Diffusion Processes on $p$-Wasserstein Space over Banach Space
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918071485071360 |
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| author | Ren, Panpan Wang, Feng-Yu Wittmann, Simon |
| author_facet | Ren, Panpan Wang, Feng-Yu Wittmann, Simon |
| contents | To study diffusion processes on the p-Wasserstein space $\mathscr P_p$ for $p\in [1,\infty)$ over a separable, reflexive Banach space $X$, we present a criterion on the quasi-regularity of Dirichlet forms in $L^2(\mathscr P_p,Λ)$ for a reference probability $Λ$ on $\mathscr P_p$. 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 $\mathscr P_p$ by using images of Dirichlet forms on the tangent space $L^p(X\to X,μ_0)$ at a reference point $μ_0\in\mathscr P_p$. The Ornstein--Uhlenbeck type Dirichlet form and process on $\mathscr P_2$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_15130 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Diffusion Processes on $p$-Wasserstein Space over Banach Space Ren, Panpan Wang, Feng-Yu Wittmann, Simon Probability 60J60, 60J25, 60J46 To study diffusion processes on the p-Wasserstein space $\mathscr P_p$ for $p\in [1,\infty)$ over a separable, reflexive Banach space $X$, we present a criterion on the quasi-regularity of Dirichlet forms in $L^2(\mathscr P_p,Λ)$ for a reference probability $Λ$ on $\mathscr P_p$. 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 $\mathscr P_p$ by using images of Dirichlet forms on the tangent space $L^p(X\to X,μ_0)$ at a reference point $μ_0\in\mathscr P_p$. The Ornstein--Uhlenbeck type Dirichlet form and process on $\mathscr P_2$ 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. |
| title | Diffusion Processes on $p$-Wasserstein Space over Banach Space |
| topic | Probability 60J60, 60J25, 60J46 |
| url | https://arxiv.org/abs/2402.15130 |