Asymptotic stability and spiraling properties for solutions of stochastic equations

Abstract

We consider a system of ItSequations in a domain in R .The boundary consists of points and closed surfaces.The coefficients are such that, starting for the exterior of the domain, the process stays in the exterior.We give sufficient conditions to ensure that the process converges to the boundary when i-oo.In the case of plane domains, we give conditions to ensure that the process "spirals"; the angle obeys the strong law of large numbers.J together with a "stable manifold" dG.The set G will consist of a finite number of points together with a finite number of closed domains.The coefficients a- b. ate such that if the process starts on dG then it stays forever on dG.Our first result (Theorem 1.1) gives a set of sufficient conditions for the nonattainability of G, starting from the exterior.If G consists of points and convex bodies, it suffices that the normal components of the diffusion and the drift vanish on dG; in general we need to impose an additional "convexity" relation between dG, the drift and the diffusion coefficients to ensure the nonattainability of dG.The next result (Theorem 2.1) gives sufficient conditions that xit) - dG when t - o.This theorem contains local stability conditions (near dG and near oo) reminiscent of the linear case [4], as well as a certain nondegeneracy condition.None of these conditions can be relaxed.