Diffusion Processes on p-Wasserstein Space over Banach Space

arXiv:2402.15130 · math.PR · Submitted 2024-02-23 · Read on arXiv

math.PR

Submitted: 2024-02-23

Updated: 2026-10-29

Comments: Sect. 3.2 & Sect. 4 revised

License: http://creativecommons.org/licenses/by/4.0/

The gist: To study diffusion processes on the p-Wasserstein space P p for p in [1, infinity) over a separable, reflexive Banach space X, we present a criterion on the quasi-regularity of Dirichlet forms in L

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Abstract

To study diffusion processes on the p-Wasserstein space P p for p in [1, infinity) over a separable, reflexive Banach space X, we present a criterion on the quasi-regularity of Dirichlet forms in L 2(P p,Λ) for a reference probability Λ on 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 P p by using images of Dirichlet forms on the tangent space L p(X to X,μ 0) at a reference point μ 0 in P p. The Ornstein--Uhlenbeck type Dirichlet form and process on P 2 are an important example in this class. We derive an L squared-estimate for the corresponding heat kernel and an integration by parts formula for the invariant measure.

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