Dynamics and large deviations for fractional stochastic partial differential equations with Lévy noise
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| Format: | Preprint |
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2025
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| _version_ | 1866910799247704064 |
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| author | Xu, Jiaohui Caraballo, Tomás Valero, José |
| author_facet | Xu, Jiaohui Caraballo, Tomás Valero, José |
| contents | This paper is mainly concerned with a kind of fractional stochastic evolution equations driven by Lévy noise in a bounded domain. We first state the well-posedness of the problem via iterative approximations and energy estimates. Then, the existence and uniqueness of weak pullback mean random attractors for the equations {are} established by defining a mean random dynamical system. Next, we prove the existence of invariant measures when the problem is autonomous by means of the fact that $H^γ(\mathcal{O})$ is compactly embedded in $L^2(\mathcal{O})$ with $γ\in (0,1)$. Moreover, the uniqueness of this invariant measure is presented which ensures the ergodicity of the problem. Finally, a large deviation principle result for solutions of SPDEs perturbed by small Lévy noise and Brownian motion is obtained by a variational formula for positive functionals of a Poisson random measure and Brownian motion. Additionally, the results are illustrated by the fractional stochastic Chafee-Infante equations |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_14843 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dynamics and large deviations for fractional stochastic partial differential equations with Lévy noise Xu, Jiaohui Caraballo, Tomás Valero, José Probability 35R11, 35Q30, 65F08, 60H15, 65F10 This paper is mainly concerned with a kind of fractional stochastic evolution equations driven by Lévy noise in a bounded domain. We first state the well-posedness of the problem via iterative approximations and energy estimates. Then, the existence and uniqueness of weak pullback mean random attractors for the equations {are} established by defining a mean random dynamical system. Next, we prove the existence of invariant measures when the problem is autonomous by means of the fact that $H^γ(\mathcal{O})$ is compactly embedded in $L^2(\mathcal{O})$ with $γ\in (0,1)$. Moreover, the uniqueness of this invariant measure is presented which ensures the ergodicity of the problem. Finally, a large deviation principle result for solutions of SPDEs perturbed by small Lévy noise and Brownian motion is obtained by a variational formula for positive functionals of a Poisson random measure and Brownian motion. Additionally, the results are illustrated by the fractional stochastic Chafee-Infante equations |
| title | Dynamics and large deviations for fractional stochastic partial differential equations with Lévy noise |
| topic | Probability 35R11, 35Q30, 65F08, 60H15, 65F10 |
| url | https://arxiv.org/abs/2501.14843 |