Stochastic differential equations with multiple invariant measures and related problems

Abstract

We study problems related to SDEs which admit multiple invariant measures. The main problem we address is determining the long time behaviour of a large class of diffusion processes on R N , generated by second order differential operators of (possibly) degenerate type. The operators that we consider need not satisfy the Hormander condition and need not admit a unique invariant measure. Instead, we consider the so-called UFG condition, introduced by Hermann, Lobry and Sussmann in the context of geometric control theory and later by Kusuoka and Stroock, this time with probabilistic motivations. In this thesis we will demonstrate the importance of UFG diffusions in several respects: We show that UFG processes constitute a family of SDEs which exhibit multiple invariant measures and for which one is able to describe a systematic procedure to determine the basin of attraction of each invariant measure (equilibrium state). We show that our results and techniques, which we devised for UFG processes, can be applied to the study of the long-time behaviour of non-autonomous hypoelliptic SDEs. We prove that there exists a change of coordinates such that every UFG diffusion can be, at least locally, represented as a system consisting of an SDE coupled with ODE, where the ODE evolves independently of the SDE part of the dynamics. As a result, UFG diffusions are inherently “less smooth” than hypoelliptic SDEs; more precisely, we prove that UFG processes do not admit a density with respect to Lebesgue measure on the entire space, but only on suitable time-evolving submanifolds, which we describe. We introduce a novel pathwise approach to obtain (long-time) derivative estimates for Markov semigroups. The content of this thesis has resulted in two long papers [1] and [2], both submitted for publication.