Average dissipation for stochastic transport equations with Lévy noise
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909105166221312 |
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| author | Flandoli, Franco Papini, Andrea Rehmeier, Marco |
| author_facet | Flandoli, Franco Papini, Andrea Rehmeier, Marco |
| contents | We show that, in one spatial and arbitrary jump dimension, the averaged solution of a Marcustype SPDE with pure jump Lévy transport noise satisfies a dissipative deterministic equation involving a fractional Laplace-type operator. To this end, we identify the correct associated Lévy measure for the driving noise. We consider this a first step in the direction of a non-local version of enhanced dissipation, a phenomenon recently proven to occur for Brownian transport noise and the associated local parabolic PDE by the first author. Moreover, we present numerical simulations, supporting the fact that dissipation occurs for the averaged solution, with a behavior akin to the diffusion due to a fractional Laplacian, but not in a pathwise sense. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08461 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Average dissipation for stochastic transport equations with Lévy noise Flandoli, Franco Papini, Andrea Rehmeier, Marco Probability 60H15, 76F55, 76F25, 35Q49, 35K10, 60J76 We show that, in one spatial and arbitrary jump dimension, the averaged solution of a Marcustype SPDE with pure jump Lévy transport noise satisfies a dissipative deterministic equation involving a fractional Laplace-type operator. To this end, we identify the correct associated Lévy measure for the driving noise. We consider this a first step in the direction of a non-local version of enhanced dissipation, a phenomenon recently proven to occur for Brownian transport noise and the associated local parabolic PDE by the first author. Moreover, we present numerical simulations, supporting the fact that dissipation occurs for the averaged solution, with a behavior akin to the diffusion due to a fractional Laplacian, but not in a pathwise sense. |
| title | Average dissipation for stochastic transport equations with Lévy noise |
| topic | Probability 60H15, 76F55, 76F25, 35Q49, 35K10, 60J76 |
| url | https://arxiv.org/abs/2402.08461 |