Reflecting Diffusion Process on Time-Inhomogeneous Manifolds with Boundary

Abstract

Let $L_t:={\Delta }_t+Z_t$ for a $C^{1,1}$-vector field $Z$ on a differential manifold $M$ with boundary $\partial M$, where ${\Delta }_t$ is the Laplacian induced by a time dependent metric $g_t$ differentiable in $t\in [0,T_c)$. We first introduce the reflecting diffusion process generated by $L_t$ and establish the derivative formula for the associated diffusion semigroup; then construct the couplings for the reflecting $L_t$-diffusion processes by parallel and reflecting displacement, which implies the gradient estimates of the associated heat semigroup; and finally, present a number of equivalent inequalities for the curvature lower bound and the convexity of the boundary, including the gradient estimations, Harnack inequalities, transportation-cost inequalities and other functional inequalities for diffusion semigroup.

arXiv admin note: text overlap with arXiv:0911.1644 by other authors

References in corpus (5)

• Formulae for the derivatives of heat semigroups
• Harnack Inequalities for Functional SDEs with Multiplicative Noise and Applications
• Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on nonconvex manifolds
• Semigroup Properties for the Second Fundamental Form
• Stochastic Analysis on Path Space over Time-Inhomogeneous Manifolds with Boundary

Cited by in corpus (1)

• A Probabilistic Approach for Gradient Inequalities on Time-Inhomogeneous Manifolds