Diffusion semigroup on manifolds with time-dependent metrics

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.

References in corpus (2)

• Formulae for the derivatives of heat semigroups
• A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality