Diffusion semigroup on manifolds with time-dependent metrics

TL;DR

This paper studies diffusion processes on manifolds with evolving metrics, providing criteria for non-explosion, derivative formulas, and inequalities related to curvature bounds, advancing understanding of time-dependent geometric analysis.

Contribution

It introduces explicit non-explosion criteria, derivative formulas, and semigroup inequalities for diffusions on manifolds with time-dependent metrics, extending existing theories.

Findings

Explicit non-explosion criteria for diffusion processes

Derivative formula for the associated semigroup

Semigroup inequalities under curvature lower bounds

Abstract

Let $L_t:=\Delta_t +Z_t $, $t\in [0,T_c)$ on a differential manifold equipped with time-depending complete Riemannian metric $(g_t)_{t\in [0,T_c)}$, where $\Delta_t$ is the Laplacian induced by $g_t$ and $(Z_t)_{t\in [0,T_c)}$ is a family of $C^{1,1}$-vector fields. We first present some explicit criteria for the non-explosion of the diffusion processes generated by $L_t$; then establish the derivative formula for the associated semigroup; and finally, present a number of equivalent semigroup inequalities for the curvature lower bound condition, which include the gradient inequalities, transportation-cost inequalities, Harnack inequalities and functional inequalities for the diffusion semigroup.