Harnack Inequalities and Ergodicity of Stochastic Reaction-Diffusion Equation in $L^p$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866914151603896320 |
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| author | Liu, Zhihui |
| author_facet | Liu, Zhihui |
| contents | We derive Harnack inequalities for a stochastic reaction-diffusion equation with dissipative drift driven by additive irregular noise in the $L^p$-space for any $p \ge 2$. These inequalities are utilized to investigate the ergodicity of the corresponding Markov semigroup $(P_t)$. The main ingredient of our method is a coupling by the change of measure. Applying our results to the stochastic reaction-diffusion equation with a super-linear growth drift having a negative leading coefficient, perturbed by a Lipschitz term, indicates that $(P_t)$ possesses a unique and thus ergodic invariant measure in $L^p$ for all $p \ge 2$, which is independent of the Lipschitz term. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_01335 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Harnack Inequalities and Ergodicity of Stochastic Reaction-Diffusion Equation in $L^p$ Liu, Zhihui Probability Primary 60H15, 60H10, 37H05 We derive Harnack inequalities for a stochastic reaction-diffusion equation with dissipative drift driven by additive irregular noise in the $L^p$-space for any $p \ge 2$. These inequalities are utilized to investigate the ergodicity of the corresponding Markov semigroup $(P_t)$. The main ingredient of our method is a coupling by the change of measure. Applying our results to the stochastic reaction-diffusion equation with a super-linear growth drift having a negative leading coefficient, perturbed by a Lipschitz term, indicates that $(P_t)$ possesses a unique and thus ergodic invariant measure in $L^p$ for all $p \ge 2$, which is independent of the Lipschitz term. |
| title | Harnack Inequalities and Ergodicity of Stochastic Reaction-Diffusion Equation in $L^p$ |
| topic | Probability Primary 60H15, 60H10, 37H05 |
| url | https://arxiv.org/abs/2008.01335 |