Some problems concerning stability under small stochastic perturbations

A. D. Venttsel' ; M. I. Freidlin

Teoriâ veroâtnostej i ee primeneniâ, Tome 17 (1972) no. 2, pp. 281-295

Résumé

Let $x_0$ be a stable equilibrium point of a dynamic system $\dot x=b(x)$ in $R^r$; a Markov stochastic process $x_t^\varepsilon$ appears as a result of small stochastic perturbations of this system: $dx_t^\varepsilon=b(x_t^\varepsilon)\,dt+\varepsilon\sigma(x_t^\varepsilon)\,d\xi_t$ where $\xi_t$ is a Wiener process. Problems concerning stability of the point $x_0$ with respect to the stochastic process $x_t^\varepsilon$ are considered. All trajectories of the process $x_t^\varepsilon$ sooner or later, leave each neighbourhood of the equilibrium point. The problem arises how to choose a region of a given area for which the mean exit time is maximum? Another problem setting: suppose that one can control the process $x_t^\varepsilon$ by chosing a drift vector $b(x)$ at each point $x$ of some set of vectors $\Pi(x)$. How should one control the process so that the mean exit time of a given region would be maximum (minimum)? Asymptotically optimal solutions to these questions are given: the control proposed by the authors is not worse (not essentially worse) than any other control if $\varepsilon$ is sufficiently small; the mean exit time of any other region $G$ of a given area is less than that of the region the authors point at if $\varepsilon$ is small. The way of solving these problems is to estimate the main term of the mean exit time of a given region $G$ when $\varepsilon\to0$. This main term is $\exp\Bigl\{\frac1{2\varepsilon^2}\min\limits_{y\in\partial G}V(x_0,y)\Bigr\}$ where $V(x_0,x)$ is a function that does not depend on the region and can be found as a solution of a specific problem for the differential equation $$ \sum a^{ij}(x)\frac{\partial V}{\partial x^i}\frac{\partial V}{\partial x^j}+4\sum b^i(x)\frac{\partial V}{\partial x^i}=0,\quad(a^{ij}(x))=\sigma(x)\sigma^*(x). $$ In order to solve the optimal control problem, a non-linear partial differential equation is considered. In the case of shift-invariance this equation can be solved by means of a certain geometrical construction.