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\document
\centerline{\bigbf Couplings and asymptotic exponentiality of}
\centerline{\bigbf exit times}
\vglue .3in
\centerline{S. Brassesco\footnote{Instituto Venezolano de Investigaciones Cientificas, Caracas, Venezuela},
E. Olivieri\footnote{II Universit\`a di Roma
``Tor Vergata'', Roma, Italy}, M.E. Vares\footnote{IMPA, Rio de Janeiro, Brasil.}
\footnote{Partially supported by FINEP (Pronex project)}}
\vglue .5in
\topmatter
\abstract{The goal of this note is simply to call attention to the resulting simplification in the proof of asymptotic exponentiality of exit times in
Freidlin-Wentzell regime (as proved in [MOS]) by using
the coupling proposed by T. Lindvall and C. Rogers (cf. [LR]).}
\endabstract
\endtopmatter
\flushpar
{\bf Key words:} Exit times, Exponentiality, Metastability, Couplings.
\vglue .5in
\openup 4pt
\flushpar
{\bf Introduction.}
\bigskip
In this note we examine a classical problem in the
framework of the theory of small random
perturbations of dynamical systems: the first exit from
a domain $G$ positively invariant with
respect to the unperturbed flow.
\par
In particular, for a class of It\^o equations,
we address the question of the asymptotic
exponentiality, in the limit of small noise,
of the suitably normalized first exit time from $G$.
We are interested in the general case of $G$
containing many attractors of the unperturbed system.
This problem is, on one side, interesting in itself;
it amounts to considerably strengthen the
classical
Freidlin-Wentzell results on the asymptotics of the first
exit time from a domain $G$. It is, on
the other side, also related to the so-called metastable
behavior of the particular stochastic
dynamics described by our It\^o equations,
in the framework of the pathwise approach to
metastability introduced in [CGOV].
From a probabilistic point of view the asymptotic
exponentiality (or asymptotic
unpredictability) of the exit time
is related to a particular exit mechanism: the
repetition of a large number of almost independent
trials. Among the various different ways the
large deviation theory is able to select
a particularly efficient one.
So the heuristic explanation of the asymptotic
exponentiality is based on a long sequence of
recurrences inside $G$ together with a loss of
memory and eventually a successful exit
attempt.
In [GOV], very sophisticated analytical results
due to Day (cf.[D]) were used to extend to a ``tunneling"
problem the previous results relative to the case of
a domain $G$ completely attracted by a unique
asymptotically stable point. In [MOS], for a
general class of domains $G$, the analytical
methods of Day were replaced by probabilistic
arguments based on contraction properties of the stochastic
map (depending on the noise) which
associates to the initial datum of our stochastic
equation the solution at a given time $T$.
Again the ingredient of loss of memory,
necessary for the asymptotic exponentiality is based on
delicate and highly non-trivial arguments developed in [MS].
\par
In the present note, in the general [MOS] context,
we give another proof of the asymptotic exponentiality by using a
simple and beautiful coupling argument due to Lindvall
and Rogers (cf. [LR]). The goal is to stress the resulting simplicity.
Coupling methods have been also successfully used to show asymptotic exponentiality for
an infinite dimensional case, as the stochastically perturbed non-linear heat equation,
also considered in [MOS]. Using a coupling introduced by
Mueller in [M], Brassesco (cf. [B]) was able to treat escape times which were not treatable with the
techniques considered in [MOS].
\vglue .5in
{\bf The result.}
Let $X_t^{x,\ve}$ be the Markov process obtained as the unique solution of the following It\^o equation:
$$
\aligned
dX_t^{x,\ve} &= b(X_t^{x,\ve})dt + \ve\,dW_t\\
X_0^{x,\ve} & = x
\endaligned \tag 1
$$
where $(W_t)$ is a standard $d$-dimensional
Brownian motion, $x \in \re^d$, $\ve > 0$, and the vector
field $b$ is assumed to be globally Lipschitz.
Let us in fact, and to simplify, assume $b$ to
be of class $C^1$ with bounded gradient.
In particular, as it is well known, this
implies strong uniqueness of the solution of (1),
for any given Brownian motion $(W_t)$, as well
as the strong Markov property for $(X_t^{x,\ve})\_t$\,.
Of course, more general assumptions on the field $b$ can be taken,
and an extension to varying diffusion coefficients is also possible,
cf. Remark 4.
\vglue .2in
\noindent{\bf Notation}: Though everything is done on any probability space $(\Om,\A,P)$ where $(W_t)$ is defined, through a pathwise (and continuous) transformation, sometimes it is more convenient to relax the notation,
eliminating the superscript $x$ on $X^{x,\ve}$
and using $P_x$ to denote the condition $X_0^\ve = x$.
Our goal is to discuss the asymptotic behavior, as $\ve \to 0$,
of the first exit time $\tau^\ve \overset{\text{def}}\to{=} \inf\{t > 0; X_t^\ve \notin G\}$, when $X_0^\ve = x \in G$, and where $G$
is a bounded domain verifying certain conditions.
A possible set of assumptions would be, similarly to [MS] and [MOS]:
\item{(a1)} $G$ is a bounded domain in $\re^d$,
with a smooth boundary $\po G$, taken as of class $C^2$.
If $I_{0,T}(\vr)$ denotes the rate functional
$$
I_{0,T}(\vr) =
\left\{
\alignedat{2}
&\frac 12 \int_0^T |\dot\vr_t - b(\vr_t)|^2\,dt &&\quad\text{if $\vr$ is absolutely continuous}\\
&+\infty &&\quad\text{otherwise}
\endalignedat \right.\tag 2
$$
defined on the space $C([0,T], \re^d)$ and corresponding
to the large deviation principle associated to the family of
laws of $\big(X_t^{x,\ve}\big)\_{0\le t \le T}$ on
this space, $V(x,y)$ is the associated quasi potential
of Freidlin and Wentzell:
$$
V(x,y) = \inf_{\Sb \vr\colon \vr(0)=x, \vr(T)=y\\ T > 0\endSb}
\{I_{0,T}(\vr)\} \tag 3
$$
and one considers the equivalence relation
$$
x \sim y \quad\text{iff}\quad V(x,y) = V(y,x) = 0 \tag 4
$$
then one assumes:
\item{(a2)} There are finitely many compact
sets $K_1,\dots,K_m$\,, equivalence classes for \,\,$\sim$\,, and such that:
\itemitem{(i)} each $w$-limit set of the deterministic system given by $\dot x(t) = b(x(t))$ is contained in some $K_i$\,.
\itemitem{(ii)} The stable classes are $K_1,\dots,K_\ell$ \,\,$(\ell < m)$ and each of them consist of a fixed
point of the deterministic system. These are denoted by $x_i$\,, $i=1,\dots,\ell$.
Here the notion of a ``stable''
class is that coming from Freidlin and Wentzell
theory:
\vglue .2in
\noindent{\bf Definition 1}: An equivalence class $K$ is said to be stable if $V(x,y) > 0$ for all $x \in K$, all $y \notin K$.
\vglue .1in
We know that $V(x,y)$ is constant for all $x \in K_i$, all $y \in K_j$\,.
Let $V_{i,j}$ denote this constant, so that $K_i$ is stable iff
$$
\inf_{j\ne i} V_{i,j} > 0.
$$
Let $1\le k \le \ell$ be such that $\{x_1,\dots,x_\ell\} \cap G = \{x_1,\dots,x_k\}$
and let $\delta > 0$ such that $B_\delta(x_i)$ is contained in the basin of
attraction of $x_i$ as well as in $G$, for
$i=1,\dots,k$. Let
$$
D_i = B_\delta(x_i)\quad D = \bigcup_{i=1}^k D_i\,.
$$
We assume further
\item{(a3)} Among $x_1,\dots,x_k$ at least one of them is a hyperbolic fixed point i.e. there exists $i_0 \in \{1,\dots,k\}$ such all the eigenvalues of the Jacobian matrix $\bigg(\dfrac{\po b_r}{\po x^s}\bigg)_{r,s}\bigg|_{x=x_{i_0}}$ have negative real
part.
The last assumption concerns the ``cycle'' property:
\item{(a4)} Let $V = \dsize\max_{i,j\le k} V_{i,j}$ and
$V_G = \dsize\min_{1\le i \le k} \, \dsize\min_{y \in \po G}\, V(x_i,y)$.
We assume that $V_G > V$.
We may now state
\vglue .2in
\noindent{\bf Theorem 1}. Under above assumptions, and if we define $\be_\ve$ through the relation
$$
\sup_{x\in D} P_x(\tau^\ve > \be_\ve) = e^{-1} \tag 5
$$
then:
i)\quad $\dsize\lim_{\ve\to0} P_x(\tau^\ve > t\be_\ve) = e^{-t}$,
\vglue .1in
\noindent for each $x \in D$, each $t > 0$
ii)\quad $\dsize\lim_{\ve\to0} \dfrac{E_x(\tau^\ve)}{\be_\ve} = 1$, \quad $\forall\, x \in D$.
\vglue .2in
\noindent{\bf Remark 1}. If $G$ is confining, i.e., $\lg b(x), n(x)\rg < 0$ for each $x \in \po G$, where $n(x)$ indicates the outward unit normal vector to $\po G$, at the point $x$, then we may take any $x \in G$ in (i) and (ii) of Theorem 1.
\vglue .2in
\noindent{\bf Remark 2}. Contrarily to what happens in the case of a domain contained in the basin of attraction of a single fixed point or a periodic orbit, we do not always have asymptotic equivalence (even logarithmically) between a quantile of the d
istribution of $\tau^\ve$ under
$P_x$\,\, $(x \in D)$ and $E_x\,\tau^\ve$. For a counterexample see eg. [FW] pg. 197.
\par
Nevertheless, if $\be_\ve$ is defined through equation (5), as observed in [MOS], the bound
$$
\sup_{x \in D} \frac{E_x\,\tau^\ve}{\be_\ve} \le C < +\infty \tag 6
$$
for some finite constant $C$, holds independently of (i) of Theorem 1.
Moreover, from equation (6) and the known results of Freidlin
and Wentzell on the asymptotic behaviour of
$\ve^2\,\log\,E_x\,\tau_\ve$\,, we get
$$
\varliminf_{\ve\to0} \ve^2\,\log\,\be_\ve \ge V_G\,. \tag 7a
$$
On the other side, and this is the reason for the name ``cycle'',
if $x \in D_i$ one has
$$
\varlimsup_{\ve\to0} \ve^2\,\log\,E_x\,\tau^\ve(D_j) \le V. \tag 7b
$$
For convenience of the reader let us recall the verification of equation (6), as in [MOS]:
$$
\align
\frac{E_x\tau^\ve}{\be_\ve} &= \frac{1}{\be_\ve} \int_0^{+\infty}
P_x(\tau^\ve > t)\,dt \tag 8\\
&= \int_0^{+\infty} P_x(\tau^\ve > t\be_\ve)\,dt\\
&\le \int_0^{+\infty} g_\ve(t)\,dt
\endalign
$$
where $g_\ve(t) \overset{\text{def}}\to{=} \dsize\sup_{x\in G} P_x(\tau^\ve > t\be_\ve)$. But the Markov property implies that
$$
g_\ve(t+s) \le g_\ve(t)g_\ve(s)
$$
so that
$$
g_\ve(2k) \le (g_\ve(2))^k.
$$
As in [MOS] we can see that $g_\ve(2) \le r < 1$ for $\ve$ small,
and so we get (6).
In fact,
$$
\aligned
g_\ve(2) &\le \sup_{x\in G} P_x(\tau_\ve(D) > \be_\ve)\\
&+ \sup_{x \in D} P_x(\tau^\ve > \be_\ve)
\endaligned \tag 9
$$
the second term on the r.h.s. of equation (9) is $e^{-1}$,
and using Freidlin and Wentzell estimates we see that the
first term goes to zero, so that we get the claimed upper bound.
\vglue .2in
\noindent{\bf Remark 3}. The argument just described allows
also to make use of the Dominated Convergence Theorem
and, from equation (8), to get (ii) of Theorem 1, once part (i) is proved.
Moreover, and as in [GOV], for the proof of part (i) in Theorem 1 in the case $x=x_{i_0}$\,, it suffices to prove the following:
\vglue .2in
\noindent{\bf Lemma 1}. Under the assumptions of Theorem 1, with $\be_\ve > 0$ given by equation (5) and letting
$$
f_\ve(t) = P_{x_{i_0}}(\tau^\ve > t \be_\ve)
$$
for $t > 0$, $\ve > 0$, then there exist positive numbers $\delta_\ve$\,,
which tend to zero as $\ve \to 0$, and such that for each $s,t > 0$:
$$
f_\ve(s+\delta_\ve)f_\ve(t+\delta_\ve)-o_t(1) \le f_\ve(t+s) \le f_\ve(s)f_\ve(t-\delta_\ve) + o_t(1), \tag 10
$$
where $o_t(1)$ is a function of $t$ and $\ve$, which tends to zero as
$\ve \to 0$, uniformly on $t \ge t_0$\,, for any given $t_0 > 0$.
\
The proof of Lemma 1, as presented below, is similar to that of
Lemma 4 in [GOV] with assumption (a4) and the Freidlin and Wentzell theory being used to control the time needed to arrive to a suitably small neighborhood of $x_{i_0}$\,, and using the coup
ling method proposed by Lindvall and Rogers (cf. [LR] sections 2 and 3) to ensure the loss of memory.
For this, let $\delta_0 > 0$ be taken so that if $x, x'\in B_{2\delta_0}(x_{i_0})$ are distinct then
$$
\lg x-x', b(x)-b(x')\rg < 0. \tag 11
$$
($\langle\,\, \cdot,\cdot\,\,\rangle$ denotes the euclidean scalar product.)
To achieve this we need to recall assumption (a3) and the fact that $b(\,\cdot\,)$ is assumed of class $C^1$.
The coupling proposed in [LR] may thus be used to replace
the analytical results of [Day] used in [GOV], or the exponential joining
proposed by [MS], and used in [MOS], and allows us to write the following
\vglue .2in
\noindent{\bf Lemma 2}. If $x \in B_{\delta_0}(x_{i_0})$ then
$$
P_x(\tau^\ve> t\be_\ve) - P_{x_{i_0}}(\tau^\ve > t\be_\ve)
\to 0\qquad \text{as } \ve \to 0 \tag 12
$$
uniformly on $t \ge t_0$\,, for any given $t_0 > 0$.
\vglue .1in
\noindent{\bf Proof}. For the proof of (12) it suffices
to present a coupling of the two processes
$X_t^{x,\ve}$ and $X_t^{x_{i_0,\ve}}$ in such a
way that with probability tending to one they will meet before leaving $B_{2\delta_0}(x_{i_0})$, and this in time of order shorter
than $\be_\ve$\,.
In order to do so, we consider the coupling proposed by Lindvall
and Rogers (sections 2 and 3 of [LR]), which is particularly
simple in the case of constant diffusion coefficient
(example 5 of [LR]).
The processes $X_t^{x,\ve}$ and $X_t^{x_{i_0},\ve}$ are
constructed using the same noise, as follows:
take $X_t^{x,\ve}$ and $X_t^{x_{i_0},\ve}$ as solutions of the
It\^o equations
$$
\align
dX_t^{x,\ve} &= b(X_t^{x,\ve})dt + \ve\,dW_t;
\qquad X_0^{x,\ve} = x\\
dX_t^{x_{i_0},\ve} &= b(X_t^{x_{i_0},\ve})dt +
\ve H(X_t^{x,\ve},X_t^{x_{i_0},\ve}) \,dW_t;
\qquad X_0^{x_{i_0},\ve} =x_{i_0},\tag 13
\endalign
$$
where $W_t$ is a standard $d$-dimensional Brownian motion and
$H(x,y)$ is the $d\times d$ orthogonal matrix with determinant $-1$ given by
$$
H(x,y)=\Bbb I-2\,\frac{(x-y)}{|x-y|}\Big[\frac{(x-y)}{|x-y|}\Big]^T.
\tag 14
$$
(We are using $^T$ for transposition, and $\Bbb I$ to
denote the identity $d\times d$ matrix .)
The geometric idea behind this construction is clear:
Consider $x\neq y\in \Bbb R^d$. From (14), we have
$$
H(x,y)\big(\frac{x-y}{|x-y|}\big)= -\big(\frac{x-y}{|x-y|}\big),\tag 15
$$
and, acting on vectors that belong to the plane orthogonal to $x-y$,
$H(x,y)$ is just the identity. Thus, $H(x,y)$ is simply the specular
reflexion
through the plane (by the origin) orthogonal to
the vector $x-y$, and has determinant $-1$. Then, given
$z\in \Bbb R ^d$, $z=x+b$, consider $z'=H(x,y)b+y$.
Then, $z'$ is the reflexion of $z$ by the plane orthogonal
to $x-y$, that passes by the middle point between $x$ and $y$.
In particular, if
$Z_t$ is a $d$-dimensional Brownian motion starting at $x$,
then $Z'_t$ as obtained
by the above described reflexion (for each point in the path),
is a $d$-dimensional Brownian motion starting at $y$.
Then, the processes $X_t^{x,\ve}$ and $X_t^{x_{i_0,\ve}}$ are both solutions
of our original It\^o equation, and
if one considers the function $g:\Bbb R ^{2d}\to \Bbb R$,
$g(x,y)=|x-y|$, then, It\^o's formula (which is valid as long as
$|x-y|>0$), yields for the one-dimensional process $Y_t$, given by
$$
Y_t=|X_t^{x,\ve}-X_t^{x_{i_0},\ve}|:
$$
$$
\align
dY_t&=\langle b(X_t^{x,\ve})- b(X_t^{x_{i_0},\ve}),
\frac{(X_t^{x,\ve}-X_t^{x_{i_0},\ve})}{|X_t^{x,\ve}-X_t^{x_{i_0},\ve}|}
\rangle \,dt\\&+
\ve \langle
[Id-H(X_t^{x,\ve},X_t^{x_{i_0},\ve})]
\frac{(X_t^{x,\ve}-X_t^{x_{i_0},\ve})}{|X_t^{x,\ve}-X_t^{x_{i_0},\ve}|}
,\,dW_t\rangle\,; \qquad
Y_0=|x-x_{i_0}| \tag 16
\endalign
$$
Recall that from (11) it follows that the drift part in
(16) is negative, as long as
$X_t^{x,\ve} $ and $X_t^{x_{i_0},\ve} $ remain in
$B_{2\delta_0}(x_{i_0})$. From (15),
$$
\langle
[Id-H(X_t^{x,\ve},X_t^{x_{i_0},\ve})]
\frac{X_t^{x,\ve}-X_t^{x_{i_0},\ve}}{|X_t^{x,\ve}-X_t^{x_{i_0},\ve}|}
,dW_t\rangle=2\langle
\frac{X_t^{x,\ve}-X_t^{x_{i_0},\ve}}{|X_t^{x,\ve}-X_t^{x_{i_0},\ve}|}
,dW_t\rangle
$$
which implies that the process $Y_t$ satisfies
$$
Y_t=Y_0+\int^t_0 \langle b(X_s^{x,\ve})- b(X_s^{x_{i_0},\ve}),
\frac{X_s^{x,\ve}-X_s^{x_{i_0},\ve}}{|X_s^{x,\ve}-X_s^{x_{i_0},\ve}|}
\rangle\, ds+2\ve B_t,
$$
for $B_t$ a standard one dimensional Brownian motion.
Next, let $S^{\ve}$ be the coupling time, $T^{\ve}(y)$ the exit time from $B_{2\delta_0}(x_{i_0})$
of the solution $X_t^{y,\ve}$ and $\tilde S^{\ve}$
the time it takes for $Y(0)+2\ve B_t$ to hit zero:
$$
\align
S^{\ve}&=\inf\{t\ge 0:|Y_t|=0\} \\
T^{\ve}(y)&=\inf\{t\ge 0\ :X_t^{y,\ve}\notin B_{2\delta_0}(x_{i_0}) \}\\
\tilde S^{\ve}&=\inf\{t\ge 0:Y_0+2\ve B_t=0\}
\endalign
$$
Now, from (11) and the above remarks ,
$$
\align
P\Big(S^{\ve}<\ve^{-3}\Big)&\ge P\Big(S^{\ve}<\ve^{-3}, T^{\ve}(x)\land T^{\ve}(x_{i_0})>\ve^{-4}\Big)\\
&\ge P\Big(\tilde S^{\ve}<\ve^{-3},T^{\ve}(x)\land T^{\ve}(x_{i_0})>\ve^{-4}\Big)\\
&\ge P\Big(\tilde S^{\ve}<\ve^{-3}\Big)-P \Big(T^{\ve}(x)\land T^{\ve}(x_{i_0})\le\ve^{-4}\Big),\tag 17
\endalign
$$
where we denoted by $t\land s$ the minumum between $t$ and $s$.
But, from the Freidlin and Wentzell theory we know
that there exists
$a,b> 0$ so that
$$
P \Big(T^{\ve}(x)\land T^{\ve}(x_{i_0})\le\ve^{-4}\Big)\le 2
\sup_{y\in B_{\delta_0}(x_{i_0})}
P\Big(T^{\ve}(y)\le \exp^{-a/\ve^2}\Big)
\le\exp^{-b/\ve^2} \tag 18
$$
For the other term, we have
$$
P\Big(\tilde S<\ve^{-3}\Big)=
1-\int\bold 1_{\{|x|\le Y_0\ve^{1/2}\}}
\frac{\exp^{-x^2/2}}{\sqrt{2\pi}}\ge 1-\delta_{0}\ve^{1/2}, \tag 19
$$
>From (17), (18) and (19), it follows that
$P(S^{\ve}<\ve ^{-3})\to 1 $ as $\ve\to 0$, which implies Lemma 2 from (7a).
\vglue .1in
\noindent{\bf Proof of Lemma 1}. As in [GOV], the point
is to show the existence of $\eta_\ve > 0$ such that
$\eta_\ve/\be_\ve \to 0$ and such that
$$
\lim_{\ve\to0} \sup_{x\in G} P_x(\tau^\ve > \eta_\ve, \tau_\ve(B_{\delta_0}(x_{i_0})) > \eta_\ve) = 0. \tag 20
$$
To verify (20) let us take
$$
V < \al < V_G
$$
and let $\eta_\ve = e^{\al/\ve^2}$.
Since $\al > 0$, $G\backslash D$ is bounded, and all stable classes are contained in $D$, from Freidlin and Wentzell theory we know that
$$
\lim_{\ve\to0} \sup_{x\in G\backslash D} P_x(\tau_\ve(G^c \cup D) > \eta_\ve) = 0.
$$
On the other side, by assumption (a4) and since $V < \al$,
Freidlin and Wentzell theory implies that
$$
\sup_{x\in D} P_x(\tau_\ve(B_{\delta_0}(x_{i_0})) > \eta_\ve/2) \to 0.
$$
Thus we get:
$$
\align
&\,\,\,\sup_{x\in G} P_x(\tau^\ve > \eta_\ve, \tau_\ve(B_{\delta_0}(x_{i_0})) > \eta_\ve)\\
&\le \sup_{x\in G} P_x(\tau^\ve > \eta_\ve, \tau_\ve(D) > \eta_\ve/2)\\
&+ \sup_{x\in D} P_x(\tau_\ve(B_{\delta_0}(x_{i_0})) > \eta_\ve/2)
\endalign
$$
which both tend to zero, yielding equation (20).
To complete the proof of Lemma 1, we proceed as in [GOV]: Let $s > 0$ and
$$
R^s = \inf\{u > s\beta_\ve: X_u^\ve \in B_{\delta_0}(x_{i_0})\}
$$
then
$$
\sup_{x\in G} P_x(\tau^\ve > s\be_\ve+\eta_\ve,\,\, R^s > s\be_\ve + \eta_\ve) \le \sup_{x\in G} P_x(\tau^\ve > \eta_\ve, \tau_\ve(B_{\delta_0}(x_{i_0})) > \eta_\ve)
$$
which tends to zero as $\ve \to 0$.
Due to equation (6) (cf. Remark 2) and the choice of $\al$,\,\, $\eta_\ve/\be_\ve \to 0$, and we have that uniformly on $(s,t) \in [0,+\infty) \times [t_0,+\infty)$ for any given $t_0 > 0$:
$$
\sup_{x\in G} P_x(\tau^\ve > (s+t)\be_\ve, R^s \ge s\be_\ve + \eta_\ve) \to 0. \tag 21
$$
But, as in equation (2.16) of [GOV]:
$$
\aligned
P_{x_{i_0}}&
(\tau^\ve > (s+t)\be_\ve, R^s \le s\be_\ve + \eta_\ve)\\
&\le P_{x_{i_0}}(\tau^\ve > s\be_\ve) \sup_{y\in B_{\delta_0}(x_{i_0})} P_y(\tau^\ve > t\be_\ve - \eta_\ve)
\endaligned \tag 22
$$
and
$$
\aligned
P_{x_{i_0}}&(\tau^\ve > (s+t)\be_\ve, R^s \le s\be_\ve + \eta_\ve)\\
&\ge P_{x_{i_0}}(\tau^\ve > s\be_\ve + \eta_\ve, R^s \le s\be_\ve + \eta_\ve)
\inf_{y\in B_{\delta_0}(x_{i_0})} P_y(\tau^\ve > t\be_\ve)
\endaligned \tag 23
$$
so that Lemma 1 follows easily from equations (21)-(23), and Lemma 2.
\vglue .2in
\noindent{\bf Proof of Theorem 1}.
As already noticed it suffices us to prove part (i).
Also if $x = x_{i_0}$, (i) follows at once from Lemma 1. Using Lemma 2 we extend to any $x \in B_{\delta_0}(x_{i_0})$. To conclude we need to recall, as in equation (7b)
$$
\varlimsup_{\ve\to0} \ve^2\,\ell n\,E_x\,\tau_\ve(B_{\delta_0}(x_{i_0})) \le V
$$
so that if $V < \al < V_G$ and $\eta_\ve = e^{\al/\ve^2}$ then
$$
P_x(\tau_\ve(B_{\delta_0}(x_{i_0})) < \eta_\ve) \to 1.
$$
Using the strong Markov property at $\tau_\ve(B_{\delta_0}(x_{i_0}))$ we then conclude the proof as before.
\vglue .2in
\noindent{\bf Remark 4}. The case of constant diffusion coefficient and $b(\,\cdot\,)$ satisfying equation (11) makes the coupling time of the two processes $X_.^{x,\ve}$,\,\, $X_.^{y,\ve}$ -- if we use the coupling designed in [LR] -- particularly easy t
o
evaluate, and directly comparable with $\tilde S$ where $\tilde S$ is the time for a one dimensional Brownian motion starting at some point $r =\frac{|x-y|}{2\ve} $ to reach the origin.
On the other hand if $\sigma(\,\cdot\,)$ is not constant, one needs to examine condition (23) of [LR] to verify if coupling occurs. Since we not only want to see the finiteness of the coupling time, but also its $\ve$-dependence, we need to make a further
comparison, and we do not enter this.
\vglue .5in
\centerline{\bf REFERENCES}
\bigskip
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\item{[D]} M. Day. On the exponential exit law in the small parameter exit problem Stochastcs. {\bf 8} (1989), 297--\quad .
\item{[FW]} M. Freidlin, A.P. Wentzell. Random perturbations of dynamical systems. Springer-Verlag (1984).
\item{[GOV]} H. Galves, E. Olivieri, M.E. Vares. Metastability for a class of dynamical systems subject to small random perturbations. Ann. Prob. {\bf 15} (1987), 1288--1305.
\item{[LR]} T. Lindvall, L.C.G. Rogers. Couplings of multidimensional diffusions by reflexion. Ann. Prob. {\bf 14} (1986), 860--872.
\item{[M]} C. Mueller. Coupling and invariant measures for the heat equation with noise. Ann. Prob. {\bf 21} (1993), 2189--2198.
\item{[MOS]} F. Martinelli, E. Olivieri, E. Scoppola. Small random perturbations of finite and infinite dimensional dynamical systems: unpredictability of exit times, J. Stat. Phys. {\bf 55} (1989), 478--503.
\item{[MS]} F. Martinelli, E. Scoppola. Small random perturbations of dynamical systems: exponential loss of memory of the initial condition CMP {\bf 120} (1988), 25--69.
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