Diffusion Processes on $p$-Wasserstein Space over Banach Space

To study diffusion processes on the p-Wasserstein space ${\mathcal{P}}_p$ for $p\in [1,\mathrm{\infty })$ over a separable, reflexive Banach space $X$, we present a criterion on the quasi-regularity of Dirichlet forms in $L^2({\mathcal{P}}_p,\mathrm{\Lambda })$ for a reference probability $\mathrm{\Lambda }$ on ${\mathcal{P}}_p$. It is formulated in terms of an upper bound condition with the uniform norm of the intrinsic derivative. We find a versatile class of quasi-regular local Dirichlet forms on ${\mathcal{P}}_p$ by using images of Dirichlet forms on the tangent space $L^p(X\to X,{\mu }_0)$ at a reference point ${\mu }_0\in {\mathcal{P}}_p$. The Ornstein–Uhlenbeck type Dirichlet form and process on ${\mathcal{P}}_2$ are an important example in this class. We derive an $L^2$-estimate for the corresponding heat kernel and an integration by parts formula for the invariant measure.