Attractivity, invariance and ergodicity for SDEs on Riemannian manifolds

TL;DR

This paper establishes conditions under which solutions to stochastic differential equations on compact Riemannian manifolds converge to a unique invariant measure, and characterizes ergodic measures for such equations.

Contribution

It provides a sufficient condition for weak convergence of solution laws to the Riemannian volume measure and characterizes invariant and ergodic measures on manifolds.

Findings

Solutions converge weakly to the Riemannian volume measure

Characterization of invariant measures for SDEs on manifolds

Conditions for ergodicity of solutions

Abstract

We give a sufficient condition on nonlinearities of an SDE on a compact connected Riemannian manifold $M$ which implies that laws of all solutions converge weakly to the normalized Riemannian volume measure on $M$. This result is further applied to characterize invariant and ergodic measures for various SDEs on manifolds.