Stochastic Analysis on Path Space over Time-Inhomogeneous Manifolds with Boundary

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Let Lₜ:=Δₜ+Zₜ for a C¹,¹-vector field Z on a differential manifold M with possible boundary ∂ M, where Δₜ is the Laplacian induced by a time dependent metric gₜ differentiable in t∈ [0,Tc). We first introduce the damp gradient operator, defined on the path space with reference measure P, the law of the (reflecting) diffusion process generated by Lₜ on the base manifold; then establish the integration by parts formula for underlying directional derivatives and prove the log-Sobolev inequality for the associated Dirichlet form, which is further applied to the free path spaces; and finally, establish numbers of transportation-cost inequalities associated to the uniform distance, which are equivalent to the curvature lower bound and the convexity of the boundary.