Diffusion Processes on p-Wasserstein Space over Banach Space

To study diffusion processes on the p-Wasserstein space Pₚ for p∈ [1,∞) over a separable Banach space X, we present a criterion on the quasi-regularity of Dirichlet forms in L²( Pₚ,Λ) for a reference probability Λ on Pₚ by using an upper bound condition with the uniform norm of intrinsic derivative, which is easy to check in applications. As a consequence, a class of quasi-regular local Dirichlet forms are constructed on Pₚ by using the image of Dirichlet forms on the tangent space Lᵖ(X→ X,μ₀) at a reference point μ₀∈ Pₚ. In particular, the quasi-regularity is confirmed for Ornstein-Uhlenbeck type Dirichlet forms, and an explicit heat kernel estimate is derived by using eigenvalues of the covariance operator for the underlying Gaussian measure.