Additive-multiplicative stochastic heat equations, stationary solutions, and Cauchy statistics
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909589112356864 |
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| author | Dunlap, Alexander Mukherjee, Chiranjib |
| author_facet | Dunlap, Alexander Mukherjee, Chiranjib |
| contents | We study long-term behavior and stationary distributions for stochastic heat equations forced simultaneously by a multiplicative noise and an independent additive noise with the same distribution. We prove that nontrivial space-time translation-invariant measures exist for all values of the parameters. We also show that if the multiplicative noise is sufficiently strong, the invariant measure has Cauchy-distributed marginals. Using the same techniques, we prove a similar result on Cauchy-distributed marginals for a logarithmically attenuated version of the problem in two spatial dimensions. The proofs rely on stochastic analysis and elementary potential theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_02907 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Additive-multiplicative stochastic heat equations, stationary solutions, and Cauchy statistics Dunlap, Alexander Mukherjee, Chiranjib Probability 60H15, 37L55 We study long-term behavior and stationary distributions for stochastic heat equations forced simultaneously by a multiplicative noise and an independent additive noise with the same distribution. We prove that nontrivial space-time translation-invariant measures exist for all values of the parameters. We also show that if the multiplicative noise is sufficiently strong, the invariant measure has Cauchy-distributed marginals. Using the same techniques, we prove a similar result on Cauchy-distributed marginals for a logarithmically attenuated version of the problem in two spatial dimensions. The proofs rely on stochastic analysis and elementary potential theory. |
| title | Additive-multiplicative stochastic heat equations, stationary solutions, and Cauchy statistics |
| topic | Probability 60H15, 37L55 |
| url | https://arxiv.org/abs/2402.02907 |