Reflecting Diffusion Processes on Manifolds Carrying Geometric Flow

Let $$L_t:=\Delta _t+Z_t$$ L t : = Δ t + Z t for a $$C^{\infty }$$ C ∞ -vector field Z on a differentiable manifold M with boundary $$\partial M$$ ∂ M , where $$\Delta _t$$ Δ t is the Laplacian operator, induced by a time dependent metric $$g_t$$ g t differentiable in $$t\in [0,T_\mathrm {c})$$ t ∈ [ 0 , T c ) . We first establish the derivative formula for the associated reflecting diffusion semigroup generated by $$L_t$$ L t . Then, by using parallel displacement and reflection, we construct the couplings for the reflecting $$L_t$$ L t -diffusion processes, which are applied to gradient estimates and Harnack inequalities of the associated heat semigroup. Finally, as applications of the derivative formula, we present a number of equivalent inequalities for a new curvature lower bound and the convexity of the boundary. These inequalities include the gradient estimates, Harnack inequalities, transportation-cost inequalities and other functional inequalities for diffusion semigroups.