Stochastic Analysis on Path Space over Time-Inhomogeneous Manifolds with Boundary

TL;DR

This paper develops stochastic analysis tools on path spaces over time-inhomogeneous manifolds with boundary, including gradient operators, integration by parts, log-Sobolev, and transportation inequalities, linking geometric properties to probabilistic inequalities.

Contribution

It introduces a damp gradient operator on path space, establishes integration by parts, log-Sobolev, and transportation inequalities on manifolds with boundary and time-dependent metrics.

Findings

Established integration by parts formula for directional derivatives.

Proved log-Sobolev inequality for the associated Dirichlet form.

Derived transportation-cost inequalities linked to curvature and boundary convexity.

Abstract

Let $L_t:=\Delta_t+Z_t$ for a $C^{1,1}$-vector field $Z$ on a differential manifold $M$ with possible 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 damp gradient operator, defined on the path space with reference measure $\mathbb{P}$, the law of the (reflecting) diffusion process generated by $L_t$ on the base manifold; then establish the integration by parts formula for underlying directional derivatives and prove the log-Sobolev inequality for the associated Dirichlet form, which is further applied to the free path spaces; and finally, establish numbers of transportation-cost inequalities associated to the uniform distance, which are equivalent to the curvature lower bound and the convexity of the boundary.