Degenerate semigroups and stochastic flows of mappings in foliated manifolds

TL;DR

This paper investigates the behavior of stochastic flows on foliated manifolds, focusing on degenerate semigroups that preserve leaves, and applies these findings to an averaging principle with convergence estimates.

Contribution

It characterizes degenerate semigroups that generate foliated stochastic flows and establishes an averaging principle with convergence rates in this geometric setting.

Findings

Foliated stochastic flows preserve leaves almost surely.

Degeneracy of semigroups creates a geometrical obstruction for trajectory coalescence across leaves.

An averaging principle with explicit convergence estimates is proved.

Abstract

Let $(M, \mathcal{F})$ be a compact Riemannian foliated manifold. We consider a family of compatible Feller semigroups in $C(M^n)$ associated to laws of the $n$-point motion. Under some assumptions (Le Jan and Raimond, \cite{Le Jan-Raimond}) there exists a stochastic flow of measurable mappings in $M$. We study the degeneracy of these semigroups such that the flow of mappings is foliated, i.e. each trajectory lays in a single leaf of the foliation a.s, hence creating a geometrical obstruction for coalescence of trajectories in different leaves. As an application, an averaging principle is proved for a first order perturbation transversal to the leaves. Estimates for the rate of convergence are calculated.