%{\normalsize\hfill\jobname.tex}\\ On large deviations in the averaging principle for SDE's with a ``full dependence'', correction %\footnote{Ann.Probab. 1999, 27, no.~1, 284--296; %corrected and expanded version Dec. 2004. %, due to a %request by Yuri Kifer } %\footnote{Received June 1998} %\footnote{Supported by grants RFBR 98-01-00062, INTAS 93-0378, %INTAS 93-1585-ext.} \author{A. Yu. Veretennikov \\ {(School of Mathematics, University of Leeds, UK} \\ {\& Institute for Information Transmission Problems, Russia)} }
A. Yu. Veretennikov
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\title{%{\normalsize\tt\hfill\jobname.tex}\\
On large deviations in the averaging principle for SDE's with a
``full dependence'', correction
%\footnote{Ann.Probab. 1999, 27, no.~1, 284--296;
%corrected and expanded version Dec. 2004. %, due to a
%request by Yuri Kifer }
%\footnote{Received June 1998}
%\footnote{Supported by grants RFBR 98-01-00062, INTAS 93-0378,
%INTAS 93-1585-ext.}
\author{A. Yu. Veretennikov
\\
{(School of Mathematics, University of Leeds, UK} \\
{\& Institute for Information Transmission Problems, Russia)} } }

\begin{document}
\maketitle
\newcommand{\eps}{\varepsilon}
%\newcommand{\argmin}{\mathop{\rm argmin}\nolimits}
\newcommand{\argmax}{\mathop{\rm argmax}\nolimits}
\newtheorem{Theorem}{Theorem}
\newtheorem{Lemma}{Lemma}


{\footnotesize
We establish the large deviation principle for stochastic
differential equations with averaging in the case when all
coefficients of the fast component depend on the slow one,
including diffusion.}
\footnote{{\it AMS} 1991 {\it subject classifications.}
60F10, 60J60.}
\footnote{{\it Key words and phrases.} Large deviations, averaging,
stochastic differential equation.}


%\footnotemark{Key words and phrases: large deviations, averaging,
%stochastic differential equation
%AMS 1991 subject classifications. 60F10, 60J60.}


\section{Introduction}
This is a corrected version of the paper Veretennikov (1999). We
consider the SDE system
\begin{eqnarray}\label{eqAA}
& dX_t = f(X_t,Y_t)dt, \quad X_0=x_0, \nonumber
 \\
& dY_t = \eps^{-2}B(X_t,Y_t)dt + \eps^{-1}C(X_t,Y_t)dW_t, \quad
Y_0=y_0.
\end{eqnarray}
Here $X_{t}\in E^{d}, \, Y_{t}\in M$, $M$ is a compact manifold
of dimension $\ell$ (e.g. torus $T^\ell$), $f$ is a function
with values in $d$--dimensional Euclidean space $E^{d}$, $B$ is
a function with values in $T M$, $C$ is a function with values
in $(T M)^\ell$ (i.e., in local coordinates an $\ell\times \ell$
matrix), $W_{t}$ is an $\ell$--dimensional Wiener process on some
probability space $(\Omega, F, P)$, $\eps>0$ is a small
parameter.  Concerning SDE's on manifolds we refer to Watanabe
and Ikeda (1989).  %Priouret?


The large deviation principle (LDP) for such systems with a ``full
dependence'', that is, $C(X_t,Y_t)$, was not treated earlier. Only
the case $C(Y_t)$ was considered in papers by Freidlin (1976),
Freidlin (1978), Freidlin and Wentzell (1984) for a compact state
space and by Veretennikov (1994) for a non-compact one. There are,
as well, recent papers on more general systems with small additive
diffusions by Liptser and by the author which also only concern
the case $C(Y_t)$.


The LDP for systems like (\ref{eqAA}) is important in averaging
and homogenization, in the KPP equation theory, for stochastic
approximation algorithms with averaging and so forth. The problem
of an LDP for the case $C(X_t,Y_t)$ has arisen since Freidlin
(1976), Freidlin (1978). Intuitively, the scheme used for $C(Y_t)$
should work; at least, almost all main steps go well. Indeed,
there was only one lacuna; the use of Girsanov's transformation
did not allow freezing of $X_t$ if $C$ depended on the slow motion
while it worked well and very naturally for the drift
$B(X_t,Y_t)$. Yet the problem remained unresolved for years and
the answer was not clear at all. Notice that this difficulty does
not appear in analogous discrete-time systems [see Gulinsky and
Veretennikov (1993), Chapter 11].

It turned out that the use of Girsanov's transformation in some
sense prevented resolving the problem.
Our approach in this paper is based on a new technical lemma,
Lemma \ref{L-main} below. The main idea is to use two different
scales of partitions of the interval $[0,T]$, a ``first-order
partition'' by points $\Delta,\, 2\Delta,\, \ldots$, which do not
depend on the small parameter $\eps$ and ``second-order
partitions'' which depend on $\eps$ in a special way, by points
$\eps^2 t(\eps),\, 2\eps^2 t(\eps), \ldots\,$.  Then
the exponential estimates needed for the proof of the
result can be established by two steps. First, the estimates
for a ``small'' partition interval are derived
using the uniform bound of Lemma \ref{L-lim} (see
below) and the estimates for stochastic integrals. It is
important that in the ``second'' scale the fast motion is still close
enough to its frozen version [the bound (\ref{eqYY}) below].
Second, the bounds for ``small'' partitions and induction give
one the estimate for a ``large'' partition interval.


{\it The original proof in the paper 1999 contains a gap. It
relates to a boundedness of some auxiliary constant $b$ in the
proof, -- in the original version this constant may depend
implicitly on the partition size $\Delta$, hence generating a
vicious circle.
%[The second gap???]
The main aim of this version of the paper is to present the
``patch''. The correction uses improved approximations that keep
this constant $b$ bounded in the lower and upper bounds, and it
uses also a truncated Legendre transformation in the upper bound.
The author is deeply indebted to Professor Yuri Kifer for
discovering this vicious circle in the original version of the
paper. The main technical tool remains the Lemma \ref{L-main}. All
standing assumptions are the same as in the original version.}




The main result is stated in Section 2. In Section 3 we expose
auxiliary lemmas, among them the main technical Lemma \ref{L-main}
with its proof and a version of an important lemma from Freidlin
and Wentzell (1984) (see Lemma \ref{L-FW}) which requires certain
comments. Those comments along with other related remarks are
given in the Appendix, the latter has been also slightly extended.
The proof of the main theorem is presented in Section 4.







\section{Main result}
We make the following assumptions.

\begin{description}
\item[$(A_f)$]
The function $f$ is bounded and satisfies the Lipschitz condition.

\item[$(A_C)$]
The function $CC^*$ is bounded, uniformly nondegenerate, $C$
satisfies the Lipschitz condition.

\item[$(A_B)$]
The function $B$ is bounded and satisfies the Lipschitz
condition.
\end{description}

Some conditions may be relaxed; for example, $B$ can be locally
bounded, $C$ locally (w.r.t. $x$) nondegenerate and so on.

The family of processes $X^\eps$ satisfies a large deviation
principle in the space $C([0,T];R^d)$ with a normalizing
coefficient $\eps^{-2}$ and a rate function $S(\varphi)$ if
three conditions are satisfied:
\begin{equation}\label{eq4}
\limsup_{\eps\to 0} \eps^2 \log P_x(X^\eps \in F) \leq - \inf_F
S(\varphi), \quad \forall F \mbox{ closed },
\end{equation}
\begin{equation}\label{eq5}
\liminf_{\eps\to 0} \eps^2 \log P_x(X^\eps\in G) \geq - \inf_G
S(\varphi), \quad \forall G \mbox{ open },
\end{equation}
and $S$ is a ``good'' rate function; that is, for any $s\geq 0$,
the set
\begin{displaymath}
\Phi(s) : = (\varphi\in C([0,T];R^d):
\,\, S(\varphi)\leq s, \,\, \varphi(0)=x)
\end{displaymath}
is compact in $C([0,T];R^d)$. In fact, we will establish the
following equivalent set of assertions due to Freidlin and
Wentzell:
\begin{equation}\label{eq4a}
%\limsup_{\delta\to 0}
\limsup_{\delta\to 0} \limsup_{\eps\to 0} \eps^2 \log
P_x(\rho_{0,T}(X^\eps, \Phi(s)\ge \delta) \leq - s, \quad \forall
s>0,
\end{equation}
where $\Phi(s) := \{\varphi\in C[0,T;R^d], \, S(\varphi)\le s\}$,
and
\begin{equation}\label{eq5a}
\liminf_{\delta\to 0}\liminf_{\eps\to 0} \eps^2 \log
P_x(\rho_{0,T}(X^\eps,\varphi)<\delta) \geq - S(\varphi), \quad
\forall \varphi,
\end{equation}
where $S$ is a ``good'' rate function (see above).


~



Let $\tilde W_t = \eps^{-1} W_{t\eps^{2}}$, $y_t =
Y_{t\eps^{2}}$, $x_t = X_{t\eps^{2}}$ and let $y^x_t$ denote a
solution of SDE,
\begin{equation}\label{eqBB}
dy^x_t = B(x,y^x_t)dt + C(x,y^x_t)d\tilde W_t, \quad y^x_0=y_0.
\end{equation}

\begin{Theorem}\label{Theorem1}
Let $(A_f)$, $(A_B)$, $A_C)$ be satisfied. Then the family
$(X^\eps_t = X_t,\; 0\leq t\leq T)$ satisfies the LDP as $\eps
\to 0$ in the space $C([0,T];R^d)$ with an action function
$$
  S(\varphi) = \int_0^T L(\varphi_t,\dot\varphi_t)dt,
$$
where
$$
  L(x,\alpha) = \sup_\beta (\alpha\beta - H(x,\beta)),
$$
$$
  H(x,\beta) = \lim_{t\to\infty} t^{-1} \log E\exp\left(\int_0^t
f(x,y^x_s)ds\right).
$$
The limit $H$ exists and is finite for any $\beta$, the
functions $H$ and $L$ are convex in their last arguments $\beta$
and $\alpha$ correspondingly, $L\geq 0$ and $H$ is continuously
differentiable in $\beta$.
\end{Theorem}

The differentiability of $H$ at any $\beta$ is provided by the
compactness of the state space of the fast component.


\section{Auxiliary lemmas}
Let us consider the semigroup of operators $T^{\beta }_{t}, t\geq
0$,  on $C(M)$ defined by the formula
$$
T^{x',x,\beta}_{t}g(y)= T^{\beta }_{t}g(y) =
E_{y}g(y^{x}_{t})\exp \left(\int^{t}_{0}
\beta f(x',y^{x}_{s})ds\right),
$$
where $\beta\in E^d$, $\beta f$ is a scalar product.


\begin{Lemma} \label{L-comp}
Let assumptions $(A_f)$, $(A_B)$, $(A_C)$ be satisfied.
Then for any $\beta$ the operator $T^{\beta }_{1}$ is compact in
the space $C([0,T];R^d)$.
\end{Lemma}


\begin{Lemma} \label{L-kls}
Let assumptions $(A_f)$, $(A_B)$, $(A_C)$ be satisfied. Then
the spectral radius $r(T^{\beta }_{1})$ is a simple eigenvalue
of $T^{\beta }_{1}$ separated from the rest of the spectrum
and its eigen--function $e_{\beta }$ belongs to the cone
$C^+(M)$. Moreover, function $r(T^{\beta }_{1})$ is smooth (of
$C^\infty$) in $\beta$ and the function $e_{\beta }$ is bounded
and separated away from zero uniformly in $|\beta|<b$ and any
$x', x$.
\end{Lemma}



\begin{Lemma} \label{L-lim}
Let $\beta \in E^{d}$, and let assumptions $(A_f)$, $(A_B)$,
$(A_C)$ be satisfied. Then there exists a limit
$$
H(x',x,\beta ) = \lim_{t\to 0} t^{-1}\log E_{y}\exp\left(\beta\int
^{t}_{0} f(x',y^{x}_{s})ds\right);
$$
moreover, $H(x',x,\beta ) = \log r(T^{x',x,\beta}_1)$.
The function $H(x',x,\beta )$ is of $C^\infty$ in
$\beta $ and convex in $\beta $. For any $b>0$ there
exists $C(b)$ such that, for any $y$, $|\beta|<b$
\begin{equation}\label{eqUU}
|t^{-1}\log E_{y}\exp \left(\beta \int^t_0
f(x',y^{x}_{s})ds\right)-H(x',x,\beta )|\le
C(b)t^{-1}.
\end{equation}
Notice that $|H(x',x,\beta)|\leq \|f\|_C |\beta|$.
\end{Lemma}

\begin{Lemma} \label{L-dif}
Let assumptions $(A_f)$, $(A_B)$, $(A_C)$ be satisfied. Then
for any $b>0$ the functions $H$ and $\nabla_\beta H$ are
uniformly continuous in $(x',x,\beta )$, $|\beta |<b$.
\end{Lemma}


Lemmas 1-4 are standard [cf. Veretennikov (1994) or (1992)]. They
are based on Frobenius-type theorems for positive compact
operators [see Krasnosel'skii, Lifshitz and Sobolev (1989)] and the
theory of perturbations of linear operators [see Kato (1976),
Chapter 2]. Denote $\tilde F_t = F_{t\eps^{2}}$.


\begin{Lemma}\label{L-main}
Let assumptions $(A_f)$, $(A_B)$, $(A_C)$, $b>0$, $t(\eps) \to
\infty$ and $t(\eps) = o(\log \eps^{-1})$ as $\eps \to 0$. Then
for any $\nu>0$ there exist $\delta(\nu)>0$, $\eps(\nu)>0$ such
that for $\eps \leq \eps(\nu)$ uniformly w.r.t.
$t_0,\,x',\,x,\,x_{0},\,y_{0}$, $x_{t_0}=x_0$, $|\beta|\leq b$,
 the inequality holds on the set
$\{|x_{t_{0}} - x| < \delta(\nu)\}$,

\begin{equation}\label{eqCC}
  \left|\log E(\exp(\beta
\int\limits_{t_{0}}^{t_{0}+t(\eps)}
f(x',y_s)ds)|\tilde F_{t_{0}})
  - t(\eps) H(x',x,\beta)\right| \leq \nu t(\eps).
\end{equation}
Moreover, if $\Delta \leq \Delta(\nu) =
(1+\|f\|_C)^{-1}\delta(\nu)/2$ and $\eps$ is small enough, then
uniformly w.r.t. $t_0,\,x',\,x,\,x_{0},\,y_{0}$,
$\delta\le\delta(\nu)$, $|x_0-x|<\delta$, and $|\beta|\leq b$,
\begin{eqnarray}\label{eqDD}
& \exp(\eps^{-2}\Delta H(x',x,\beta) -
\nu\Delta\eps^{-2})
  \nonumber \\
&\leq E\left( \exp(\beta\eps^{-2}
\int\limits_{t_{0}}^{t_{0}+\Delta} f(x',Y_s)ds ) |
F_{t_{0}}\right)
  \nonumber \\
&\leq \exp(\eps^{-2} \Delta H(x',x,\beta) + \nu\Delta\eps^{-2}).
\end{eqnarray}
\end{Lemma}
{\bf Remark.} We reiterate and emphasize that any couple
$(\Delta,\delta)$ satisfying only $\Delta \le \Delta(\nu)$ and
$\delta \le \delta(\nu)$ would do.


\noindent {\bf Proof.} {\it Step} 1. It is sufficient to prove
(\ref{eqCC}) and (\ref{eqDD}) for $t_0=0$. Moreover, since $H$ is
continuous, it suffices to check both inequalities for $x=x_0$.
Indeed, the bound
$$
\left|\log E \exp\left(\beta \int\limits_{0}^{t(\eps)}
f(x',y_s)ds\right) - t(\eps) H(x',x_0,\beta)\right| \leq \nu
t(\eps)
$$
implies
\begin{eqnarray*}
&\left|\log E \exp\left(\beta \int\limits_{0}^{t(\eps)}
f(x',y_s)ds\right) - t(\eps) H(x',x,\beta)\right|
 \\
&\leq t(\eps)(\nu + |H(x',x,\beta)-H(x',x_0,\beta)|),
\end{eqnarray*}
and we use the uniform continuity of the function $H$ on compact
sets (remind that $|\beta|\le b$).  The same arguments are
applicable to the second inequality of the assertion of the lemma.
So, in the sequel we consider the case $x_0=x$.




Let us show first that
\begin{equation}\label{eqEE}
\sup_{x',x_{0}}\left| t(\eps)^{-1} \log E\exp\left(\beta
\int_0^{t(\eps)} f(x',y_s)ds\right) - H(x',x,\beta)\right|
\leq \nu
\end{equation}
if $\eps$ is small enough. Due to Lemma \ref{L-lim}, it would be
correct if $y_s$ were replaced by $y^x_s$ and $t(\eps)\geq
\nu^{-1}C(b)$. We will also use the bounds
\begin{equation}\label{eqXX}
\sup_{0\leq s\leq t} |x_s-x_0| \leq \|f\|_C \eps^2t \quad
\forall C, \;\;\; \exp(Ct(\eps))t(\eps)^2 \eps^2 \to 0, \;\;
\eps\to 0.
\end{equation}
Let $|f(x',y)-f(x',y')|\leq L_f |y-y'|$ for all $y,y',x'$,
$L_f>0$, $C_f=\|f\|_C$. We estimate for $t(\eps) > \nu^{-1}
C(b)/4$,
\begin{eqnarray}\label{eqFF}
E\exp\left(\beta\int\limits_0^{t(\eps)} f(x',y_s)ds\right)
 \nonumber\\
\times
\left\{I\left(\sup_{0\leq t\leq t(\eps)} |y_t-y^x_t|
\leq \nu/(4L_fb)\right)
+ I\left(\sup_{0\leq t\leq t(\eps)} |y_t-y^x_t| > \nu/(4L_fb)
\right)\right\}
  \nonumber \\
\leq E\exp\left(\beta\int\limits_0^{t(\eps)} f(x',y^x_s)ds +
t(\eps)\nu/4\right)
I\left(\sup_{0\leq t\leq t(\eps)} |y_t-y^x_t|\leq \nu/(4L_fb)
\right)
  \nonumber \\
+ \exp(C_f b t(\eps)\nu) E I\left(\sup_{0\leq t\leq t(\eps)}
|y_t-y^x_t| > \nu/(4L_fb)\right)
  \nonumber \\
\leq E\exp\left(\beta\int\limits_0^{t(\eps)} f(x',y^x_s)ds\right)
\exp(t(\eps)\nu/4)
  \nonumber \\
+ \exp\left(C_f b t(\eps)\nu\right)\nu^{-2} E\sup_{t\leq t(\eps)}
|y_t-y^x_t|^2.
\end{eqnarray}
By virtue of Lemma \ref{L-lim} we have
$$
E\exp\left(\beta\int\limits_0^{t(\eps)} f(x',y^x_s)ds\right)
\leq \exp(t(\eps)(H(x',x,\beta)+\nu/4))
$$
if $\eps$ is small enough. A similar lower bound holds true also.

Let us estimate the second term. By virtue of the inequalities
for the It\^o and Lebesgue integrals, we have
\begin{eqnarray*}
E\sup_{t'\leq t}|y_{t'}-y^x_{t'}|^2
 \\
\leq C E \int\limits_0^t |C(x_s,y_s) - C(x,y^x_s))|^2 ds
 \\
+ Ct E\int\limits_0^t |B(x_s,y_s) - B(x,y^x_s))|^2 ds
 \\
 \leq C\int\limits_0^t E|x_s-x|^2ds
+ C\int\limits_0^t E\sup_{u\leq s} |y_s-y^x_s|^2ds
 \\
\leq Ct^2\eps^2 + C\int\limits_0^t E\sup_{u\leq s} |y_u-y^x_u|^2
ds.
\end{eqnarray*}
By virtue of Gronwall's lemma, one gets
$$
E\sup_{t'\leq t} |y_{t'}-y^x_{t'}|^2 \leq Ct^2\eps^2\exp(Ct).
$$
In particular,
\begin{equation}\label{eqYY}
E\sup_{t'\leq t(\eps)} |y_{t'}-y^x_{t'}|^2 \leq
Ct(\eps)^2\eps^2\exp(Ct(\eps)).
\end{equation}
So the second term in (\ref{eqFF}) does not exceed the value
$\exp(C_f b t(\eps)\nu)\nu^{-2}Ct(\eps)^2\eps^2$ which is
$o(\exp(K t(\eps)))$ for any $K>0$. Indeed,
$\exp(t(\eps)(C_fb\nu-K))\nu^{-2}Ct(\eps)^2\eps^{2} \to 0$ due to
the assumption $t(\eps) = o(\log \eps^{-1}), \; \eps \to 0$. This
proves (\ref{eqEE}).

Notice that the bound (\ref{eqEE}) is uniform w.r.t.
$|\beta|\leq b$ and $x',\, x,\, y_0$. Since the function $H$ is
continuous, we get on the set $\{|x_{t_{0}} - x| <
\delta(\nu)\}$,

\begin{eqnarray}\label{eqG}
\sup\limits_{x',x,y_{0},t_{0},|\beta|\leq b} \left| \log
E\left(\exp\left(\beta \int\limits_{t_{0}}^{t_{0}+t(\eps)}
f(x',y^x_s)ds\right) \mid \tilde F_{t_{0}}\right)
%\right. \nonumber \\ \nonumber \\ \left.
- t(\eps) H(x',x,\beta)\right| \leq \nu t(\eps)
\end{eqnarray} if
$\delta(\nu)$ is small enough.

\noindent
{\it Step} 2.
Let $\Delta \leq (1+\|f\|_C)^{-1} \delta (\nu)/2 = \Delta(\nu)$
and $N=\Delta
\eps^{-2} t(\eps)^{-1}$. Then $\sup_{0\leq s\leq Nt(\eps)} |x_s
- x_0| \leq \delta(\nu)/2$. Let $|x-x_0|<\delta(\nu)/2$. So,
$\sup_{0\leq s\leq Nt(\eps)} |x_s - x| < \delta(\nu)$. In
particular, $|x_{k t(\eps)} - x| < \delta(\nu)$ for any $1 \leq
k \leq N$. By induction, we get from (\ref{eqG}) for such $k$,

\begin{eqnarray*}
\exp(k t(\eps) H(x',x,\beta) - \nu k t(\eps))
 \\
\leq E \exp\left(\beta \int\limits_0^{k t(\eps)} f(x',y_s)ds
\right)
 \\
\leq \exp(k t(\eps) H(x',x,\beta) + \nu k t(\eps) ),
\end{eqnarray*}
or, after the time change,
\begin{eqnarray*}
\exp(k t(\eps) H(x',x,\beta) - \nu k t(\eps))
 \\
\leq E \exp\left(\beta\eps^{-2}
\int\limits_0^{k t(\eps)\eps^{-2}}
f(x',Y_s)ds\right)
 \\
\leq \exp(k t(\eps) H(x',x,\beta)
+\nu k t(\eps)).
\end{eqnarray*}
Since $H$ is continuous then we obtain for $k=N$,
\begin{eqnarray}\label{eq8}
\exp(\eps^{-2}\Delta H(x',x,\beta) -
\nu\Delta\eps^{-2})
  \nonumber \\ \nonumber \\
\leq E \exp\left(\beta\eps^{-2}
\int\limits_0^\Delta f(x',Y_s)ds\right)
 \nonumber \\ \nonumber \\
\leq \exp(\eps^{-2}\Delta H(x',x_0,\beta)
+ \nu\Delta\eps^{-2}).
\end{eqnarray}
Lemma \ref{L-main} is proved. \hfill QED


%In the sequel we denote $\Delta = \Delta(\nu)$.
%An important point is that $\delta(\nu)/\Delta(\nu) = const > 0$.
The next Lemma is an improved version of the Lemma 7.5.2 from
Freidlin and Wentzell. Although we will not use it, its technique
is essential.

\begin{Lemma}\label{L-FW}[Freidlin (1978), Freidlin and
Wentzell (1984)]. Let $S(\varphi)<\infty$. If $\psi^n$ is a
sequence of step functions tending uniformly to $\varphi$ in
$C[0,T];R^d)$ as $n\to\infty$, then there exists a sequence of
piecewise linear functions $\chi^n$ (with the same partitions)
which also tend uniformly to $\varphi$ and such that
$$
  \limsup_{n\to\infty} \int_0^T L(\psi^n_s,
\dot\chi^n_s) ds\leq S(\varphi).
$$
Moreover, one may assume without loss of generality that for any
$s$ there exists a value
$$
\beta_s = \argmax\limits_{\beta} (\beta \dot\chi^n_{s+} -
H(\psi^n_s,\psi^n_s,\beta))
$$
and
$$
L(\psi^n_s,\alpha) > L(\psi^n_s,\dot\chi^n_{s+}) +
(\alpha-\dot\chi^n_{s+})\beta_s \quad \forall
\alpha\not=\dot\chi^n_s.
$$
If $\hat\psi$ is close enough to $\psi^n_s$ then there exists
a value
$$
\hat\beta_s = \argmax\limits_{\beta} (\beta \dot\chi^n_{s+} -
H(\psi^n_s,\hat\psi,\beta)),
$$
$$
L(\psi^n_s,\hat\psi,\alpha)
> L(\psi^n_s,\hat\psi,\dot\chi^n_{s+}) +
(\alpha-\dot\chi^n_{s+})\hat\beta_s \quad \forall
\alpha\not=\dot\chi^n_s
$$
and
$$
L(\psi^n_s,\hat\psi,\dot\chi^n_{s+}) \to
L(\psi^n_s,\psi^n_s,\dot\chi^n_{s+}), \quad \hat\psi\to\psi^n_s.
$$
\end{Lemma}

We added to the original assertion the property
that $\chi^n_t$ may be chosen piecewise
linear. Indeed, such functions are used in the proof; see
Freidlin and Wentzell (1984), Section 7.5.
The existence of $\beta_s$ asserted in the lemma also follows
from the proof; see Freidlin and Wentzell (1984) or Freidlin
(1978).
Assertions about $\hat\psi$ and $\hat\beta_s$ also added to
the original assertion can be deduced from the proof using
similar arguments.

In fact, there is a little gap in the original proof, namely, an
additional assumption was used which was not formulated
explicitly. This is why we have to present a precise statement and
give necessary comments on it in the Appendix.

%~

\section{Proof of theorem 1} %: non-degenerate case}
\begin{enumerate}
\item
%{\it Step} 0.
\underline{\bf First part of the proof:}  the lower bound. Let
$S(\varphi)<\infty$, and $\nu > 0$. To establish the lower bound,
we will show the inequality: given any $\nu>0$, and {\it any}
$\delta>0$, we have for $\eps
>0$ small enough,
$$% \begin{equation}\label{eq5a}\liminf_{\delta\to 0}\liminf_{\eps\to 0}
\eps^2 \log P_x(\rho_{0,T}(X^\eps,\varphi)<\delta) \geq -
S(\varphi) - \nu.
$$

%\item %Step 1
Denote $H(x,\beta) = H(x,x,\beta)$. The existence of the limit
$H(x,x',\cdot)$ for any $x, x'$, and its differentiability and
continuity are asserted in Lemmas 3 and 4. Throughout the proof,
we may and will assume that for any $s$,
$L(\varphi_s,\dot\varphi_s)<\infty$. Indeed, this may be violated
only on a set of $s$ of Lebesgue measure zero. Notice that due to
the boundedness of the function $f$, this inequality implies
$\sup_s|\dot\varphi_s|\le \|f\|_C$. Indeed, for any $|\alpha|>
\|f\|_C$, we have $L(x,\alpha)=+\infty$.

In the sequel, both $X_0=x$ and $Y_0=y$ are fixed, hence, the
probability symbol $P$ will be used without indices.

\item %2
We are going to reduce the problem of estimation from below the
probability
$$
P(\rho(X,\varphi)<\delta)
$$
to that for the probability
$$
P(\rho(X^\varphi,\varphi)<\delta'), \quad  \mbox{where} \quad
X^\psi_t := x_0 + \int_0^t f(\psi_s,Y_s)ds, \; \forall \psi,
$$
and further to
$$
P(\rho(X^{\psi},\chi)<\delta'),
$$
where both $\psi,\chi$ approximate $\varphi$. The rough idea is
eventually to choose a step function as $\psi$ and piecewise
linear one as $\chi$, however we are going to perform these
approximations gradually. A step function is needed because we
only have a technical tool -- the Lemma 5 -- established for this
very case. A piecewise linear $\psi$ is not necessary, but
convenient. Eventually we consider a finite-dimensional subset of
the set $\{\rho(X,\varphi)<\delta\}$, of the form (slightly
abusing notations which will be explained in the sequel)
$$
\{\rho(X^\psi_\Delta,\chi_\Delta)<\delta'_1,
\rho(X^\psi_{2\Delta},\chi_{2\Delta)}<\delta'_2, \ldots,
\rho(X^\psi_T,\chi_T)<\delta'_{T/\Delta}\},
$$
with appropriately chosen $\Delta$,  $X^\psi$, deterministic
curves $\psi, \chi$, and constants $\delta'_k$: in particular, we
will choose $\delta'_1 << \delta'_2 << \ldots <<
\delta'_{T/\Delta} << \delta$.

While performing all these approximations, we need to establish
simultaneously a special property: at any point $s$, the
Fenchel-Legendre adjoint to the $\dot \chi_s$ variable $\beta_s =
\beta_s[\psi_s,\dot \chi_s]$ (see below) can be chosen uniformly
bounded.

There will be several, -- actually, a lot of, -- small constants
chosen consequently throughout this proof. In the beginning,
$\delta>0$ and $\nu>0$ are fixed; due to the Lemma 5, we have also
$\Delta(\nu)$ and $\delta(\nu)$. Here we prompt in advance the
{\it order of the choice}\, for most of them (by this diagram we
do not claim that every following constant only depends on the
previous one, just the order):
$$
b\, \& \,\tilde\delta' \mapsto \delta' \mapsto \Delta\, \&\,
\delta'' \mapsto \delta'_m \mapsto \delta'_{m-1} \mapsto z_{m-1}
\mapsto \nu\,'_{m-1} \ldots \mapsto \delta'_1 \mapsto z_1 \mapsto
\nu\,'_{1}.
$$

{\bf Remark.} {\it Emphasize that the final set (above) will be a
subset of the $\{\rho(X,\varphi)<\delta\}$, however, it is
\underline{not} necessary that}
$$
\{\rho(\varphi_\Delta,\chi_\Delta)<\delta'_1,
\rho(\varphi_{2\Delta},\chi_{2\Delta)}<\delta'_2, \ldots,
\rho(\varphi_T,\chi_T)<\delta'_{T/\Delta}\},
$$
{\it that is, the curve $\varphi$ itself \underline{does not have}
to belong to this subset.}


\item %4
For any nonrandom curve $\psi$, -- although we will apply this
firstly to $\varphi$, but other functions are also necessary for
the analysis below, -- we have, due to the Lipschitz condition on
$f$,
\begin{equation}\label{eqC}
  \{\rho(X,\varphi)<\delta\} \supset
\{\rho(X^\psi,\chi)<\delta'\}
\end{equation}
if $\delta'$ and $\lambda : = \rho_{0,T}(\varphi,\psi)$ are small
enough with respect to $\delta$. (A small constant $\lambda>0$ is
used just within this step.) E.g., $ \delta' < \delta(e^{CT}CT +
1)^{-1}/2, \quad \lambda < \delta(e^{CT}CT + 1)^{-1}/2 $ suffice,
see below. Indeed,
$$
X_t = x + \int_0^t f(X_s,Y_s)ds, \quad X^\psi_t = x + \int_0^t
f(\psi_s,Y_s)ds,
$$
thence,
\begin{eqnarray*}
|X_t-X^\psi_t| \le \int_0^t |f(X_s,Y_s)ds - f(\psi_s,Y_s)|ds \le
C\int_0^t |X_s-\psi_s|ds
\\ \\
\le C\int_0^t |X_s-X^\psi_s|ds + C\int_0^t |X^\psi_s-\chi_s|ds
+C\int_0^t |\chi_s - \psi_s|ds;
\end{eqnarray*}
so on the set $\{\rho(X^\psi,\chi)<\delta'\}$,
\begin{eqnarray*}
|X_t-X^\psi_t| \le C\int_0^t |X_s-X^\psi_s|ds + C\delta' t
+C\lambda t,
\end{eqnarray*}
and, moreover (on the same set),
$$
\sup_{0\le t'\le t} |X_{t'}-X^\psi_{t'}| \le C\int_0^t \sup_{0\le
s'\le s} |X_{s'}-X^\psi_{s'}|ds + C(\delta' + \lambda) t,
$$
which implies by Gronwall's inequality that on the same set,
$$
\rho(X,X^\psi) \le e^{CT}C(\delta'+\lambda)T.
$$
Now,
\begin{eqnarray*}
\rho(X,\varphi) \le \rho(X,X^\psi) + \rho(X^\psi,\chi) +
\rho(\chi,\varphi) \\ \\
\le e^{CT}C(\delta'+\lambda)T + \delta' + \lambda =
(\delta'+\lambda)(e^{CT}CT + 1).
\end{eqnarray*}
Therefore, (\ref{eqC}) holds true. E.g.,
$$
\delta' < \delta(e^{CT}CT + 1)^{-1}/2, \quad \lambda <
\delta(e^{CT}CT + 1)^{-1}/2
$$
suffice. %{\it However, the smaller the better}.
In particular, it is true that
$$%\begin{equation}\label{eqC}
  \{\rho(X,\varphi)<\delta\} \supset
\{\rho(X^\varphi,\chi)<\delta'\},
$$%\end{equation}
if $\delta'$ and $\lambda$ are small enough with respect to
$\delta$. This bound will be used while establishing a lower
bound.

\item %4
While establishing an upper bound, an opposite inclusion will be
useful,
\begin{equation}\label{eqCa}
  \{\rho(X,\varphi)<\delta\} \subset
  \{\rho(X^\psi,\chi)<2\delta(KT+1)\},
\end{equation}
if $\lambda =
\max\left(\rho(\varphi,\psi),\rho(\varphi,\chi)\right) \le
\delta$. Indeed,
\begin{eqnarray*}
|X_t-X^\psi_t| \le \int_0^t |f(X_s,Y_s)ds - f(\psi_s,Y_s)|ds \le
K\int_0^t |X_s-\psi_s|ds
\\ \\
\le K\int_0^t |X_s-\varphi_s|ds + K\int_0^t |\psi_s-\varphi_s|ds;
\end{eqnarray*}
so on the set $\{\rho(X,\varphi)<\delta\}$,
\begin{eqnarray*}
|X_t-X^\psi_t| \le K\delta t +K\lambda t,
\end{eqnarray*}
and, moreover (on the same set),
$$
\rho(X,X^\psi) \le K(\delta+\lambda)T.
$$
Now, (\ref{eqCa}) follows from the inequalities,
\begin{eqnarray*}
&\rho(X^\psi,\chi) \le \rho(X,X^\psi) + \rho(X,\varphi) +
\rho(\chi,\varphi) \\ \\
&\le K(\delta+\lambda)T + \delta + \lambda.
\end{eqnarray*}



\item Our next goal is the choice of an appropriate $\chi =
\varphi^b$; essential is to keep the integral $\int\limits_0^T
L(\varphi_s,\dot\varphi^b_s)\,ds$ close to $S(\varphi)$.
%\item
Suppose for some $s\in [0,T]$, the set $\{\alpha: \,
L(\varphi_s,\alpha)<\infty\}$ has a non-empty interior, -- for the
latter we will use a notation ${\cal L}^\circ[f,\varphi_s]$, --
with respect to its linear hull ${\cal L}[f,\varphi_s]$. Since
$L(\varphi_s,\dot\varphi_s)<\infty$, this value is attained as a
$\,\liminf\,$ of the values $L(\varphi_s,\alpha)$, $\alpha \in
{\cal L}^\circ[f,\varphi_s]$,
as $\alpha\to\dot\varphi$, see %Freidlin and Wentzell and/or
Rockafellar (1970). It is a property of any such $\alpha$ that
there exists a finite adjoint vector $\beta = \argmax_\beta
(\alpha\beta - H(\varphi_s,\beta))$ given $\alpha$, although this
adjoint may be not unique which we will discuss shortly. Notice
that, in particular, we have
$$
H(\varphi_s,\beta) = (\alpha\beta - L(\varphi_s,\alpha)), \;\;
\mbox{and}\;\; L(\varphi_s,\alpha) = (\alpha\beta -
H(\varphi_s,\beta)).
$$
Then we choose a vector $\dot{\tilde\varphi_s}:= \alpha \in {\cal
L}^\circ[f,\varphi_s]$ so that the value
$L(\varphi_s,\dot{\tilde\varphi_s})$ is close enough to
$L(\varphi_s,\dot{\varphi_s})$.
%, and denote the corresponding
%$\beta$-value by $\beta[\varphi_s,\dot{\tilde\varphi_s}]$.

The adjoint $\beta$-value is unique iff $L$ is differentiable at
$\alpha$, which is also equivalent to strict convexity of $H$ at
$\beta$ (see Rockafellar (1970)). If this is not a case, that is,
$L$ is non-differentiable at $\alpha$, then there is a
sub-gradient to the graph of $L$ at $\alpha$ which is a
non-trivial cone (although its dimension may be less than $d$). In
this case one can choose various adjoint vectors $\beta$'s.
Although not unique, both $\dot{\tilde\varphi_s}$ and
$\beta[\varphi_s,\dot{\tilde\varphi_s}]$ can be chosen as Borel
functions of $s$ (due to the Measurable Choice Theorem). Denote
thus chosen adjoint by $\beta[\varphi_s,\dot{\tilde\varphi_s}]$.

If the set ${\cal L}^\circ[f,\varphi_s]$ is empty, one can choose
$\beta[\varphi_s,\dot{\tilde\varphi_s}]=0$, see Appendix~A.


Now set
$$
\tilde\varphi_t := x + \int_0^t \dot{\tilde\varphi_s}\,ds.
$$
We can choose this new curve to be as close to $\varphi$ as we
like, and the values $\int_0^T
L(\varphi_s,\dot{\tilde\varphi_s})\,ds $ and $\int_0^T
L(\varphi_s,\dot\varphi_s)\,ds $ are arbitrarily close to each
other, too, say,
$$
\left|\int\limits_0^T L(\varphi_s,\dot{\tilde\varphi_s})\,ds -
S(\varphi) \right| \le \nu/3.
$$


\item For any $s$, let us choose a (measurable) vector $\hat\alpha_s$ such that
$L(\varphi_s,\hat\alpha_s) = 0$.

Notice that  $L(\varphi_s,\cdot)\ge 0$ since $H(\varphi_s,0)=0$;
moreover, $H(\varphi_s,\cdot)$ is convex at the origin, hence,
there does exist $\alpha$ such that $L(\varphi_s,\alpha)=0$. If
not unique, this vector can be still chosen measurable due to the
Measurable Choice Theorem. Moreover, $|\hat\alpha_s|\le \|f\|_C$
for any vector where $L$ is finite.





It is shown in the Appendix C that $\hat\alpha_s \in {\cal
L}^\circ[f,\varphi_s]$, if the latter set is not empty. A
corresponding adjoint vector $\beta$ (i.e. $\beta =
\mbox{argsup}\,[\langle \cdot \,,\hat\alpha_s\rangle -
H(\varphi_s, \cdot)]$) does exist, and may be chosen as a
zero-vector in $R^d$ that we denote by $\bar 0$. This vector is,
indeed, one of (if not unique) adjoint vectors for the vector
$\hat\alpha_s$, since $H(\varphi_s,\bar 0)=0$, and $\bar
0\hat\alpha_s-H(\varphi_s,\bar 0)=0$. If unique (that depends on
the differentiability of the function $L$, or, equivalently, on
strict convexity of the function $H$), this adjoint is
precisely~$\bar 0$.


\item Next, given $b$, define
\begin{eqnarray*}
\varphi^b_t := x + \int_0^t \left(\dot{\tilde\varphi_s}\,
1(|\beta[\varphi_s,\dot{\tilde\varphi_s}]|\le b) + \hat\alpha_s
\,1(|\beta[\varphi_s,\dot{\tilde\varphi_s}]| > b)\right)\,ds.
\end{eqnarray*}
Since $|\hat\alpha_s|\leq \|f\|_C$, we can find such a $b$ that
the curve $\varphi^b$ is still as close to $\varphi$ as we like in
the uniform norm, and the values $\,\int_0^T
L(\varphi_s,\dot{\varphi^b_s})\,ds \,$ and $\,\int_0^T
L(\varphi_s,\dot\varphi_s)\,ds \,$ are arbitrarily close to each
other.



At the same time, if for some $s$ we have
$|\beta[\varphi_s,\dot{\tilde\varphi_s}]| > b$, then the
$\beta$-value is transformed into
$\beta[\varphi_s,\dot{\varphi_s^b}]=\beta[\varphi_s,\hat\alpha_s]
= 0$, so that for {\it any} $s\in [0,T]$ the inequality holds
true,
$$
|\beta[\varphi_s,\dot{\varphi^b_s}]|\le b.
$$
We have,
$$%\begin{equation}\label{eqCb}
\fbox{ \mbox{$ \{\rho(X,\varphi)<\delta\} \supset
\{\rho(X,\varphi^b )<\delta/2\}$,}}
$$%\end{equation}
if $b$ is large enough.

\item Next, the values of rate functions
$S^\varphi_s(\tilde\varphi)$ and $S^\varphi(\varphi^b)$ are
arbitrarily close, say, $|S^\varphi_s(\tilde\varphi) -
S^\varphi_s(\varphi^b)|< \nu/3$, so that
$$
\left|S^\varphi(\varphi^b) - S(\varphi) \right| \le 2\nu/3,
$$
if $b$ is large enough. Moreover, in addition,
\begin{equation}\label{eqCb} \fbox{ \mbox{$
\{\rho(X,\varphi^b)<\delta/2\} \supset
\{\rho(X^{\varphi},\varphi^b )<\tilde\delta'\}$,}}
\end{equation}
if $\tilde\delta'$ and $\lambda = \rho(\varphi,\varphi^b)$ are
small enough with respect to $\delta$; hence, we can {\it fix the
values $b$ and $\tilde\delta'$ here.}

\item\label{9} The next transform is the change of both $\varphi$
and $\varphi^b$ so that the first becomes a step function, while
the second becomes piecewise linear on $[0,T]$. In the meantime,
all adjoint $\beta$-values will remain bounded.

Consider the approximations
$$
\psi_s = \varphi_{\kappa_m(s+a)- a}, \quad \dot\chi_s =
\dot{\varphi^b}_{\kappa_m(s+ a)- a},
$$
where $m=T/\Delta$ is a positive integer, $\kappa_m(s) :=
[s/\Delta]\Delta$, and $a$ is `almost any' value from $[0,T]$; we
assume $\varphi_s \equiv \varphi_0, \, s<0$, and
$\dot{\varphi^b}_s \equiv \dot{\varphi^b}_0, \, s<0$. Since
$\sup_t|\dot\varphi_t|<\infty$, we have $\rho(\varphi,\psi)\to 0,
\, m\to\infty$. Next, it is well-known\footnote{Possibly this
folklore fact can be found in N.~V.~Krylov's Lectures on
Stochastic Processes (Moscow State University, 1980s, Russian,
vol. 2; the English translation of this book has been published
recently by the AMS). For the reader's convenience we provide a
folklore proof here. Indeed, due to a standard technique, it
suffices to show for the first integral that
$$\int_0^T\int_0^T |\dot{\varphi^b}_{\kappa_m(s+ a)-
a}-\dot{\varphi^b}_s| \,da\,ds\to 0, \quad m\to \infty.$$ Let
$g_s, \, 0\le s\le T,$ be a smooth bounded function such that
$\int|\dot{\varphi^b}_s-g_s|\,ds < \nu$; all functions are
extended to $-\infty<s<\infty$ so that all vanish outside $[0,T]$.
Then \begin{eqnarray*}\int_0^T\,da
\int|\dot{\varphi^b}_{\kappa_m(s+ a)- a}-g_{\kappa_m(s+ a)-
a}|\,ds = \int\,ds \int_0^T|\dot{\varphi^b}_{\kappa_m(s+ a)-
a}-g_{\kappa_m(s+ a)- a}|\,da  \\ = \sum_{i=1}^{m}\Delta
\left(\Delta^{-1}\int_{(i-1)\Delta}^{i\Delta}|\dot{\varphi^b}_s-g_s|\,ds\right)
\equiv \int|\dot{\varphi^b}_s-g_s|\,ds < \nu.\end{eqnarray*}
Hence, it suffices to show that
$$\int_0^T\int_0^T |g_{\kappa_m(s+ a)- a}- g_s| \,da\,ds\to 0,
\quad m\to \infty,$$ which for smooth functions follows from the
Lebesgue dominated convergence theorem, or even from uniform
convergence under the integrals (of course, the smooth function
$g$ with a compact support is bounded). In turn, the existence of
a smooth function $g$ claimed above is due to the property that
smooth functions are dense in $L_1[0,T]$; this is, e.g., because
so are step functions, while the indicator of any Borel set may be
approximated in this function space by a finite set of intervals,
and any indicator of an open interval can be smoothed. In turn,
the `standard technique' above means that one chooses next $m_n$
so that the Lebesgue measure of the set $\{a:\,
\int|\dot{\varphi^b}_{\kappa_m(s+ a)- a} -
\dot{\varphi^b}_{s}|\,ds > 2^{-n}\}$ does not exceed $2^{-n}$,
then finally the almost sure convergence is due to the
Borel-Cantelli Lemma.  The reasoning for the second integral is
similar: we treat the integrand as a function of $s$ and use the
same freezing.} that there exists such a subsequence $m \to
\infty$ (we keep notation $m$ for this subsequence) that almost
surely w.r.t. $a$, the values of the two integrals are close to
zero,
\begin{equation}\label{kr}%$$
\int_0^T |\dot\varphi_s - \dot\chi_s| \,ds + \int_0^T
|L(\varphi_s,\dot{\varphi^b}_s) - L(\psi_s,\dot\chi_s)| \,ds
\approx 0.
\end{equation}%$$
So, we can choose a value $m$ from this subsequence ($m\to\infty$)
so that, firstly, $\Delta\le\Delta(\nu)$ (a value from the Lemma
5); secondly, $\int_0^T |L(\varphi_s,\dot{\varphi^b}_s) -
L(\psi_s,\dot\chi_s)| \,ds \le \nu/3$, so that
\begin{equation}\label{s-s}
\left|S^\psi(\chi) - S(\varphi) \right| \le \nu;
\end{equation}
thirdly, $\int_0^T |\dot\varphi^b_s - \dot\chi_s| \,ds \le
\tilde\delta'/10$, and finally,
\begin{equation}\label{eqD}
\fbox{ \mbox{$ \{\rho(X^{\varphi},\varphi^b )<\tilde\delta'\}
\supset \{\rho(X^{\varphi}, \chi)<\tilde\delta'\times 9/10\}$,}}
\end{equation}
and (see above (\ref{eqC})), for $\delta'$ small enough (i.e. we
can {\it fix the value $\delta'$ here}),
\begin{equation}\label{eqE}
\fbox{ \mbox{$ \{\rho(X^{\varphi},\chi )<\tilde\delta'\times
9/10\} \supset \{\rho(X^{\psi}, \chi)<\delta'\}$,}}
\end{equation}
for the latter we need $\rho(\varphi,\chi)+\rho(\varphi,\psi)$ to
be small enough which means, in particular, $m$ large enough.



\item For simplicity of presentation, we assume that $a=0$;
otherwise the discretisations below should be read $\varphi^\Delta
= (\varphi_{\Delta-\tilde a},\varphi_{2\Delta -\tilde a}, \ldots,
\varphi_{m\Delta -\tilde a}, \varphi_{T})$, where $\tilde a = a -
[a/\Delta]$; this general case is considered similarly.
%
%\item
So, we assume $a=0$, and denote $\varphi^\Delta = (\varphi_\Delta,
\varphi_{2\Delta}, \ldots, \varphi_{m\Delta})$, $m\Delta=T$. Since
$\|f\|_C < \infty$, we have (see Freidlin and Wentzell, proof of
the Lemma 7.5.1),
\begin{equation}\label{eqGG}
\fbox{$\{\rho(X^\psi,\chi)<\delta'\} \supset
\{\rho((X^\psi)^\Delta,\chi^\Delta)<\delta''\},$}
\end{equation}
if $\delta''$ and $\Delta$ are small enough,
\begin{equation}\label{discr}
\delta'' < \delta''(\delta') \quad \mbox{and} \quad \Delta\le
\Delta(\delta')
\end{equation}
(however, $\Delta\le \Delta(\delta'')$ is \underline{not}
required!), and assuming all our curves start at $x_0$ at time
zero (hence, we do not include the starting point into the
definition of $\varphi^\Delta$). Here for discretised curves we
use the metric,
$$
\rho(\psi^\Delta,\chi^\Delta) := \sup_{k} %\Delta\le T}
|\psi_{k\Delta}-\chi_{k\Delta}|.
$$
Now, we are going to estimate from below the value in the right
hand side of the inequality,
\begin{equation}\label{delta''}%$$
\fbox{$P(\rho((X^\psi)^\Delta,\chi^\Delta) < \delta'') \ge E
\prod_{k=1}^m I( |X^\psi_{k\Delta} - \chi_{k\Delta}| <
\delta'_i),$}
\end{equation}%$$
where $\delta'_1<\delta'_2<\ldots < \delta'_m =
\min(\delta(\nu),\delta'')$, % =: \tilde\delta$,
$i = 1,\ldots, m$,
and $\delta(\nu)$ is from the Lemma~5; here all values $\delta'_i$
and certain auxiliary values $z_i$ will be chosen in the next two
steps as follows:
$$
m_{\nabla H}(\delta'_{k-1}+z_{k-1}) + \frac{\kappa}{2}\,
\delta'_{k-1}\le \frac{\kappa}{2}\,\delta'_k, \quad \& \quad
\delta'_{k-1}\le \frac{\delta'_k}{2}, \quad
 \& \quad m_H(\delta'_{k}) \le \nu,
$$
where $0<\kappa\le 1$, and $m_{g}$ stands for the modulus of
continuity of any function $g$ with respect to all its variables
restricted to $|\beta|\le b+1$. Emphasize that $\delta''$ and
$\Delta$ can be chosen arbitrarily small at this stage, in
particular, in addition to they should satisfy the conditions of
the Lemma 5 that will be used in the sequel, that is, we require
also $\delta''\le \delta(\nu)$ and $\Delta\le \Delta(\nu)$. Hence,
{\it both $\delta''$ and $\Delta$ are fixed at this stage.}



\item\label{item11} Now everything is prepared for the lower
estimate. We start with the estimation of the conditional
expectation $E(I(|X^\psi_{m\Delta} - \chi_{m\Delta}| < \delta'_m)
\mid F_{(m-1)\Delta})$ on the set $\{|X^\psi_{(m-1)\Delta} -
\chi_{(m-1)\Delta}| < \delta'_{m-1} \}$, or, in the other words,
on the set $\{X^\psi_{(m-1)\Delta} = \hat\psi_{(m-1)\Delta}\}$
with $|\hat\psi_{(m-1)\Delta} -
\chi_{(m-1)\Delta}|~<~\delta'_{m-1}$. Let us apply the Cram\'er
transformation of measure. Let $|\beta|\leq b$, we will choose
this vector a bit later (as
$\beta[\psi_{(m-1)\Delta},\dot\chi_{(m-1)\Delta+}]$). We get,
\begin{eqnarray*}
&E\left( I(|X^\psi_{m\Delta} - \chi_{m\Delta}| < \delta'_m)
|F_{(m-1)\Delta} \right) = E^\beta \left(I(|X^\psi_{m\Delta} -
\chi_{m\Delta}| < \delta'_m) \times \right.
 \\ \\
&\left. \times \exp\left( -\eps^{-2} \beta (X^\psi_{m\Delta}
-X^\psi_{(m-1)\Delta}) +\eps^{-2}\Delta
H^{\eps,\psi}_m(\hat\psi_{m-1},\beta) \right)|F_{(m-1)\Delta}
\right),
\end{eqnarray*}
where $E^\beta$ is the (conditional) expectation with respect to
the measure $P^\beta$ defined on the sigma-field $F_{m\Delta}$
given $F_{(m-1)\Delta}$, by its density
$$
  \frac{dP^\beta}{dP}(\omega) = \exp\left(\eps^{-2}\beta(X^\psi_{m\Delta}
-X^\psi_{(m-1)\Delta})-\eps^{-2}\Delta
H^{\eps,\psi}_m(\hat\psi_{(m-1)\Delta},\beta)\right),
$$
where
$$
\eps^{-2}\Delta H^{\eps,\psi}_m(\hat\psi_{m-1},\beta) = \log E
\left( \exp\left( \eps^{-2}\beta (X^\psi_{m\Delta}
-X^\psi_{(m-1)\Delta})\right) |F_{(m-1)\Delta} \right).
$$
(We remind that $\hat\psi_{(m-1)\Delta} = X^\psi_{(m-1)\Delta}$.)
By virtue of the Lemma \ref{L-main}, on the set
$\{|X^\psi_{(m-1)\Delta} - \chi_{(m-1)\Delta}| < \delta'_{m-1} \}$
we estimate,
\begin{eqnarray}\label{eq17}
&E\left[ I(|X^\psi_{m\Delta} -\chi_{m\Delta}| < \delta'_m) \mid
F_{(m-1)\Delta}\right] \nonumber
 \\ \nonumber \\
&=E^\beta \left[ I(|X^\psi_{m\Delta} -\chi_{m\Delta}| < \delta'_m)
\right.\nonumber \\ \left.\right.\nonumber \\  &\left. \times
\exp\left(\eps^{-2}\beta(X^\psi_{m\Delta}
-X^\psi_{(m-1)\Delta})-\eps^{-2}\Delta
H^{\eps,\psi}_m(\hat\psi_{(m-1)\Delta},\beta)\right) \mid
F_{(m-1)\Delta} \right]
\nonumber \\ \nonumber \\
& \geq E^\beta \left( I(|X^\psi_{m\Delta} -\chi_{m\Delta}| <
\delta'_m) \exp\left( -\eps^{-2}\Delta \beta
\left((\chi_{m\Delta}-\chi_{(m-1)\Delta})/\Delta\right)
\nonumber \right. \right. \\ \nonumber \\
& \left. \left. - \frac{\Delta}{\eps^{2}} (H(\psi_{(m-1)\Delta},
\hat \psi_{(m-1)\Delta},\beta) + \nu) -
\frac{(\delta'_m+\delta'_{m-1})}{\eps^{2}} \right) |
F_{(m-1)\Delta}\right).
\end{eqnarray}


Let us choose $\beta = \beta(m) =
\beta[\psi_{(m-1)\Delta},\dot\chi_{(m-1)\Delta+}] \;\; \{=
\argmax_{\beta}(\beta\dot\chi_{(m-1)\Delta+} -
H(\psi_{(m-1)\Delta},\beta))\}$. As was explained above,
$|\beta(m)|\leq b$, moreover,
\begin{eqnarray*}
\beta(m)\dot\chi_{(m-1)\Delta+} -
H(\psi_{(m-1)\Delta},\beta(m)) %\\\\
= L(\psi_{(m-1)\Delta}, \dot\chi_{(m-1)\Delta+}),
\end{eqnarray*}
and
$$
\dot\chi_{(m-1)\Delta+} = \nabla_\beta
H(\psi_{(m-1)\Delta},\beta(m)).
$$
So (\ref{eq17}) implies (with $\beta=\beta(m)$),
\begin{eqnarray}\label{eq-low}
&E\left( I(|X^\psi_{m\Delta} -\chi_{m\Delta}| < \delta'_m) |
F_{(m-1)\Delta}\right)
 \nonumber \\\nonumber \\
&\geq \exp\left(-\eps^{-2}\Delta(L(\psi_{(m-1)\Delta},
\dot\chi_{(m-1)\Delta+})+\nu) - \eps^{-2}(\delta'_m+\delta'_{m-1})
\right)\times
 \nonumber \\\nonumber \\
&\times  \exp\left(- \eps^{-2}\Delta (H(\psi_{(m-1)\Delta}, \hat
\psi_{(m-1)\Delta},\beta) -
H(\psi_{(m-1)\Delta},\psi_{(m-1)\Delta},\beta))\right)
 \nonumber  \\ \nonumber \\
&\times E^{\beta(m)}\left( I(|X^\psi_{m\Delta} - \chi_{m\Delta}| <
\delta'_m)|F_{(m-1)\Delta}\right)
 \nonumber \\\nonumber \\
&\geq \exp\left(-\eps^{-2}\Delta\left(L(\psi_{(m-1)\Delta},
\dot\chi_{(m-1)\Delta+})+ 2\nu\right)  - \eps^{-2}
(\delta'_m+\delta'_{m-1}) \right)\times
 \nonumber  \\ \nonumber \\
&\times E^{\beta(m)}\left( I(|X^\psi_{m\Delta} - \chi_{m\Delta}| <
\delta'_m)|F_{(m-1)\Delta}\right).
\end{eqnarray}
We have used the uniform continuity of $H(x,\cdot,\beta)$ over
$|\beta|\le b$ and $x~\in~R^d$:
\begin{eqnarray*}
|H(\psi_{(m-1)\Delta}, \hat \psi_{(m-1)\Delta},\beta) -
H(\psi_{(m-1)\Delta},\psi_{(m-1)\Delta},\beta)| \\\\
\le m_H(|\hat \psi_{(m-1)\Delta} - \psi_{(m-1)\Delta}|)\le
m_H(\delta'_{m-1}) \le \nu
\end{eqnarray*}
(here $m_H$ stands for the modulus of continuity of $H$ for
$|\beta|\le b$ with $b$ fixed), as $\delta'_{m-1}$ is small
enough.

\item\label{item12} Let us show the bound
\begin{equation}\label{eqLLN}
  E^{\beta(m)}\left( I(|X^\psi_{m\Delta} - \chi_{m\Delta}| <
\delta'_m)|F_{(m-1)\Delta}\right) \geq 1 -
\exp(-C_m\Delta\eps^{-2})
\end{equation}
with some $C_m>0$ which may depend on $\nu$, on the set
$\{|\hat\psi_{(m-1)\Delta} - \chi_{(m-1)\Delta}| <
\delta'_{m-1}\}$, if $\eps$ is small enough. There exists a finite
number of vectors $v_1,\, v_2,\, \ldots, v_N$ such that $\|v_k\|=1
\; \forall k$, $N = 2d$ (any orthonormal basis would do
accomplished by its ``symmetric'' transformation, i.e. with each
coordinate vector $v$ we consider $-v$ as well), and
\begin{eqnarray*}
&E^{\beta(m)} (I(|X^\psi_{m\Delta} - \chi_{m\Delta}| > \delta'_m)
\mid F_{(m-1)\Delta})
\\ \\
&\leq \sum^N_{k=1} E^{\beta(m)} ( I((X^\psi_{m\Delta} -
X^\psi_{(m-1)\Delta} - \chi_{m\Delta} + \chi_{(m-1)\Delta}) v_k
\\\\
&> \kappa (\delta'_m-\delta'_{m-1}) \mid F_{(m-1)\Delta}),
\end{eqnarray*}
given $\{|X^\psi_{(m-1)\Delta} - \chi_{(m-1)\Delta}| <
\delta'_{m-1}\}$, where $\kappa = (2/N)^{1/2}$ (notice that $\kappa\le 1$). %??
%Remind that $\tilde\delta = \min(\delta(\nu),\delta'')$ (see (\ref{delta''})).
Given any $\nu\,'_{m-1}>0$ (a new constant which has nothing to do
with $\nu$ and will be fixed shortly; we need it so to say only
locally, while establishing the inequality (\ref{eqLLN})), we
estimate, for any $v := v_k$, $0\leq z\leq 1$,
\begin{eqnarray}\label{eq20}
&E^{\beta(m)}\left( I((X^\psi_{m\Delta}-X^\psi_{(m-1)\Delta}
-\chi_{m\Delta}+\chi_{(m-1)\Delta})v
> \kappa (\delta'_m-\delta'_{m-1}))|F_{(m-1)\Delta}\right)
 \nonumber \\ \nonumber \\
&\leq \exp(-(\delta'_m-\delta'_{m-1})\Delta z \kappa
\eps^{-2})\exp\left( \Delta\eps^{-2} [- z v\dot\chi_{(m-1)\Delta+}
\right.
 \nonumber \\ \nonumber \\
&\left. + H^{\eps,\psi}(\hat\psi_{(m-1)\Delta},\beta(m)+ v z) -
H^{\eps,\psi}(\hat\psi_{(m-1)\Delta},\beta(m)) +
2\nu\,'_{m-1}]\right)
\nonumber \\ \nonumber \\
&\leq \exp(-(\delta'_m-\delta'_{m-1})\Delta z \kappa
\eps^{-2})\exp\left( \Delta\eps^{-2} [- z v\dot\chi_{(m-1)\Delta+}
\right.
 \nonumber \\ \nonumber \\
&\left. + H(\psi_{(m-1)\Delta},\hat\psi_{(m-1)\Delta},\beta(m)+ v
z) \right.
 \nonumber \\ \nonumber \\
&\left. - H(\psi_{(m-1)\Delta},\hat\psi_{(m-1)\Delta},\beta(m)) +
2\nu\,'_{m-1}]\right),
\end{eqnarray}
if $\eps$ is small enough. Denote
\begin{eqnarray*}
&h(z) :=  (\delta'_m-\delta'_{m-1})\kappa z +
\dot\chi_{(m-1)\Delta+} v z
 \\ \\
&-[H(\psi_{(m-1)\Delta},\hat\psi_{(m-1)\Delta},\beta(m)+ v z) -
H(\psi_{(m-1)\Delta},\hat\psi_{(m-1)\Delta},\beta(m))].
\end{eqnarray*}
We have, $h(0)=0$. Moreover,
\begin{eqnarray*}
&h'(0) = (\delta'_m-\delta'_{m-1}) \kappa +
\dot\chi_{(m-1)\Delta+}v - \nabla_\beta
H(\psi_{(m-1)\Delta},\hat\psi_{(m-1)\Delta},\beta(m))v
 \\ \\
&= (\delta'_m-\delta'_{m-1})\kappa  + \nabla_\beta
H(\psi_{(m-1)\Delta},\psi_{(m-1)\Delta},\beta(m))v
 \\ \\
&- \nabla_\beta
H(\psi_{(m-1)\Delta},\hat\psi_{(m-1)\Delta},\beta(m))v
\\\\
&\ge (\delta'_m-\delta'_{m-1})\kappa - m_{\nabla H}(\delta'_{m-1})
=:C_{m-1} > 0
\end{eqnarray*}
(where $m_{\nabla H}$ stands for the modulus of continuity of the
function $\nabla_\beta H$ given $|\beta(m)|\le b+1$ ($b+1$ is
needed for the sequel, although here $b$ would be enough)),
because $\dot\chi_{(m-1)\Delta+} =\nabla_\beta
H(\psi_{(m-1)\Delta},\psi_{(m-1)\Delta},\beta(m))$. The latter
inequality, $C_{m-1}=(\delta'_m-\delta'_{m-1})\kappa - m_{\nabla
H}(\delta'_{m-1}) > 0$, holds true provided $\delta'_{m-1}$ is
small enough in compare to $(\delta'_m-\delta'_{m-1})$, e.g.,
$$
m_{\nabla H}(\delta'_{m-1}) \le
\frac{\kappa}{2}(\delta'_m-\delta'_{m-1}),
$$
or, equivalently,
\begin{equation}\label{modulus1}
m_{\nabla H}(\delta'_{m-1}) + \frac{\kappa}{2} \,\delta'_{m-1}\le
\frac{\kappa}{2}\,\delta'_m.
\end{equation}
Moreover, since $\nabla_\beta H$ is bounded and continuous due to
the Lemma \ref{L-dif}, then $h'(z) \geq C_{m-1}/2$ for small $z$,
say, for $0\leq z\leq z_{m-1}$ (thus, $z_{m-1}$ has been chosen
here). Indeed,
\begin{eqnarray*}
h'(z) = (\delta'_m-\delta'_{m-1})\kappa  +
\dot\chi_{(m-1)\Delta+}v
\\ \\
- \nabla_\beta
H(\psi_{(m-1)\Delta},\hat\psi_{(m-1)\Delta},\beta(m)+vz)v
 \\ \\
= (\delta'_m-\delta'_{m-1})\kappa + \nabla_\beta
H(\psi_{(m-1)\Delta},\psi_{(m-1)\Delta},\beta(m))v
\\ \\
- \nabla_\beta
H(\psi_{(m-1)\Delta},\hat\psi_{(m-1)\Delta},\beta(m)+vz)v
\\\\
\ge (\delta'_m-\delta'_{m-1})\kappa - m_{\nabla
H}(\delta'_{m-1}+z). %% - m_{\nabla H}(z).
\end{eqnarray*}
So, $h(z_{m-1}) \geq C_{m-1} z_{m-1} /2$, provided $z_{m-1}$ along
with $\delta'_{m-1}$ are small in compare to
$(\delta'_m-\delta'_{m-1})$, e.g.,
\begin{equation}\label{modulus2} m_{\nabla
H}(\delta'_{m-1}+z_{m-1}) \le (\delta'_m-\delta'_{m-1})\kappa/2,
\end{equation}
rather than (\ref{modulus1}). Hence, under assumption of
(\ref{modulus2}), the r.h.s. in (\ref{eq20}) with $z=z_m$ does not
exceed the value
$$
\exp(\Delta\eps^{-2}(2\nu\,'_{m-1}- h(z))) \leq \exp(-
C_{m-1}\Delta z_{m-1}\eps^{-2}/4])
$$
if we choose
\begin{equation}\label{nure}
\nu\,'_{m-1} < C_{m-1} z_{m-1}/8.
\end{equation}
Remind that the constant $\nu\,'_{m-1}$ should be fixed in the
beginning of this step of the proof; hence, we can do it now, once
we have chosen $z_{m-1}$, since the latter does not require any
knowledge of $\nu\,'_{m-1}$. This gives the bound, given
$\{|X^\psi_{(m-1)\Delta} - \chi_{(m-1)\Delta}| < \delta'_{m-1}\}$,
$$
E^{\beta(m)}\left( I(|X^\psi_{m\Delta} - \chi_{m\Delta}| \geq
\delta'_m)|F_{(m-1)\Delta}\right) \leq \exp(- C_{m-1}
\Delta\eps^{-2}/4),
$$
which is equivalent to (\ref{eqLLN}). In turn, (\ref{eqLLN})
implies the estimate
\begin{eqnarray*}
&P(|X^\psi_{m\Delta} - \chi_{m\Delta}| < \delta'_m |
F_{(m-1)\Delta})
 \nonumber \\ \nonumber \\
&\geq \exp\left(-\eps^{-2}\Delta(L(\psi_{(m-1)\Delta},
\dot\chi_{(m-1)\Delta+}) + 3\nu) -
\eps^{-2}(\delta'_m+\delta'_{m-1}) \right),
\end{eqnarray*}
if $\eps$ is small enough. Indeed, $\nu$ being fixed, one can
choose $\eps$ so that
$$
1-\exp(- C_{m-1} \Delta\eps^{-2}) \ge \exp(-1) \ge
\exp(-\nu(\Delta\eps^{-2})).
$$


\item By ``backward'' induction from $k=m$ to $k=1$, choosing at
each step $\delta'_{k-1}$ and $z_{k-1}$ small enough in compare to
$\delta'_{k}-\delta'_{k-1}$,
\begin{equation}\label{modulus3}
m_{\nabla H}(\delta'_{k-1}+z_{k-1}) + \frac{\kappa}{2}
\delta'_{k-1}\le \frac{\kappa}{2}\delta'_k, \;\; \& \;\;
\delta'_{k-1}\le\delta'_k/2, \;\; \& \;\; m_H(\delta'_{k-1})<\nu
\end{equation}
(cf. (\ref{modulus2})), as well as all auxiliary values $C_{k-1}$,
we get the for $\varepsilon$ small enough the desired lower bound:
\begin{eqnarray*}
&P(| X^{\psi}_{\Delta m}-\varphi_{\Delta m}|<\delta'_m,\ldots,
| X^{\psi}_{\Delta}-\varphi_{\Delta}|<\delta'_1)
 \\ \\
& \ge\exp\left(-\eps^{-2}\Delta \sum^m_{i=1}
(L(\psi_{(m-i)\Delta}, \dot\chi_{(m-i)\Delta+})+3\nu ) -
2\eps^{-2}\sum_{k=1}^{m}\delta'_k\right)
 \\ \\
&=\exp\left(-\eps^{-2}(\int_0^T L(\psi_s,\dot\chi_s)\,ds + 3\nu T)
- 4\eps^{-2}\delta'_m\right)
 \\\\
&\ge \exp \left(-\eps^{-2}(S_{0T}(\varphi ) + \nu
(3T+2))\right),\qquad\eps \to 0,
\end{eqnarray*}
provided $4\delta'_m<\nu$, and due to (\ref{s-s}). This is
equivalent to (\ref{eq5a}). This bound is uniform in $x\in E^d,\,
|y|\leq r$, and $\varphi\in\Phi_x(s)$ for any $r,s>0$ (similar to
the Lemma 7.4.1 from Freidlin and Wentzell (1984)).

For the reader's convenience we remind the order of all choices by
repeating the diagram in the third step of the proof. The (any)
constants $\delta>0$ and $\nu>0$ are fixed; due to the Lemma 5, we
have also $\Delta(\nu)$ and $\delta(\nu)$; all other constants are
chosen consequently as follows:
$$
b\,\&\, \tilde\delta' \mapsto \delta' \mapsto \Delta \,\&\,
\delta'' \mapsto \delta'_m \mapsto \delta'_{m-1} \mapsto z_{m-1}
\mapsto \nu\,'_{m-1} \ldots \mapsto \delta'_1 \mapsto z_1 \mapsto
\nu\,'_{1};
$$
one may say that all these values have been constructed via $\nu$.
Some other constants (namely, $C_k$'s) are constructed via
$\delta'_k$'s; the value $m$ is defined as soon as $\Delta$ is
chosen, as $\Delta = T/m$; and eventually $\eps$ should be small
enough to ensure all our asymptotic inequalities.

\item
The property of the rate function $S$ to be a ``good rate
function'' can be shown as in Freidlin and Wentzell (1984), using
the semi-continuity of the function$L(x,y)$ w.r.t. $y$ and
continuity w.r.t. $x$ variable (see Lemma 7.4.2 from Freidlin and
Wentzell).



\item \underline{\bf Second part of the proof:}  the
upper bound. Assume that the assertion is not true, that is, there
exist $s$ and $\nu>0$ with the following properties:
$$
\forall \bar\delta>0, \;
\mbox{there exists} \;\delta_0<\bar\delta, \; \forall
\bar\varepsilon, \; \mbox{there exists} \;
\varepsilon<\bar\varepsilon:
$$
$$
P(\rho(X,\Phi_x(s))>\delta_0) > \exp(-\varepsilon^{-2}(s - \nu)).
$$
In the other words, for some (hence, actually, for any)
$\delta_0>0$ arbitrarily close to zero, there exists a sequence
$\varepsilon_n\to 0$ such that
\begin{equation}\label{contra}%$$
P(\rho(X,\Phi_x(s))>\delta_0)
> \exp(-\varepsilon_n^{-2}(s - \nu)).
\end{equation}%$$
We fix any such $\delta_0>0$.


\item Since $f$ is bounded, all possible paths $X^\psi$ for any
$\psi$ belong to some compact $F\subset C[0,T;R^d]$. Due to
semi-continuity of the functional $S^\psi(\varphi)$ w.r.t. $\psi$,
for any $\nu>0$ there exists  a value $\delta_\nu(\varphi)
> 0$ such that $\rho(\varphi,\psi)<\delta_\nu(\varphi)$ and
$S(\varphi)>s$ imply $S^\psi(\varphi)>s-\nu/2$.

Since $S^\psi(\varphi)$ is lower semi-continuous w.r.t.
$\varphi\,$, too, it follows that $\delta_\nu(\varphi)$ is also
lower semi-continuous w.r.t. $\varphi$. Hence, it attains its
minimum on any compact.

Consider $F_1$ = the compact obtained from $F$ by dropping the
$\delta_0/2$-neighbourhood of the set $\Phi_x(s) = \{\varphi \in
C[0,T;R^d]: \, \varphi_0=x, \, S(\varphi)\le s\}$. Denote
$\bar\delta_\nu = \inf_{\varphi\in F_1} \delta_\nu(\varphi)$, and
take any $\delta' \le
\min\left(\bar\delta_\nu/(4KT+2),\delta_0/2\right)$ where $K$ is a
Lipschitz constant of $f$.

Choose a $\delta'$-net in $F_1$, let $\varphi^1, \ldots,
\varphi^N$ be its elements. All of them do not belong to
$\Phi_x(s)$, hence, $S(\varphi^i)\ge s'>s$.

\item
Note that
$$
\{\rho(X,\Phi_x(s))>\delta_0\} \subset \bigcup_{i=1}^N
\{\rho(X,\varphi^i)<\delta'\}.
$$


Then, by the Dirichlet principle, for any $n$ there exists an
index\footnote{Indeed, otherwise for any $i$,
$$
P(\rho(X,\varphi^i)\le \delta') \le
N^{-1}\exp(-\varepsilon_n^{-2}(s - \nu)),
$$
and
$$
P(\rho(X,\Phi_x(s))>\delta_0) \le P\left(\bigcup_{i=1}^N
\{\rho(X,\varphi^i)<\delta'\}\right) \le
\exp(-\varepsilon_n^{-2}(s - \nu)),
$$
which does contradict (\ref{contra}).} $i$ such that
\begin{equation}\label{5.14}
%P(\rho(X,\Phi_x(s))>\delta) \le \sum_{i=1}^{N_\delta}
P(\rho(X,\varphi^i)\le \delta') > N^{-1}\exp(-\varepsilon_n^{-2}(s
- \nu)).
\end{equation}



There is a finite number of $i = 1, \ldots, N$. Thus, there exists
at least one $i$ such that (\ref{5.14}) holds true for this $i$
for some subsequence $n'\to\infty$ and correspondingly
$\varepsilon_{n'}\to 0$; however, we will keep a notation $n$ for
simplicity.

We can rewrite (\ref{5.14}) as
\begin{equation}\label{5.14a}
P(\rho(X,\varphi^i)\le \delta') > \exp(-\varepsilon_n^{-2}(s -
\nu)),
\end{equation}
since $N$ does not depend on $\varepsilon_n$, strictly speaking
with a new $\nu>0$; however, it is convenient to keep the same
notation. Denote $\varphi^i =: \varphi(\delta')$.

\item Consider a sequence $\delta'\to 0$ such that a corresponding
function $\varphi(\delta')$ does exist for any $\delta'$ from this
sequence. Recall that $\delta_0$ is fixed. All these functions
satisfy the inequality
$$
S(\varphi(\delta'))\ge s'> s,
$$
since $\rho(\varphi^i,\Phi_x(s))\ge \delta_0/2$, and also
$$
S(\varphi(\delta'))<\infty,
$$
which implies
$$
\sup_t |\dot\varphi_t(\delta')|\le C.
$$
Due to the Arcela-Ascoli Theorem, it is possible to extract from
this set of functions  a subsequence which converges in
$C[0,T;R^d]$ to some limit, $\bar\varphi$. Since
$\rho(\varphi(\delta'),\Phi_x(s))\ge \delta_0/2$, we have,
$\rho(\bar\varphi,\Phi_x(s))\ge \delta_0/2$, hence,
$$
S(\bar\varphi) > s,
$$
and, in particular, the lower bound (\ref{eq5a}) can be applied.
However, due to the construction, the function $\bar\varphi$
satisfies one more lower bound,
\begin{equation}\label{better}%$$
\liminf_{\delta'\to 0} \limsup_{\varepsilon\to 0} \varepsilon^2\ln
P(\rho(X,\bar\varphi)<\delta') \ge - s+\nu.
\end{equation}%$$
Indeed, the latter follows from (\ref{5.14a}) because, e.g.,
$$
P\left(\rho(X,\bar\varphi)\le
\delta'+\rho(\bar\varphi,\varphi(\delta'))\right) \ge
P(\rho(X,\varphi(\delta'))\le \delta') >
\exp(-\varepsilon_n^{-2}(s - \nu)).
$$

Due to (\ref{better}), there exists $\hat\delta'>0$ such that for
smaller $\delta'$'s (a sequence)
$$
\limsup_{\varepsilon\to 0} \varepsilon^2\ln P(\rho(X,\bar
\varphi)<\delta') \ge - s+\nu/2.
$$
In fact, this implies the same inequality for any $\delta'>0$,
because with any $\delta'$ for which the inequality holds true,
each greater value would do as well. Therefore, for any $\delta'$,
there exists $\varepsilon>0$ (arbitrarily small) such that

\begin{equation}\label{ge}%$$
\varepsilon^2\ln P(\rho(X,\bar\varphi)<\delta') \ge - s +\nu/3 =
-(s -\nu/3).
\end{equation}%$$
We are going to show that this leads to a contradiction.

\item Consider the case $S(\bar\varphi)<\infty$. Remind that
$ S(\bar\varphi) > s. $ Denote
\begin{eqnarray*}%$$
L^b(x,y) = \sup_{|\beta|\le b}(\beta y - H(x,\beta)),
\\\\
\ell^b(x,y) := L(x,y) - L^b(x,y)
\\\\
\equiv \sup_{\beta}(\beta y - H(x,\beta)) - \sup_{|\beta|\le
b}(\beta y - H(x,\beta)).
\end{eqnarray*}
Consider the function $\ell^b(\bar\varphi_t,\dot{\bar\varphi}_t)$.
We have, $$0\le \ell^b(\bar\varphi_t,\dot{\bar\varphi}_t)\le
L(\bar\varphi_t,\dot{\bar\varphi}_t).
$$
Moreover,
$$
\ell^b(\bar\varphi_t,\dot{\bar\varphi}_t)\to 0, \quad b\to\infty,
$$
and the function $\ell$ is decreasing with $b\to\infty$. Hence,
given $\nu>0$, one can choose a $b>0$ such that
$$
\int_0^T \ell^b(\bar\varphi_t,\dot{\bar\varphi}_t)\,dt < \nu/20.
$$
Notice that we have chosen $b$. Moreover, one can also choose a
discretisation step $\Delta$ (see above, item \ref{9} of the
proof) such that
$$
\int_0^T
\ell^b(\bar\varphi_{\kappa_m(t+a)-a},\dot{\bar\varphi}_{\kappa_m(t+a)-a})\,dt
< \nu/10,
$$
and, correspondingly,
\begin{equation}\label{elb}
\int_0^T
L^b(\bar\varphi_{\kappa_m(t+a)-a},\dot{\bar\varphi}_{\kappa_m(t+a)-a})\,dt
> s - \nu/10;
\end{equation}
again assume for simplicity of presentation that $a=0$. In
addition, we require $\Delta\le \Delta(\nu/20)$ (this notation is
from the Lemma \ref{L-main}). Hence, we have chosen $\Delta$ and
$m=T/\Delta$.


\item So, with $a=0$, let
$$
\psi_t := \bar\varphi_{\kappa_m(t)}, \quad \dot\chi_t
:= \dot{\bar\varphi}_{\kappa_m(t)}, \quad \chi_0=x.
$$

We have, with a unique $C = 2(KT+1)$ (see (\ref{eqCa})) and for
any $\delta'$,
$$%\begin{equation}\label{eqC}
 P(\rho(X,\bar\varphi)<\delta') \le
 P(\rho(X^{\psi},\chi)<C\delta') \le
 P(\rho(X^{\psi,\Delta},\chi^\Delta)<C\delta').
$$%\end{equation}
Denote $\delta''=C\delta'$. Let us choose $\delta''\le
\delta(\nu/20)$ (the notation from the Lemma 5 is used), and
consider the following inequality, with the sequence $(\delta'_i,
\ 1\le i\le m)$, $\delta'_m = \delta''$, constructed via the value
$\nu/20$ instead of $\nu$ (compare to (\ref{modulus3}), where we
can now drop the requirement related to $m_{\nabla H}$),
$$%\begin{equation}\label{eqC}
 P(\rho(X,\bar\varphi)<\delta'_1) \le
 E \prod_{i=1}^{m} 1(|X^{\psi,\Delta}-\chi^\Delta| < \delta'_i).
$$%\end{equation}
In particular, we require $4\delta'' = 4\delta'_m \le \nu/20$, and
$\sum_{i=1}^{m}\delta'_i\le 2\delta''$. Then, due to the Lemma 5
and using the calculus as in the steps
(\ref{item11}-\ref{item12}), -- the only change is that now we
need an upper bound, and an estimation of the indicator function
can be skipped, -- we get on the set $\{|X^\psi_{(m-1)\Delta} -
\chi_{(m-1)\Delta}| < \delta'_{m-1}\}$ and for any $|\beta|\le b$,
\begin{eqnarray}\label{eq17a}
&E\left( I(|X^\psi_{m\Delta} -\chi_{m\Delta}| < \delta'_m) |
F_{(m-1)\Delta}\right) \nonumber
 \\ \nonumber \\
& \leq E^\beta \left( I(|X^\psi_{m\Delta} -\chi_{m\Delta}| <
\delta'_m) \exp\left( -\eps^{-2}\Delta \beta
\left((\chi_{m\Delta}-\chi_{(m-1)\Delta})/\Delta\right)
\nonumber \right. \right. \\ \nonumber \\
& \left. \left. - \eps^{-2}\Delta (H(\psi_{(m-1)\Delta}, \hat
\psi_{(m-1)\Delta},\beta) - \nu/20) +
\frac{\delta'_m+\delta'_{m-1}}{\varepsilon^{2}}\right) |
F_{(m-1)\Delta}\right)
\end{eqnarray}
(compare to (\ref{eq17})). Estimate here $I(|X^\psi_{m\Delta}
-\chi_{m\Delta}| < \delta'_m)\le 1$ and drop the expectation sign,
then on the set $\{|X^\psi_{(m-1)\Delta} - \chi_{(m-1)\Delta}| <
\delta'_{m-1}\}$, for any $|\beta|\le b$,
\begin{eqnarray}\label{eq17b}
&E\left(I(|X^\psi_{m\Delta} -\chi_{m\Delta}| < \delta'_m) |
F_{(m-1)\Delta}\right) \nonumber
 \\ \nonumber \\
& \leq \exp\left( -\eps^{-2}\Delta \beta
\left((\chi_{m\Delta}-\chi_{(m-1)\Delta})/\Delta\right)
\nonumber \right. \\ \nonumber \\
& \left. - \eps^{-2}\Delta (H(\psi_{(m-1)\Delta}, \hat
\psi_{(m-1)\Delta},\beta)) +
\frac{\delta'_m+\delta'_{m-1}}{\varepsilon^{2}}\right),
\nonumber \\ \nonumber \\
& \leq \exp\left( -\eps^{-2}\Delta \beta
\left((\chi_{m\Delta}-\chi_{(m-1)\Delta})/\Delta\right)
\nonumber \right. \\ \nonumber \\
& \left. - \eps^{-2}\Delta (H(\psi_{(m-1)\Delta},
\psi_{(m-1)\Delta},\beta) - \nu/20) +
\frac{\delta'_m+\delta'_{m-1}}{\varepsilon^{2}}\right),
\end{eqnarray}
once we have chosen $m_H(\delta'')\le \nu/20$ (remind that $m_H$
is the modulus of continuity of the function $H$ on the set
$|\beta|\le b$), because of the inequality $|\hat
\psi_{(m-1)\Delta} - \psi_{(m-1)\Delta}| \le \delta'_{m-1} \le
\delta''$.


Let $\beta$ satisfy a condition,
\begin{eqnarray*} &\beta
(\chi_{m\Delta}-\chi_{(m-1)\Delta})/\Delta - H(\psi_{(m-1)\Delta},
\psi_{(m-1)\Delta},\beta) \\\\
&= \sup_{|\beta|\le b}\left(\beta
(\chi_{m\Delta}-\chi_{(m-1)\Delta})/\Delta - H(\psi_{(m-1)\Delta},
\psi_{(m-1)\Delta},\beta)\right)\\\\ &=
L^b(\psi_{(m-1)\Delta},\dot\chi_{(m-1)\Delta +}).
\end{eqnarray*}
Then, on the set $\{|X^\psi_{(m-1)\Delta} - \chi_{(m-1)\Delta}| <
\delta'_{m-1}\}$,
\begin{eqnarray}\label{eq17c}
&E\left(I(|X^\psi_{m\Delta} -\chi_{m\Delta}| < \delta'_m) |
F_{(m-1)\Delta}\right) \nonumber
 \\ \nonumber \\
& \leq \exp\left( -\eps^{-2}\Delta
(L^b(\psi_{(m-1)\Delta},\dot\chi_{(m-1)\Delta +}) +\eps^{-2}\Delta
\frac{\nu}{20} +
\frac{\delta'_m+\delta'_{m-1}}{\varepsilon^{2}}\right).
\end{eqnarray}
Similarly, using induction and due to (\ref{elb}), we get
\begin{eqnarray}\label{eq17d}
&P(\rho(X,\chi)<\delta'_1) \nonumber
 \\ \nonumber \\
& \leq \exp \left(-\eps^{-2} \int\limits_0^T
L^b(\bar\varphi_{\kappa_m(t)},\dot{\bar\varphi}_{\kappa_m(t)})\,dt
+ \eps^{-2}\nu/20 + 4\varepsilon^{-2}\delta'_m \right) \nonumber
 \\ \nonumber \\
&\le \exp\left( -\eps^{-2}(s  - \nu/5)\right).
\end{eqnarray}
This evidently contradicts (\ref{ge}) where $\delta'$ may be
arbitrarily small.

\item Consider the case $\bar\varphi$ absolute continuous, and
$S(\bar\varphi)=\infty$. In this case, due to monotone convergence
$L^b\to L$, there exist $b>0$, $m$ and $a\in [0,T]$ such that
$$
\int_0^T L^b(\bar\varphi_t,\dot{\bar\varphi_t})\,dt \ge s -
\nu/20, \;  \int_0^T
L^b(\bar\varphi_{\kappa_m(t+a)-a},\dot{\bar\varphi}_{\kappa_m(t+a)-a})\,dt
\ge s - \nu/10.
$$
The rest is similar to the main case, $S(\bar\varphi)<\infty$, and
leads again to
$$
P(\rho(X,\bar\varphi)<\delta'_1)\le \exp\left( -\eps^{-2}(s  -
\nu/5)\right).
$$
This contradicts (\ref{ge}).

\item Consider the last possible case, $\bar\varphi$ not absolute
continuous. In this case, for any constant $c$, in particular, for
$c=\|f\|_C+1$, there exist two values $0\le t_1 < t_2\le T$, such
that $|\bar\varphi_{t_2}-\bar\varphi_{t_1}|>c(t_2-t_1)$; indeed,
otherwise $\bar\varphi$ must be Lipschitz with
$|\dot{\bar\varphi}|\le c$. Therefore, for $\delta<(t_2-t_1)/2$,
the probability $P(\rho(X,\bar\varphi)<\delta)$ necessarily equals
zero, because the event $\{\rho(X,\bar\varphi)<\delta\}$ is empty.
This evidently contradicts (\ref{ge}). In all possible cases, we
got to contradictions. Hence, the assumption is wrong, that is,
the upper bound (\ref{eq4a}) holds true. The Theorem is proved.

\end{enumerate}


~

%\vspace{1cm}

\pagebreak
\begin{center}{\bf APPENDIX}\end{center}

~

{\bf A. Comments on the Lemma \ref{L-FW}}. To explain that the
Lemma \ref{L-FW} is valid without additional assumptions, we have
to review very briefly its proof and show those assumptions.

Let $0 = t_0 < t_0 < \ldots < t_m = T$ be a partition,
$\gamma_k(\beta) := \int_{t_{k-1}}^{t_{k}} H(\varphi_s,\beta)ds$,
$\ell_k(\alpha) = \sup_\beta (\alpha\beta - \gamma_k(\beta))$,
$A_k = \{\alpha: \; \ell_k(\alpha) < \infty\}$, $A_k^\circ$ its
interior w.r.t. the linear hull $L_{A_{k}}$.

The inequality $S(\varphi) = \int_0^T L(\varphi_t,\dot\varphi_t)dt
< \infty$ implies
$$
\sum\limits_{k=1}^m \sup\limits_\beta
\left((\varphi_{t_{k}}-\varphi_{t_{k-1}}) - \gamma_k(\beta)\right)
= \sum\limits_{k=1}^m \ell_k(\varphi_{t_{k}}-\varphi_{t_{k-1}})
\leq S(\varphi).
$$
Under additional assumption $A^\circ_k \not= \emptyset$ it is
proved in Freidlin and Wentzell (1984) using the arguments from
Rockafellar (1970) that for any $\nu > 0$, there exists a function
$\tilde \varphi$ such that $\rho(\varphi,\tilde\varphi) < \nu$ and
there exist $\beta_k$ such that

\begin{equation}\label{eq-FW1}
\ell_k(\tilde\varphi_{t_{k}} - \tilde\varphi_{t_{k-1}}) =
(\tilde\varphi_{t_{k}} - \tilde\varphi_{t_{k-1}})\beta_k -
\gamma_k(\beta_k)
\end{equation}
and
\begin{equation}\label{eq-FW21}
\tilde\varphi_{t_{k}}-\tilde\varphi_{t_{k-1}} =
\nabla\gamma_k(\beta_k).
\end{equation}
The proof goes well if $A^\circ_k \not= \emptyset \; \forall k$.

Let us show that the same is true if $A^\circ_k = \emptyset$ for
some $k$'s. The property $A^\circ_k = \emptyset$ is equivalent to
$\dim L_{A_{k}} = 0$. In this case, $\gamma_k(\beta) = c_k \beta$
with some $c_k\in R^d$. Hence, $\ell_k(\alpha_k) < \infty$ means
that $\ell_k(\alpha_k) = 0$ and for any other $\alpha$,
$\ell_k(\alpha) = + \infty$ and $\gamma_k(\beta) = \alpha_k\beta$.
So, we have
$$
\ell_k(\varphi_{t_{k}}-\varphi_{t_{k-1}}) = 0 =
(\varphi_{t_{k}}-\varphi_{t_{k-1}})\beta - \gamma_k(\beta)
$$
for {\it any} $\beta$. Let $\beta_k = 0$. Evidently,
$$
\varphi_{t_{k}}-\varphi_{t_{k-1}} = \nabla\gamma_k(\beta_k).
$$
Hence, in the case $A^\circ_k = \emptyset$, one should not just
change the curve $\varphi_s$ on the interval $(t_{k-1},t_k)$; that
is,
(\ref{eq-FW1})%%??
and (\ref{eq-FW21}) are valid in this case also.

The rest of the proof is not changed. For any step function
$\zeta$, one defines a piecewise linear $\chi$ by the formula
$$
\chi_0=\varphi_0, \quad \dot\chi_s = \nabla_\beta
H(\zeta_s,\beta_k)), \; t_{k-1}<s<t_k, \; k=1,2,\ldots ,m.
$$
Then it is shown that $\zeta^n \to \varphi$ implies $\chi^n\to
\varphi$ due to the property that the convergence of smooth convex
functions to the limit implies the convergence of their gradients.
Then there exists a partition such that this construction gives
one
$$
\int_0^T L(\zeta_t,\dot\chi_t)\,dt \leq S(\varphi)+\nu.
$$
So, the lemma holds true without additional assumptions. The
assertions about $\hat\zeta$ and $\hat\beta_s$ can be shown
similarly.


{\bf B. Comments on the property $A^\circ_k\not=\emptyset$, and
characterization of the set ${\cal L}^\circ[f,x]$}. Denote the
interior of $A(x) = \{\alpha: \, L(x,\alpha)<\infty\}$ with
respect to its linear hull $L_{A(x)}$ by $A^\circ(x)$. Then
$A^\circ_k=\emptyset \Longleftrightarrow
A^\circ(\varphi_{t_{k-1}})=\emptyset$. In this section we show the
following equivalence:
$$
card (f\in R^d:\; f=f(x,y), y\in M) = 1 \Longleftrightarrow \dim
L_{A(x)}=0 \Longleftrightarrow A^\circ(x)=\emptyset.
$$

Since $A(x)$ is convex, clearly the first two conditions are
equivalent.

If $\{f(x,\cdot)\}$ contains only one point then $H(x,\beta)$ is
linear w.r.t. $\beta$; hence, $A(x)$ consists of a unique point
and $A^\circ(x) = \emptyset$.

Now, let $\{f(x,\cdot)\}$ contain at least two different points,
say, $f(x,y_1) \not= f(x,y_2)$. Then there exists $1\leq k \leq d$
such that $(f(x,y_1)-f(x,y_2))_k \not= 0$. Denote $M_k = \sup_y
f^k(x,y), \; m_k = \inf_y f^k(x,y)$. Let $0 < \nu <
(f(x,y_1)-f(x,y_2))_k/2$. Take two points $y'$ and $y''$ such that
$f^k(x,y') < m_k + \nu/5$ and $f^k(x,y'') > M_k - \nu/5$. There
exist two open sets $B'\subset M$ and $B''\subset M$ such that
$\sup_{y\in B'} f^k(x,y) < m_k + \nu/4$ and $\inf_{y\in B''}
f^k(x,y) > M_k - \nu/4$.

Since the process $y^x_t$ is a nondegenerate ergodic diffusion,
there exists $\lambda >0$ such that
$$
P(y^x_s \in B',\; 1\leq s\leq t) \geq \lambda ^{t-1}, \quad
P(y^x_s \in B'',\; 1\leq s\leq t) \geq \lambda ^{t-1}, \quad t\to
\infty.
$$

Let $\beta = z \beta_k$ where $\beta_k\in E^d$ is a $k$th unit
coordinate vector and $z \in R$. Then for $z > 0$ we have,
\begin{eqnarray*}
z^{-1}t^{-1}\log E\exp(z\beta_k \int_0^t f(x,y_s^x)\,ds)
 \\ \\
\geq z^{-1}t^{-1}\log E\exp(z\beta_k \int_0^t f(x,y_s^x)\,ds)
I(y^x_s \in B'',\; 1\leq s\leq t)
 \\ \\
\geq z^{-1}t^{-1}\log \{\exp(z(M_k-\nu/2)t) \lambda ^{t-1}\} \\\\=
M_k-\nu/4 + \frac{t-1}{t} z^{-1}\log \lambda  \geq \frac{t-1}{t}
M_k - \nu/2,
\end{eqnarray*}
if $z$ is large enough. In the other words, for large positive $z$
one has $H(x,z\beta_k) \geq z (M_k-2\nu)$. Similarly, for large
negative $z$
%
\begin{eqnarray*}
|z|^{-1}t^{-1}\log E\exp(z\beta_k \int_0^t f(x,y_s^x)\,ds)
 \\ \\
\geq |z|^{-1}t^{-1}\log E\exp(z\beta_k \int_0^t f(x,y_s^x)\,ds)
I(y^x_s \in B'',\; 1\leq s\leq t)
 \\ \\
\geq |z|^{-1}t^{-1}\log \{\exp(z(m_k+\nu/4)t) \lambda ^{t-1}\}
 \\ \\
= -(m_k+\nu/4) + \frac{t-1}{t}|z|^{-1}\log \lambda  \geq -
\frac{t-1}{t}m_k - \nu/2,
\end{eqnarray*}
if $|z|$ is large enough. In other words, for negative $z$ with
large absolute values one has $H(x,z\beta_k) \geq z (m_k+\nu)$.
Therefore, $\{\alpha: \; \alpha = \beta_k \theta, \, m_k+\nu <
\theta < M_k - \nu\} \, \subset \, A(x)$.

On the other hand, it is obvious that if $\alpha = \beta_k
\theta$, $\theta\in R^1$, with $\theta > M_k$ or $\theta < m_k$,
then $L(x,\alpha)=\infty$, because $m_k z \le H(x,\beta_k z) \le
M_k z$, and, hence (say, if $\theta > M_k$), for $z >> 1$,
$$
\beta_k \theta \beta_k z - H(x,\beta_k z) \ge  (\theta - M_k) z
\to +\infty, \quad z\to +\infty.
$$

A similar calculus and inequalities are valid for any unit vector
$\beta_0$. This shows, in particular, that $\dim L_A(x) = \dim
L_f(x)$, and, moreover, that $L_A(x) = L_f(x)$. Since $A(x)$ is
convex, it shows also that the interior $A^\circ(x)$ w.r.t.
$L_{A(x)}$ is not empty., except only the case dim$(L_{A(x)}) =
1$. Hence, the third condition is equivalent to the second one and
the first.

So, the condition $A^\circ_k \not= \emptyset$ is always satisfied
if the set $\{f(x,\cdot)\}$ for any $x$ consists of more than one
point. In fact, if $card \{f(x,\cdot)\}=1$ for {\it any} $\,x$
then $f$ does not depend on $y$. In this case, one has nothing to
average.

Notice that our considerations above provide the following
description of the set ${\cal L}^\circ[f,x]$:
\begin{eqnarray*}
&{\cal L}^\circ[f,x] = \{\alpha\in R^d: \, m_\beta(x) < \langle
\alpha,\beta\rangle <M_\beta(x), \; \forall %\beta\in R^d,
|\beta|=1, \; \mbox{with}\; m_\beta(x) < M_\beta(x), \\
&\mbox{and}\; \langle \alpha,\beta\rangle = M_\beta(x), \; \forall
|\beta|=1, \; \mbox{with}\; m_\beta(x) = M_\beta(x) \},
\end{eqnarray*}

\vspace{2mm}

\noindent where $m_\beta(x) := \inf\limits_y \langle
\frac{\beta}{|\beta|}, f(x,y)\rangle, \, M_\beta(x) :=
\sup\limits_y \langle \frac{\beta}{|\beta|}, f(x,y)\rangle$.
Moreover, it can be shown similarly that for any $x, \tilde x$
(although we do not need it here),
$$
{\cal L}^\circ[f,x,\tilde x] = {\cal L}^\circ[f,x].
$$




~

%%%%%%%%%%%%%%%%%%%%%%%%






%%%%%%%%%%%%%%%%%%%%%%%%

{\bf C. About $\hat\alpha_s \in {\cal L}^\circ[f,\varphi_s]$}. Let
$x=\varphi_s$, $\hat\alpha = \hat\alpha[x,\dot\chi]$ as described
in the proof of the theorem 1. If we show that for any direction
$v$ (a unit vector) satisfying the property $m_v < M_v$,
%$$\inf (vf(x,y),\, y\in R^d) =: m_v <
%M_v:= \sup (vf(x,y),\, y\in R^d),
%$$ %for any $\nu>0$
the strict double inequality holds true
$$
m_v  < \partial H(x,zv)/\partial z|_{z=0} < M_v,
$$
$z\in R^1$, then it would follow $\hat\alpha_s \in {\cal
L}^\circ[f,\varphi_s]$. Let $\nu>0$ and again two open sets $B'$
and $B''$ be chosen such that $\sup_{y\in B'} vf(x,y) < m_v +
\nu/2$, and $\inf_{y\in B''} vf(x,y) > M_v- \nu/2$. Let
$\mu_{inv}(B'')$ be invariant measure for the event $\{y^x_t\in
B''\}$. We can choose $\nu$ and correspondingly $B''$ so that
$\mu_{inv}(B'') < 1$. Then, due to large deviation asymptotics for
the process $y^x_t$, for any $\mu_{inv}(B'') < \zeta < 1$ there
exists $\lambda>0$ such that
$$
P\left(t^{-1}\int_0^t 1(y^x_s\in  B'')\,ds \ge \zeta\right) \le
\exp(-\lambda t), \quad t\ge t_{\zeta}.
$$
Denote $A_\zeta = \left\{t^{-1}\int_0^t 1(y^x_s\in  B'')\,ds
<\zeta\right\}$, $A^c_\zeta = \left\{t^{-1}\int_0^t 1(y^x_s\in
B'')\,ds \ge \zeta\right\}$, then for $z>0$,
\begin{eqnarray*}
E\exp(z v \int_0^t f(x,y^x)\,ds)
 \\ \\
\le E\exp(z \int_0^t
 \left(M_v 1(y^x_s\in B'') + (M_v-\nu) 1(y^x_s\not\in B'')\right)
 \,ds)\,1(A^c_\zeta)
 \\ \\
+E\exp(z \int_0^t
 \left(M_v 1(y^x_s\in B'') + (M_v-\nu) 1(y^x_s\not\in B'')\right)
 \,ds)\, 1(A_\zeta)
  \\ \\
\le E\exp(z t M_v + z t (M_v-\nu)) \,1(A^c_\zeta)
 %\\ \\
+E\exp(z t M_v \zeta  + zt(M_v-\nu))\, 1(A_\zeta)
 \\ \\
\le \exp(z t M_v + z t (M_v-\nu) -z\lambda t/z)
 %\\ \\
+\exp(z t M_v \zeta  + zt(M_v-\nu)),
\end{eqnarray*}
hence,
$$
\limsup_{z\to 0}\,\limsup_{t\to\infty}\, (tz)^{-1} E\exp(z v
\int_0^t f(x,y^x)\,ds) < M_v.
$$
Similarly, using $B'$ one can get
$$
\liminf_{z\to 0}\,\liminf_{t\to\infty} \,(tz)^{-1} E\exp(z v
\int_0^t f(x,y^x)\,ds) > m_v.
$$
Thus,
$$
m_v < \partial H(x,zv)/\partial z|_{z=0} < M_v.
$$
Therefore, $\hat\alpha \in {\cal L}^\circ[x,f]$.

~

%\pagebreak

\begin{center}{REREFERCES}\end{center}

~

Freidlin, M.I. (1976) Fluctuations in dynamical systems with
averaging. {\it Dok. Acad. Nauk SSSR} (Soviet Doklady Acad.
Sci.) {\bf 226} 273-276 (in Russian).  %Engl?

Freidlin, M.I. (1978) Averaging principle and large deviations.
{\it Uspekhi Matem. Nauk.} {\bf 33} 107-160
(in Russian).  %Engl.?

%\bibitem{FW}
Freidlin, M.I. and Wentzell, A.D. (1984) {\it Random perturbations
of dynamical systems.} N.Y., Springer.

%\bibitem{GV}
Gulinsky, O.V. and Veretennikov, A.Yu. (1993) {\it Large
Deviations for Discrete--Time Processes with Averaging.} VSP,
Utrecht.

Ikeda, N. and Watanabe, S. (1989) {\it Stochastic differential
equations and diffusion processes.} 2nd ed. North-Holland,
Amsterdam.


Kato, T. (1976) {\it Perturbation Theory for Linear Operators.}
2nd ed. Springer, New York.

%Krylov, N.V., Controlled Diffusion Processes, Nauka, Moscow, 1977
%(in Russian).


%\bibitem{KLS}
Krasnosel'skii, M.A., Lifshitz, E.A. and Sobolev, A.V. (1989) {\it
Positive linear systems.} Helderman, Berlin.

%\bibitem{Roc}
Rockafellar, R.T. (1970) {\it Convex analysis.} Princeton Univ.
Press.

%\bibitem{V92}
Veretennikov, A.Yu. (1992) On large deviations in the averaging
principle for stochastic differential equations with periodic
coefficients {\bf 2}. {\it Math. USSR Izvestiya,} {\bf 39}
677-701.

%\bibitem{Ve-Sto}
Veretennikov, A.Yu. (1994) Large deviations in averaging principle
for stochastic differential equation systems (noncompact case).
{\it Stochastics Stochastics Rep.} {\bf 48} 83-96.

Veretennikov, A.Yu. (1999) On large deviations in the averaging
principle for SDE's with a ``full dependence'', {\it Ann. Probab.}
{\bf 27} no.~1, 284--296.
%%; corrected and expanded version Dec. 2004. %, due to a
%request by Yuri Kifer

%Watanabe, S. and Ikeda, N. (1989) {\it Stochastic differential
%equations and diffusion processes.} 2nd ed. North-Holland,
%Amsterdam.


\end{document}

math0502098.extracted_bib
\bibitem{FW}
Freidlin, M.I. and Wentzell, A.D. (1984) {\it Random perturbations
of dynamical systems.} N.Y., Springer.

%
\bibitem{GV}
Gulinsky, O.V. and Veretennikov, A.Yu. (1993) {\it Large
Deviations for Discrete--Time Processes with Averaging.} VSP,
Utrecht.

Ikeda, N. and Watanabe, S. (1989) {\it Stochastic differential
equations and diffusion processes.} 2nd ed. North-Holland,
Amsterdam.


Kato, T. (1976) {\it Perturbation Theory for Linear Operators.}
2nd ed. Springer, New York.

%Krylov, N.V., Controlled Diffusion Processes, Nauka, Moscow, 1977
%(in Russian).


%
\bibitem{KLS}
Krasnosel'skii, M.A., Lifshitz, E.A. and Sobolev, A.V. (1989) {\it
Positive linear systems.} Helderman, Berlin.

%
\bibitem{Roc}
Rockafellar, R.T. (1970) {\it Convex analysis.} Princeton Univ.
Press.

%
\bibitem{V92}
Veretennikov, A.Yu. (1992) On large deviations in the averaging
principle for stochastic differential equations with periodic
coefficients {\bf 2}. {\it Math. USSR Izvestiya,} {\bf 39}
677-701.

%
\bibitem{Ve-Sto}
Veretennikov, A.Yu. (1994) Large deviations in averaging principle
for stochastic differential equation systems (noncompact case).
{\it Stochastics Stochastics Rep.} {\bf 48} 83-96.

Veretennikov, A.Yu. (1999) On large deviations in the averaging
principle for SDE's with a ``full dependence'', {\it Ann. Probab.}
{\bf 27} no.~1, 284--296.
%%; corrected and expanded version Dec. 2004. %, due to a
%request by Yuri Kifer

%Watanabe, S. and Ikeda, N. (1989) {\it Stochastic differential
%equations and diffusion processes.} 2nd ed. North-Holland,
%Amsterdam.


