t),\phi^{\prime}(t-\eta_{r}))-L(\phi(t-\eta_{r}),\phi^{\prime}(t-\eta_{r}))$
$\displaystyle\leq
k\delta_{K}(\eta)+kC_{3}|\phi^{\prime}(t-\eta_{r})|\frac{2\delta_{K}(\eta)}{\lambda_{4}}$
$\displaystyle\leq
k\delta_{K}(\eta)+kC_{3}\frac{2\delta_{K}(\eta)L(\phi(t-\eta_{r}),\phi^{\prime}(t-\eta_{r}))}{C_{1}\lambda_{4}\log|\phi^{\prime}(t-\eta_{r})|}.$
(51)

If however $|\phi^{\prime}(t-\eta_{r})|<B$, Lemma 4.5 implies similarly as
before that $|\mu^{*}(t-\eta_{r})|\leq C_{3}B$. From (46), we deduce

$L(\phi^{\eta}(t),\phi^{\prime}(t-\eta_{r}))-L(\phi(t-\eta_{r}),\phi^{\prime}(t-\eta_{r}))\leq
k\delta_{K}(\eta)+kC_{3}B\frac{2\delta_{K}(\eta)}{\lambda_{4}}.$ (52)

Inequalities (51) and (52) imply

$\displaystyle
L(\phi^{\eta}(t),\phi^{\prime}(t-\eta_{r}))-L(\phi(t-\eta_{r}),\phi^{\prime}(t-\eta_{r}))\leq\delta_{1,K}(\eta)+\delta_{2,K}(\eta)L(\phi(t-\eta_{r}),\phi^{\prime}(t-\eta_{r}))$

with constants $\delta_{i,K}(\eta)\rightarrow 0$ as $\eta\rightarrow 0$. We
can hence choose $\eta<\eta_{4}<\eta_{3}$ small enough such that

$\sum_{r=1}^{J}\int_{t_{r-1}+\eta_{r}}^{t_{r}+\eta_{r}}L(\phi^{\eta}(t),\phi^{\prime}(t-\eta_{r}))dt-\sum_{r=1}^{J}\int_{t_{r-1}+\eta_{r}}^{t_{r}+\eta_{r}}L(\phi(t-\eta_{r}),\phi^{\prime}(t-\eta_{r}))dt<\frac{\epsilon}{2}.$

This yields the result. ∎

The following lemma is the main difference to the corresponding result of
Shwartz and Weiss (2005). We transform the LLN from Assumption 2.3 (C) to a
LDP lower bound for linear functions following the vector $v_{i}$ near the
boundary.

In the next statement, $\alpha$ is the exponent which appears in the
Assumption 2.3 (C).

######  Lemma 5.7.

Assume that Assumption 2.3 holds. Let $i\leq I_{1}$, $x\in A\cap B_{i}$ and
$x^{N}\in A^{N}\cap B_{i}$ such that

$\limsup_{N\rightarrow\infty}|x^{N}-x|N^{\alpha}<1.$

Let furthermore $\epsilon>0$ and define $\mu^{i}$, $\phi^{x}$ and $\eta_{0}$
as in Assumption 2.3 (C). Then for all $\eta$ small enough, all $\epsilon$,

$\liminf_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{P}\Big{[}\sup_{t\in[0,\eta]}|Z^{N,x^{N}}(t)-\phi^{x}(t)|<\epsilon\Big{]}\geq-\int_{0}^{\eta}\ell(\phi^{x}(t),\mu^{i})dt,$

and the above convergence is uniform in $x\in A$.

###### Proof.

The proof follows the same line of reasoning as the proof of Proposition 5.1
but is technically more involved.

For simplicity, let $N$ be large enough and $\eta<\eta_{0}$ (for $\eta_{0}$ as
in Assumption 2.3 (C)) be small enough such that $\phi^{x^{N}}(t)\in B_{i}$
for all $t\leq\eta$. We furthermore let

$\tilde{\epsilon}<\epsilon_{1}:=\epsilon\wedge\lambda_{1}\eta$

Define the set

$F^{N}:=\Big{\\{}\sup_{t\in[0,\eta]}|Z^{N,x^{N}}(t)-\phi^{x}(t)|<\tilde{\epsilon}\Big{\\}}.$

Let
$\xi_{\eta}=\xi^{N}_{\eta}=\frac{d\tilde{\mathbb{P}}}{d\mathbb{P}}|_{\mathcal{F}_{\eta}}$
be given as in Theorem A.1 for the rates
$\tilde{\beta}_{j}=\tilde{\mu}^{i}_{j}$.We note that due to Assumption 2.3
(C),

$\tilde{\mathbb{P}}[F^{N}]\geq 1-\delta(N,\tilde{\epsilon})\rightarrow
1\quad\text{as }N\rightarrow\infty.$ (53)

From Corollary A.2, (53) and Jensen’s inequality, we deduce that

$\displaystyle\liminf_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{P}\Big{[}\sup_{t\in[0,\eta]}|Z^{N,x^{N}}(t)-\phi^{x}(t)|<\epsilon\Big{]}$
$\displaystyle\qquad\geq\liminf_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{P}\big{[}F^{N}\big{]}$
$\displaystyle\qquad\geq\liminf_{N\rightarrow\infty}\frac{1}{N}\log\tilde{\mathbb{E}}\big{[}\xi_{\eta}^{-1}\mathds{1}_{F^{N}}\big{]}$
$\displaystyle\qquad=\liminf_{N\rightarrow\infty}\frac{1}{N}\log\Big{\\{}\tilde{\mathbb{P}}\big{[}F^{N}\big{]}\tilde{\mathbb{E}}_{F^{N}}\big{[}\exp(X_{\eta})\big{]}\Big{\\}}$
$\displaystyle\qquad\geq\liminf_{N\rightarrow\infty}\frac{1}{N}\log\tilde{\mathbb{P}}\big{[}F^{N}\big{]}+\liminf_{N\rightarrow\infty}\frac{1}{N}\log\tilde{\mathbb{E}}_{F^{N}}\big{[}\exp(X_{\eta})\big{]}$
$\displaystyle\qquad\geq\liminf_{N\rightarrow\infty}\tilde{\mathbb{E}}_{F^{N}}\Big{[}\frac{X_{\eta}}{N}\Big{]},$
(54)

where $\tilde{\mathbb{E}}_{F^{N}}[X]:=\tilde{\mathbb{E}}[X|F^{N}]$ and

$\displaystyle X_{\eta}^{N}:=X_{\eta}:=\log\xi_{\eta}^{-1}$
$\displaystyle=\sum_{\tau\leq\eta}\Big{[}\log\beta_{j(\tau)}(Z^{N,x^{N}}(\tau-)-\log\tilde{\mu}^{i}_{j(\tau)}(Z^{N,x^{N}}(\tau-)\Big{]}$
$\displaystyle\qquad+N\sum_{j}\int_{0}^{\eta}\big{(}\tilde{\mu}^{i}_{j}(Z^{N,x^{N}}(t)-\beta_{j}(Z^{N,x^{N}}(t))\big{)}dt.$

We have $\operatorname{dist}(\phi^{x^{N}}(t),\partial A)\geq\lambda_{1}t$ (cf.
Assumption 2.3 (A3)) and therefore on $F^{N}$,

$\operatorname{dist}(Z^{N,x^{N}}(t),\partial
A)>\lambda_{1}t-\tilde{\epsilon}\quad\text{for
}t\in\Big{[}\frac{\tilde{\epsilon}}{\lambda_{1}},\eta\Big{]}.$

Consequently

$\tilde{\mu}^{i}_{j}(Z^{N,x^{N}}(t))=\mu^{i}_{j}\quad\text{for all }j\text{
and for all }t\in\Big{[}\frac{\tilde{\epsilon}}{\lambda_{1}},\eta\Big{]}.$

We obtain

$\displaystyle\frac{1}{\tilde{\mathbb{P}}[F^{N}]}\tilde{\mathbb{E}}\Big{[}\mathds{1}_{F^{N}}\int_{0}^{\eta}\sum_{j}\tilde{\mu}^{i}_{j}(Z^{N,x^{N}}(t))dt\Big{]}$
$\displaystyle\qquad=\frac{1}{\tilde{\mathbb{P}}[F^{N}]}\Big{(}\tilde{\mathbb{E}}\Big{[}\mathds{1}_{F^{N}}\int_{\tilde{\epsilon}/\lambda_{1}}^{\eta}\sum_{j=1}^{k}\tilde{\mu}^{i}_{j}(Z^{N,x^{N}}(t))dt\Big{]}+\tilde{\mathbb{E}}\Big{[}\mathds{1}_{F^{N}}\int_{0}^{\tilde{\epsilon}/\lambda_{1}}\sum_{j=1}^{k}\tilde{\mu}^{i}_{j}(Z^{N,x^{N}}(t))dt\Big{]}\Big{)}$
$\displaystyle\qquad=:\sum_{j=1}^{k}\int_{\tilde{\epsilon}/\lambda_{1}}^{\eta}\mu^{i}_{j}dt+X_{1}^{N}(\tilde{\epsilon}),$
(55)

since $\mu^{i}_{j}(Z^{N,x^{N}}(t))=\mu^{i}_{j}$ on $F^{N}$. We note that for
all $N$,

$|X_{1}^{N}(\tilde{\epsilon})|\leq\frac{\tilde{\epsilon}}{\lambda_{1}}k\bar{\mu}\quad\text{where
}\bar{\mu}:=\max_{j=1,\dots,k}\mu^{i}_{j}.$ (56)

By the fact that the $\beta_{j}$’s are bounded and continuous and by Theorem
3.1, we have for $j=1,\dots,k$,

$\sup_{t\in[0,\eta]}|\beta_{j}(Z^{N,x^{N}}(t))-\beta_{j}(\phi^{x}(t))|\rightarrow
0\quad\text{a.s.\leavevmode\nobreak\ as }N\rightarrow\infty.$

Combined with (53), this implies

$\frac{1}{\tilde{\mathbb{P}}[F^{N}]}\tilde{\mathbb{E}}\Big{[}\mathds{1}_{F^{N}}\int_{0}^{\eta}\sum_{j=1}^{k}\beta_{j}(Z^{N,x^{N}}(t))dt\Big{]}\longrightarrow\sum_{j=1}^{k}\int_{0}^{\eta}\beta_{j}(\phi^{x}(t))dt$
(57)

as $N\rightarrow\infty$.

Let us now define the following processes. For $z\in A$, $j=1,\dots,k$ and
$0\leq s<t\leq\eta$ let $\bar{Z}^{N,z}$ solves Equation (3) with constant
rates $\mu^{i}_{j}$ under $\tilde{\mathbb{P}}$, and

$\displaystyle Y_{j}^{N,z,s,t}$ $\displaystyle:=\frac{1}{N}\cdot\\#\text{jumps
of }Z^{N,z}\text{ in direction }h_{j}\text{ in }[s,t],$
$\displaystyle\bar{Y}_{j}^{N,z,s,t}$
$\displaystyle:=\frac{1}{N}\cdot\\#\text{jumps of }\bar{Z}^{N,z}\text{ in
direction }h_{j}\text{ in }[s,t],$

We have for any event $F$, noting that
$\tilde{\mathbb{P}}[F]-\tilde{\mathbb{P}}[F]^{2}=\tilde{\mathbb{P}}[F^{c}]-\tilde{\mathbb{P}}[F^{c}]^{2}$,

$\displaystyle\tilde{\mathbb{E}}[Y_{j}^{N,z,s,t}]$
$\displaystyle\leq(t-s)\mu^{i}_{j}=\tilde{\mathbb{E}}[\bar{Y}_{j}^{N,z,s,t}],$
(58) $\displaystyle\widetilde{\operatorname{Var}}[Y_{j}^{N,z,s,t}]$
$\displaystyle\leq(t-s)\mu^{i}_{j}=\widetilde{\operatorname{Var}}[\bar{Y}_{j}^{N,z,s,t}],$
(59)
$\displaystyle|\widetilde{\operatorname{Cov}}(\mathds{1}_{F},Y_{j}^{N,z,s,t})|,\,|\widetilde{\operatorname{Cov}}(\mathds{1}_{F},\bar{Y}_{j}^{N,z,s,t})|$
$\displaystyle\leq\sqrt{\widetilde{\operatorname{Var}}[\mathds{1}_{F}]}\sqrt{\widetilde{\operatorname{Var}}[\bar{Y}_{j}^{N,z,s,t}]}$
$\displaystyle\leq\sqrt{\tilde{\mathbb{P}}[F]-\tilde{\mathbb{P}}[F]^{2}}\sqrt{(t-s)\mu^{i}_{j}}.$
(60)

We define the sets

$F_{1}^{N}:=\Big{\\{}\Big{|}Z^{N,x^{N}}\Big{(}\frac{2\tilde{\epsilon}}{\lambda_{1}}\Big{)}-\phi^{x}\Big{(}\frac{2\tilde{\epsilon}}{\lambda_{1}}\Big{)}\Big{|}<\frac{\tilde{\epsilon}}{2}\Big{\\}}\in\mathcal{F}_{2\tilde{\epsilon}/\lambda_{1}}$

and for $z\in A$ with
$|z-\phi^{x}(2\tilde{\epsilon}/\lambda_{1})|<\tilde{\epsilon}/2$,

$F_{2}^{N,z}:=\Big{\\{}\sup_{t\in[0,\eta-2\tilde{\epsilon}/\lambda_{1}]}|Z^{N,z}(t)-\phi^{z}(t)|<\frac{\tilde{\epsilon}}{2}\Big{\\}}.$

Note that

$\operatorname{dist}(\phi^{x}(t),\partial A)\geq
2\tilde{\epsilon}\quad\text{for
}t\in\Big{[}\frac{2\tilde{\epsilon}}{\lambda_{1}},\eta\Big{]}$

and whenever $|z-\phi^{x}(2\tilde{\epsilon}/\lambda_{1})|<\tilde{\epsilon}/2$,

$\Big{|}\phi^{z}(t)-\phi^{x}\Big{(}t+\frac{2\tilde{\epsilon}}{\lambda_{1}}\Big{)}\Big{|}<\frac{\tilde{\epsilon}}{2}\quad\text{for
}t\in\Big{[}0,\eta-\frac{2\tilde{\epsilon}}{\lambda_{1}}\Big{]}.$

Hence

$\operatorname{dist}(Z^{N,z}(t),\partial A)\geq\tilde{\epsilon}\quad\text{for
}t\in\Big{[}0,\eta-\frac{2\tilde{\epsilon}}{\lambda_{1}}\Big{]}$

and therefore $Z^{N,z}(t)=\bar{Z}^{N,z}(t)$ on $F_{2}^{N,z}$. This implies

$F^{N,z}_{2}=\Big{\\{}\sup_{t\in[0,\eta-2\tilde{\epsilon}/\lambda_{1}]}|\bar{Z}^{N,z}(t)-\phi^{z}(t)|<\frac{\tilde{\epsilon}}{2}\Big{\\}}.$
(61)

We now let

$\frac{2\tilde{\epsilon}}{\lambda_{1}}\leq s<t\leq\eta$

and compute (by using the Markov property of $Z^{N}$ and the fact that
$Y_{j}^{N,z,s-2\tilde{\epsilon}/\lambda_{1},t-2\tilde{\epsilon}/\lambda_{1}}=\bar{Y}_{j}^{N,z,s-2\tilde{\epsilon}/\lambda_{1},t-2\tilde{\epsilon}/\lambda_{1}}$
on the event $F_{2}^{N,z}$)

$\displaystyle\tilde{\mathbb{E}}[Y_{j}^{N,x^{N},s,t}]$
$\displaystyle=\tilde{\mathbb{E}}[\mathds{1}_{F^{N}_{1}}Y_{j}^{N,x^{N},s,t}]+\tilde{\mathbb{E}}[\mathds{1}_{(F^{N}_{1})^{c}}Y_{j}^{N,x^{N},s,t}]$
$\displaystyle\geq\tilde{\mathbb{P}}[F^{N}_{1}]\cdot\inf_{z;|z-\phi^{x}(\tilde{\epsilon}/\lambda_{1})|<\tilde{\epsilon}/2}\tilde{\mathbb{E}}[Y_{j}^{N,z,s-2\tilde{\epsilon}/\lambda_{1},t-2\tilde{\epsilon}/\lambda_{1}}]+\tilde{\mathbb{E}}[\mathds{1}_{(F^{N}_{1})^{c}}Y_{j}^{N,x^{N},s,t}]$
$\displaystyle\geq\tilde{\mathbb{P}}[F^{N}_{1}]\cdot\inf_{z;|z-\phi^{x}(\tilde{\epsilon}/\lambda_{1})|<\tilde{\epsilon}/2}\tilde{\mathbb{E}}[\mathds{1}_{F^{N,z}_{2}}Y_{j}^{N,z,s-2\tilde{\epsilon}/\lambda_{1},t-2\tilde{\epsilon}/\lambda_{1}}]$
$\displaystyle\qquad+\tilde{\mathbb{P}}[F^{N}_{1}]\cdot\inf_{z;|z-\phi^{x}(\tilde{\epsilon}/\lambda_{1})|<\tilde{\epsilon}/2}\tilde{\mathbb{E}}[\mathds{1}_{(F^{N,z}_{2})^{c}}Y_{j}^{N,z,s-2\tilde{\epsilon}/\lambda_{1},t-2\tilde{\epsilon}/\lambda_{1}}]$
$\displaystyle\qquad+\tilde{\mathbb{E}}[\mathds{1}_{(F^{N}_{1})^{c}}Y_{j}^{N,x^{N},s,t}]$
$\displaystyle=\mu_{j}^{i}(t-s)\tilde{\mathbb{P}}[F^{N}_{1}]\cdot\inf_{z;|z-\phi^{x}(\tilde{\epsilon}/\lambda_{1})|<\tilde{\epsilon}/2}\tilde{\mathbb{P}}[F^{N,z}_{2}]$
$\displaystyle\qquad+\tilde{\mathbb{P}}[F^{N}_{1}]\cdot\inf_{z;|z-\phi^{x}(\tilde{\epsilon}/\lambda_{1})|<\tilde{\epsilon}/2}\widetilde{\operatorname{Cov}}(\mathds{1}_{F^{N,z}_{2}},Y_{j}^{N,z,s-2\tilde{\epsilon}/\lambda_{1},t-2\tilde{\epsilon}/\lambda_{1}})$
$\displaystyle\qquad+\tilde{\mathbb{P}}[F^{N}_{1}]\cdot\inf_{z;|z-\phi^{x}(\tilde{\epsilon}/\lambda_{1})|<\tilde{\epsilon}/2}\tilde{\mathbb{E}}[\mathds{1}_{(F^{N,z}_{2})^{c}}Y_{j}^{N,z,s-2\tilde{\epsilon}/\lambda_{1},t-2\tilde{\epsilon}/\lambda_{1}}]$
$\displaystyle\qquad+\tilde{\mathbb{E}}[\mathds{1}_{(F^{N}_{1})^{c}}Y_{j}^{N,x^{N},s,t}]$
(62)
$\displaystyle\geq\mu_{j}^{i}(t-s)\tilde{\mathbb{P}}[F^{N}_{1}]\cdot\inf_{z;|z-\phi^{x}(\tilde{\epsilon}/\lambda_{1})|<\tilde{\epsilon}/2}\tilde{\mathbb{P}}[F^{N,z}_{2}]$
$\displaystyle\qquad-\tilde{\mathbb{P}}[F^{N}_{1}]\cdot\inf_{z;|z-\phi^{x}(\tilde{\epsilon}/\lambda_{1})|<\tilde{\epsilon}/2}\big{|}\widetilde{\operatorname{Cov}}(\mathds{1}_{F^{N,z}_{2}},Y_{j}^{N,z,s-2\tilde{\epsilon}/\lambda_{1},t-2\tilde{\epsilon}/\lambda_{1}})\big{|}$
(63)

as the third and the fourth term in (62) are non-negative. As furthermore

$\tilde{\mathbb{P}}[F^{N}_{1}]\cdot\inf_{z;|z-\phi^{x}(\tilde{\epsilon}/\lambda_{1})|<\tilde{\epsilon}/2}\tilde{\mathbb{P}}[F^{N,z}_{2}]\rightarrow
1$

and

$\tilde{\mathbb{P}}[F^{N}_{1}]\cdot\inf_{z;|z-\phi^{x}(\tilde{\epsilon}/\lambda_{1})|<\tilde{\epsilon}/2}\widetilde{\operatorname{Cov}}(\mathds{1}_{F^{N,z}_{2}},Y_{j}^{N,z,s-2\tilde{\epsilon}/\lambda_{1},t-2\tilde{\epsilon}/\lambda_{1}})\rightarrow
0$

as $N\rightarrow\infty$ by Assumption 2.3 (C), Theorem 3.1 and (60). Combinig
the resulting inequality with (58) for all $\tilde{\epsilon}<\epsilon_{1}$ and
$2\tilde{\epsilon}/\lambda_{1}\leq s<t$, we deduce that

$\lim_{N\rightarrow\infty}\tilde{\mathbb{E}}[Y_{j}^{N,x^{N},s,t}]=\mu^{i}_{j}(t-s).$

Note that for $0\leq s<t\leq\eta$, and all $\tilde{\epsilon}<\epsilon_{1}$

$Y_{j}^{N,x^{N},s,t}=Y_{j}^{N,x^{N},s,(s\vee
2\tilde{\epsilon}/\lambda_{1})\wedge t}+Y_{j}^{N,x^{N},(s\vee
2\tilde{\epsilon}/\lambda_{1})\wedge t,t}$

and hence also for $0\leq s<t\leq\eta$, since when
$s<2\tilde{\epsilon}/\lambda_{1}$,
$Y_{j}^{N,x^{N},s,2\tilde{\epsilon}/\lambda_{1}\wedge t}$ is of the order of
$\tilde{\epsilon}$,

$\lim_{N\rightarrow\infty}\tilde{\mathbb{E}}[Y_{j}^{N,x^{N},s,t}]=\mu^{i}_{j}(t-s).$
(64)

Let now $\tau_{j}\in[0,\eta]$ denote the jump times of $Z^{N,x^{N}}$ in
direction $h_{j}$. Since
$\tilde{\mu}^{i}_{j}(Z^{N,x^{N}}(\tau_{j}-))=\log\mu^{i}_{j}$
$\tilde{\mathbb{P}}$ a.s.,

$\displaystyle\frac{1}{\tilde{\mathbb{P}}[F^{N}]}\sum_{j;\mu^{i}_{j}>0}\tilde{\mathbb{E}}\Big{[}\frac{1}{N}\mathds{1}_{F^{N}}\sum_{\tau_{j}\leq\eta}\log\tilde{\mu}^{i}_{j}(Z^{N,x^{N}}(\tau_{j}-))\Big{]}$
$\displaystyle\qquad=\frac{1}{\tilde{\mathbb{P}}[F^{N}]}\sum_{j;\mu^{i}_{j}>0}\log\mu^{i}_{j}\,\,\tilde{\mathbb{E}}\big{[}\mathds{1}_{F^{N}}Y^{N,x^{N},0,\eta}_{j}\big{]}$
$\displaystyle\qquad=\frac{1}{\tilde{\mathbb{P}}[F^{N}]}\sum_{j;\mu^{i}_{j}>0}\log\mu^{i}_{j}\Big{\\{}\tilde{\mathbb{P}}[F^{N}]\cdot\tilde{\mathbb{E}}\big{[}Y^{N,x^{N},0,\eta}_{j}\big{]}+\widetilde{\operatorname{Cov}}(\mathds{1}_{F^{N}},Y^{N,x^{N},0,\eta}_{j})\Big{\\}}$
$\displaystyle\qquad\longrightarrow\sum_{j;\mu^{i}_{j}>0}\eta\mu^{i}_{j}\log\mu^{i}_{j}=\sum_{j=1}^{k}\int_{0}^{\eta}\mu^{i}_{j}\log\mu^{i}_{j}dt$
(65)

as $N\rightarrow\infty$ by (53), (60) and (64).

For the last and most extensive step of the proof, we define for
$\tilde{\epsilon}<\epsilon_{1}$ and (cf. Assumption 2.3 (C) and (11))

$\epsilon_{N}=\frac{1}{N^{\alpha}}$

and the set

$\tilde{F}^{N}:=\Big{\\{}\sup_{t\in[0,\eta]}|Z^{N,x^{N}}(t)-\phi^{x}(t)|<\epsilon_{N}\Big{\\}}.$

We assume w.l.o.g. that from now on $N$ is large enough (cf. Assumption 2.3
(C)) such that

$\tilde{\mathbb{P}}\big{[}\tilde{F}^{N}\big{]}\geq\tilde{\mathbb{P}}\Big{[}\sup_{t\in[0,\eta]}|Z^{N,x^{N}}(t)-\phi^{x^{N}}(t)|<\epsilon_{N}\Big{]}\geq
1-\delta(N,\epsilon_{N}),$

where $\delta(N,\epsilon_{N})\to 0$ as $N\rightarrow\infty$. We note that we
have for all $\tilde{\epsilon}\leq\epsilon_{1}$, $j=1,\dots,k$ with
$\mu^{i}_{j}>0$, $N\in\mathds{N}$ and
$t\in[2\tilde{\epsilon}/\lambda_{1},\eta]$,

$\log\beta_{j}(\phi^{x}(t)),\log\beta_{j}(Z^{N,x^{N}}(t))\geq\log\underline{\beta}(\tilde{\epsilon})>0\quad\text{on
}F^{N}\text{ and }\tilde{F}^{N}.$

We compute

$\displaystyle\frac{1}{\tilde{\mathbb{P}}[F^{N}]}\tilde{\mathbb{E}}\Big{[}\frac{1}{N}\mathds{1}_{F^{N}}\sum_{\tau\leq\eta}\log\beta_{j(\tau)}(Z^{N,x^{N}}(\tau-))\Big{]}$
$\displaystyle\qquad=\frac{1}{\tilde{\mathbb{P}}[F^{N}]}\sum_{j,\,\mu^{i}_{j}>0}\tilde{\mathbb{E}}\Big{[}\frac{1}{N}\mathds{1}_{\tilde{F}^{N}}\sum_{\tau_{j}\in[2\tilde{\epsilon}/\lambda_{1},\eta]}\log\beta_{j}(Z^{N,x^{N}}(\tau_{j}-))\Big{]}$
$\displaystyle\qquad\qquad+\frac{1}{\tilde{\mathbb{P}}[F^{N}]}\sum_{j,\,\mu^{i}_{j}>0}\tilde{\mathbb{E}}\Big{[}\frac{1}{N}\mathds{1}_{\tilde{F}^{N}}\sum_{\tau_{j}\in[0,2\tilde{\epsilon}/\lambda_{1}]}\log\beta_{j}(Z^{N,x^{N}}(\tau_{j}-))\Big{]}$
$\displaystyle\qquad\qquad+\frac{1}{\tilde{\mathbb{P}}[F^{N}]}\sum_{j,\,\mu^{i}_{j}>0}\tilde{\mathbb{E}}\Big{[}\frac{1}{N}\mathds{1}_{F^{N}\setminus\tilde{F}^{N}}\sum_{\tau_{j}\in[0,\eta]}\log\beta_{j}(Z^{N,x^{N}}(\tau_{j}-))\Big{]}.$
(66)

Let us first consider the first term in Equation (66). As for all $j$,

$\log\beta_{j}(\cdot):\tilde{A}(\tilde{\epsilon}):=\\{z\in
A|\operatorname{dist}(z,\partial
A)\geq\tilde{\epsilon}\\}\rightarrow\mathds{R}$

is uniformly continuous, there exit constants $\tilde{\delta}_{N}>0$ with
$\tilde{\delta}_{N}\downarrow 0$ such that

$z,\tilde{z}\in\tilde{A}(\tilde{\epsilon}),\,|\tilde{z}-z|<3\epsilon_{N}\Rightarrow|\log\beta_{j}(\tilde{z})-\log\beta_{j}(z)|<\tilde{\delta}_{N}.$
(67)

We define

$M=M(N):=\lfloor(\eta-2\tilde{\epsilon}/\lambda_{1})\epsilon_{N}^{-1}+1\rfloor$

and divide the interval $[2\tilde{\epsilon}/\lambda_{1},\eta]$ into $M$
equidistant subintervals $[t_{r},t_{r+1}]$ ($r=0,\dots,M-1$, $t_{r}=t_{r}(N)$)
of length $\Delta=\Delta(N)$, i.e. (for $N$ large enough),

$\frac{\epsilon_{N}}{2}\leq\Delta<\epsilon_{N}.$

For $j=1,\dots,k$, $r=0,\dots,M-1$ and $\tau_{j},t\in[t_{r},t_{r+1}]$ we have,

$|Z^{N,x^{N}}(\tau_{j}-)-\phi^{x}(t)|\leq 2\epsilon_{N}\quad\text{on
}\tilde{F}^{N},$

since $|\phi^{x}(\tau_{j})-\phi^{x}(t)|\leq|\tau_{j}-t|$ as $|v_{i}|\leq 1$,
and hence (cf. (67))

$\inf_{t\in[t_{r},t_{r+1}]}\log\beta_{j}(\phi^{x}(t))-\tilde{\delta}_{N}\leq\log\beta_{j}(Z^{N,x^{N}}(\tau_{j}-)).$

From this inequality, we deduce

$\displaystyle\frac{1}{\tilde{\mathbb{P}}[F^{N}]}\tilde{\mathbb{E}}\Big{[}\frac{1}{N}\mathds{1}_{\tilde{F}^{N}}\sum_{\tau\in[2\tilde{\epsilon}/\lambda_{1},\eta]}\log\beta_{j(\tau)}(Z^{N,x^{N}}(\tau-))\Big{]}$
$\displaystyle\qquad=\frac{1}{\tilde{\mathbb{P}}[F^{N}]}\sum_{j,\,\mu^{i}_{j}>0}\sum_{r=0}^{M-1}\tilde{\mathbb{E}}\Big{[}\frac{1}{N}\mathds{1}_{\tilde{F}^{N}}\sum_{\tau_{j}\in[t_{r},t_{r+1})}\log\beta_{j}(Z^{N,x^{N}}(\tau_{j}-))\Big{]}$
$\displaystyle\qquad\geq\frac{1}{\tilde{\mathbb{P}}[F^{N}]}\sum_{j,\,\mu^{i}_{j}>0}\sum_{r=0}^{M-1}\Big{(}\inf_{t\in[t_{r},t_{r+1})}\log\beta_{j}(\phi^{x}(t))-\tilde{\delta}_{N}\Big{)}\tilde{\mathbb{E}}\Big{[}\mathds{1}_{\tilde{F}^{N}}Y^{N,x^{N},t_{r},t_{r+1}}_{j}\Big{]}$
$\displaystyle\qquad\geq\frac{1}{\tilde{\mathbb{P}}[F^{N}]}\sum_{j,\,\mu^{i}_{j}>0}\sum_{r=0}^{M-1}\inf_{t\in[t_{r},t_{r+1})}\log\beta_{j}(\phi^{x}(t))\tilde{\mathbb{P}}[\tilde{F}^{N}]\tilde{\mathbb{E}}\big{[}Y^{N,x^{N},t_{r},t_{r+1}}_{j}\big{]}$
$\displaystyle\qquad\qquad-\frac{1}{\tilde{\mathbb{P}}[F^{N}]}\sum_{j,\,\mu^{i}_{j}>0}\sum_{r=0}^{M-1}\big{|}\log\underline{\beta}(\tilde{\epsilon})\big{|}\big{|}\widetilde{\operatorname{Cov}}(\mathds{1}_{\tilde{F}^{N}},Y^{N,x^{N},t_{r},t_{r+1}}_{j})\big{|}$
$\displaystyle\qquad\qquad-\frac{1}{\tilde{\mathbb{P}}[F^{N}]}\tilde{\delta}_{N}\sum_{j,\,\mu^{i}_{j}>0}\tilde{\mathbb{E}}\big{[}\mathds{1}_{\tilde{F}^{N}}Y^{N,x^{N},2\tilde{\epsilon}/\lambda_{1},\eta}].$
(68)

The second term in Inequality (68) satisfies (cf. Inequality (60) and
Assumption 2.3 (C); we assume that $N$ is sufficiently large such that $M\leq
2\epsilon_{N}^{-1}\eta$),

$\displaystyle\frac{1}{\tilde{\mathbb{P}}[F^{N}]}\sum_{j,\,\mu^{i}_{j}>0}\sum_{r=0}^{M-1}\big{|}\log\underline{\beta}(\tilde{\epsilon})\big{|}\big{|}\widetilde{\operatorname{Cov}}(\mathds{1}_{\tilde{F}^{N}},Y^{N,x^{N},t_{r},t_{r+1}}_{j})\big{|}$
$\displaystyle\qquad\leq\frac{1}{\tilde{\mathbb{P}}[F^{N}]}2k\eta\big{|}\log\underline{\beta}(\tilde{\epsilon})\big{|}\epsilon_{N}^{-1}\sqrt{\bar{\mu}\;\epsilon_{N}}\sqrt{\delta(N,\epsilon_{N})}$
$\displaystyle\qquad\rightarrow 0$ (69)

as $N\rightarrow\infty$. The third term in Equation (68) satisfies

$\frac{1}{\tilde{\mathbb{P}}[F^{N}]}\tilde{\delta}_{N}\sum_{j,\,\mu^{i}_{j}>0}\tilde{\mathbb{E}}\big{[}\mathds{1}_{\tilde{F}^{N}}Y^{N,x^{N},2\tilde{\epsilon}/\lambda_{1},\eta}]\leq\frac{1}{\tilde{\mathbb{P}}[F^{N}]}\tilde{\delta}_{N}k\bar{\mu}\eta\rightarrow
0$ (70)

as $N\rightarrow\infty$. Finally, let us consider the first term in Equation
(68). Recall that by (58) and (63), we have

$\displaystyle\mu_{j}^{i}(t_{r+1}-t_{r})$
$\displaystyle\geq\tilde{\mathbb{E}}[Y_{j}^{N,x^{N},t_{r},t_{r+1}}]$
$\displaystyle\geq\mu_{j}^{i}(t_{r+1}-t_{r})\tilde{\mathbb{P}}[F^{N}_{1}]\cdot\inf_{z;|z-\phi^{x}(\tilde{\epsilon}/\lambda_{1})|<\tilde{\epsilon}/2}\tilde{\mathbb{P}}[F^{N,z}_{2}]$
$\displaystyle\qquad\qquad-\tilde{\mathbb{P}}[F^{N}_{1}]\cdot\inf_{z;|z-\phi^{x}(\tilde{\epsilon}/\lambda_{1})|<\tilde{\epsilon}/2}\big{|}\widetilde{\operatorname{Cov}}(\mathds{1}_{F^{N,z}_{2}},Y_{j}^{N,z,t_{r}-2\tilde{\epsilon}/\lambda_{1},t_{r+1}-2\tilde{\epsilon}/\lambda_{1}})\big{|}.$

We define

$\displaystyle\alpha_{1}^{N}$
$\displaystyle:=\frac{\tilde{\mathbb{P}}[\tilde{F}^{N}]}{\tilde{\mathbb{P}}[F^{N}]},$
$\displaystyle\alpha^{N}_{2}$
$\displaystyle:=\tilde{\mathbb{P}}[F^{N}_{1}]\cdot\inf_{z;|z-\phi^{x}(\tilde{\epsilon}/\lambda_{1})|<\tilde{\epsilon}/2}\tilde{\mathbb{P}}[F^{N,z}_{2}]<1,$
$\displaystyle\alpha_{3}^{N}$
$\displaystyle:=\tilde{\mathbb{P}}[F^{N}_{1}]\cdot\inf_{z;|z-\phi^{x}(\tilde{\epsilon}/\lambda_{1})|<\tilde{\epsilon}/2}\big{|}\widetilde{\operatorname{Cov}}(\mathds{1}_{F^{N,z}_{2}},Y_{j}^{N,z,t_{r}-2\tilde{\epsilon}/\lambda_{1},t_{r+1}-2\tilde{\epsilon}/\lambda_{1}})\big{|},$
$\displaystyle\phi^{r}_{j}$
$\displaystyle:=\inf_{t\in[t_{r},t_{r+1})}\log\beta_{j}(\phi^{x}(t)),$
$\displaystyle S^{N}$
$\displaystyle:=\sum_{j,\,\mu^{i}_{j}>0}\mu^{i}_{j}\sum_{r=0}^{M-1}(t_{r+1}-t_{r})\inf_{t\in[t_{r},t_{r+1})}\log\beta_{j}(\phi^{x}(t)).$

We compute (for $N$ large enough as before)

$\displaystyle\frac{\tilde{\mathbb{P}}[\tilde{F}^{N}]}{\tilde{\mathbb{P}}[F^{N}]}\sum_{j,\,\mu^{i}_{j}>0}\sum_{r=0}^{M-1}\inf_{t\in[t_{r},t_{r+1})}\log\beta_{j}(\phi^{x}(t))\tilde{\mathbb{E}}\big{[}Y^{N,x^{N},t_{r},t_{r+1}}_{j}\big{]}$
$\displaystyle\qquad\geq\alpha_{1}^{N}\sum_{j,\,\mu^{i}_{j}>0}\sum_{r=0}^{M-1}\inf_{t\in[t_{r},t_{r+1})}\log\beta_{j}(\phi^{x}(t))\cdot\Big{\\{}\mathds{1}_{\\{\phi^{r}_{j}<0\\}}\mu_{j}(t_{r+1}-t_{r})$