Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on nonconvex manifolds

TL;DR

This paper establishes dimension-free Harnack inequalities for SDEs with multiplicative noise using coupling methods, and extends results to Neumann semigroups on nonconvex manifolds, impacting heat kernel estimates and diffusion processes.

Contribution

It introduces a novel coupling approach with unbounded time-dependent drift to derive Harnack inequalities for complex stochastic systems, including nonconvex manifold boundaries.

Findings

Proves dimension-free Harnack inequalities for SDEs with multiplicative noise.

Extends inequalities to reflecting diffusions on nonconvex manifolds.

Provides heat kernel bounds and contractivity results for associated semigroups.

Abstract

By constructing a coupling with unbounded time-dependent drift, dimension-free Harnack inequalities are established for a large class of stochastic differential equations with multiplicative noise. These inequalities are applied to the study of heat kernel upper bound and contractivity properties of the semigroup. The main results are also extended to reflecting diffusion processes on Riemannian manifolds with nonconvex boundary.