Diffusion semigroup on manifolds with time-dependent metrics

摘要

Let L-t := Delta(t) + Z(t), t is an element of [0, T-c) on a differential manifold equipped with a complete geometric flow (g(t))(t is an element of[0,Tc)),where Delta(t) is the Laplacian operator induced by the metric gt and (Z(t))(t is an element of[0,Tc)) is a family of C-1,C-infinity-vector fields. In this article, we present a number of equivalent inequalities for the lower bound curvature condition, which include gradient inequalities, transportation-cost inequalities, Harnack inequalities and other functional inequalities for the semigroup associated with diffusion processes generated by L-t. To this end, we establish derivative formulae for the associated semigroup and construct coupling processes for these diffusion processes by parallel displacement and reflection.