Estimates for the volume variation of compact submanifolds driven by a stochastic flow

Abstract

Consider a compact submanifold N without the boundary of a Riemannian manifold M, and a stochastic flow (Formula presented.) associated with a stochastic differential equation. Let (Formula presented.) be the random compact submanifold obtained by the action of the stochastic flow. In this work, we present an Itô formula for the volume of the random variable (Formula presented.) and, as a main result, we obtain estimates for its average growth assuming that Ricci curvature is bounded. We first analyse the particular case where the submanifolds are closed curves, thus obtaining estimates for the arc length, and then we study the volume variation of compact submanifolds of dimensions greater than or equal to 2. In addition, we apply our results to the special case where the vector fields of stochastic differential equation are conformal Killing.