Logarithmic Harnack inequalities for transition semigroups in Hilbert spaces
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2021
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| author | Angiuli, L. Bignamini, D. A. Ferrari, S. |
| author_facet | Angiuli, L. Bignamini, D. A. Ferrari, S. |
| contents | We consider the stochastic differential equation $$ \left\{ \begin{array}{lc} dX(t)=[AX(t)+F(X(t))]dt+C^{1/2}dW(t), & t>0;\\ X(0)=x \in \mathcal{X}; \end{array}\right. $$ where $\mathcal{X}$ is a Hilbert space, $\{W(t)\}_{t\geq 0}$ is a $\mathcal{X}$-valued cylindrical Wiener process, $A, C$ are suitable operators on $\mathcal{X}$ and $F:{\rm Dom}\,(F)\subseteq \mathcal{X}\to \mathcal{X}$ is a smooth enough function. We establish a logarithmic Harnack inequality for the transition semigroup $\{P(t)\}_{t\geq 0}$ associated with the stochastic problem above, under less restrictive conditions than those considered in the literature. Some applications to these inequalities are also shown. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_13250 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Logarithmic Harnack inequalities for transition semigroups in Hilbert spaces Angiuli, L. Bignamini, D. A. Ferrari, S. Probability Analysis of PDEs 60H10, 60J60 We consider the stochastic differential equation $$ \left\{ \begin{array}{lc} dX(t)=[AX(t)+F(X(t))]dt+C^{1/2}dW(t), & t>0;\\ X(0)=x \in \mathcal{X}; \end{array}\right. $$ where $\mathcal{X}$ is a Hilbert space, $\{W(t)\}_{t\geq 0}$ is a $\mathcal{X}$-valued cylindrical Wiener process, $A, C$ are suitable operators on $\mathcal{X}$ and $F:{\rm Dom}\,(F)\subseteq \mathcal{X}\to \mathcal{X}$ is a smooth enough function. We establish a logarithmic Harnack inequality for the transition semigroup $\{P(t)\}_{t\geq 0}$ associated with the stochastic problem above, under less restrictive conditions than those considered in the literature. Some applications to these inequalities are also shown. |
| title | Logarithmic Harnack inequalities for transition semigroups in Hilbert spaces |
| topic | Probability Analysis of PDEs 60H10, 60J60 |
| url | https://arxiv.org/abs/2111.13250 |