Markov Processes and Stochastic Extrinsic Derivative Flows on the Space of Absolutely Continuous Measures
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916811881054208 |
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| author | Ren, Panpan Wang, Feng-Yu Wittmann, Simon |
| author_facet | Ren, Panpan Wang, Feng-Yu Wittmann, Simon |
| contents | Let $E$ be the class of finite (resp. probability) measures absolutely continuous with respect to a $σ$-finite Radon measure on a Polish space. We present a criterion on the quasi-regularity of Dirichlet forms on $E$ in terms of upper bound conditions given by the uniform $(L^1+L^\infty)$-norm of the extrinsic derivative. As applications, we construct a class of general type Markov processes on $E$ via quasi-regular Dirichlet forms containing the diffusion, jump and killing terms. Moreover, stochastic extrinsic derivative flows on $E$ are studied by using quasi-regular Dirichlet forms, which in particular provide martingale solutions to SDEs on these two spaces, with drifts given by the extrinsic derivative of entropy functionals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_15687 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Markov Processes and Stochastic Extrinsic Derivative Flows on the Space of Absolutely Continuous Measures Ren, Panpan Wang, Feng-Yu Wittmann, Simon Probability 60J46, 60J60, 60J25, 60G57, 60G44 Let $E$ be the class of finite (resp. probability) measures absolutely continuous with respect to a $σ$-finite Radon measure on a Polish space. We present a criterion on the quasi-regularity of Dirichlet forms on $E$ in terms of upper bound conditions given by the uniform $(L^1+L^\infty)$-norm of the extrinsic derivative. As applications, we construct a class of general type Markov processes on $E$ via quasi-regular Dirichlet forms containing the diffusion, jump and killing terms. Moreover, stochastic extrinsic derivative flows on $E$ are studied by using quasi-regular Dirichlet forms, which in particular provide martingale solutions to SDEs on these two spaces, with drifts given by the extrinsic derivative of entropy functionals. |
| title | Markov Processes and Stochastic Extrinsic Derivative Flows on the Space of Absolutely Continuous Measures |
| topic | Probability 60J46, 60J60, 60J25, 60G57, 60G44 |
| url | https://arxiv.org/abs/2408.15687 |