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On small random perturbations of dynamical systems
A. D. Venttsel',
M. I. Freidlin
Abstract:
In this paper we study the effect on a dynamical system
$\dot x_t=b(x_t)$ of small random perturbations of the type of white noise:
$$
\dot x_t^\varepsilon=b^\varepsilon(x_t^\varepsilon)
+\varepsilon \sigma (x_t^\varepsilon)\bar\xi_t,
$$
where
$\xi_t$ is the
$r$-dimensional Wiener process and
$b^\varepsilon(x)\to b(x)$ as
$\varepsilon\to 0$. We are mainly concerned with the effect of these perturbations on long time-intervals that increase with the decreasing
$\varepsilon$. We discuss two problems: the first is the behaviour of the invariant measure
$\mu^\varepsilon$ of the process
$x_t^\varepsilon$ as
$\varepsilon\to 0$, and the second is the distribution of the position of a trajectory at the first time of its exit from a compact domain. An important role is played in these problems by an estimate of the probability for a trajectory of
$x_t^\varepsilon$ not to deviate from a smooth function
$\varphi_t$ by more than
$\delta$ during the time
$[0, T]$. It turns out that the main term of this probability for sma
$\varepsilon$ and
$\delta$ has the form
$\exp\bigl\{-\frac{1}{2\varepsilon^2}I(\varphi)\bigr\}$ where
$I(\varphi)$,
is a certain non-negative functional of
$\varphi_t$.
A function
$V(x,y)$, the minimum o
$I(\varphi)$ over the set of all functions connecting
$x$ and
$y$, is involved in the answers to both the problems.
By means of
$V(x,y)$ we introduce an independent of perturbations relation of equivalence in the phase-space. We show, under certain assumption, at what point of the phase-space the invariant measure concentrates in the limit.
In both the problems we approximate the process in question by a certain Markov chain; the answers depend on the behaviour of
$V(x,y)$ on graphs that are associated with this chain.
Let us remark that the second problem is closely related to the behaviour of the solution of a Dirichlet problem with a small parameter at the highest derivatives.
UDC:
519.2+519.9
MSC: 37J40,
60H40,
60J27 Received: 08.08.1969
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