Reflecting Diffusion Process on Time-Inhomogeneous Manifolds with Boundary

Let $L_t:={\mathrm{\Delta }}_t+Z_t$ for a $C^{1,1}$-vector field $Z$ on a differential manifold $M$ with boundary $\mathrm{\partial }M$, where ${\mathrm{\Delta }}_t$ 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.