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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.15993 |
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| _version_ | 1866916447883624448 |
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| author | Eisenhuth, Benedikt Grothaus, Martin |
| author_facet | Eisenhuth, Benedikt Grothaus, Martin |
| contents | We consider a degenerate infinite dimensional stochastic Hamiltonian system with multiplicative noise and establish the essential m-dissipativity on $L^2(μ^Φ)$ of the corresponding Kolmogorov (backwards) operator. Here, $Φ$ is the potential and $μ^Φ$ the invariant measure with density $e^{-Φ}$ with respect to an infinite dimensional non-degenerate Gaussian measure. The main difficulty, besides the non-sectorality of the Kolmogorov operator, is the coverage of a large class of potentials. We include potentials that have neither a bounded nor a Lipschitz continuous gradient. The essential m-dissipativity is the starting point to establish the hypocoercivity of the strongly continuous contraction semigroup $(T_t)_{t\geq 0}$ generated by the Kolmogorov operator. By using the refined abstract Hilbert space hypocoercivity method of Grothaus and Stilgenbauer, originally introduced by Dolbeault, Mouhot and Schmeiser, we construct a $μ^Φ$-invariant Hunt process with weakly continuous paths and infinite lifetime, whose transition semigroup is associated with $(T_t)_{t\geq 0}$. This process provides a stochastically and analytically weak solution to the degenerate infinite dimensional stochastic Hamiltonian system with multiplicative noise. The hypocoercivity of $(T_t)_{t\geq 0}$ and the identification of $(T_t)_{t\geq 0}$ with the transition semigroup of the process leads to the exponential ergodicity. Finally, we apply our results to degenerate second order in time stochastic reaction-diffusion equations with multiplicative noise. A discussion of the class of applicable potentials and coefficients governing these equations completes our analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_15993 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The essential m-dissipativity for degenerate infinite dimensional stochastic Hamiltonian systems and applications Eisenhuth, Benedikt Grothaus, Martin Probability Functional Analysis 47D07, 37K45, 35K57, 35J70 We consider a degenerate infinite dimensional stochastic Hamiltonian system with multiplicative noise and establish the essential m-dissipativity on $L^2(μ^Φ)$ of the corresponding Kolmogorov (backwards) operator. Here, $Φ$ is the potential and $μ^Φ$ the invariant measure with density $e^{-Φ}$ with respect to an infinite dimensional non-degenerate Gaussian measure. The main difficulty, besides the non-sectorality of the Kolmogorov operator, is the coverage of a large class of potentials. We include potentials that have neither a bounded nor a Lipschitz continuous gradient. The essential m-dissipativity is the starting point to establish the hypocoercivity of the strongly continuous contraction semigroup $(T_t)_{t\geq 0}$ generated by the Kolmogorov operator. By using the refined abstract Hilbert space hypocoercivity method of Grothaus and Stilgenbauer, originally introduced by Dolbeault, Mouhot and Schmeiser, we construct a $μ^Φ$-invariant Hunt process with weakly continuous paths and infinite lifetime, whose transition semigroup is associated with $(T_t)_{t\geq 0}$. This process provides a stochastically and analytically weak solution to the degenerate infinite dimensional stochastic Hamiltonian system with multiplicative noise. The hypocoercivity of $(T_t)_{t\geq 0}$ and the identification of $(T_t)_{t\geq 0}$ with the transition semigroup of the process leads to the exponential ergodicity. Finally, we apply our results to degenerate second order in time stochastic reaction-diffusion equations with multiplicative noise. A discussion of the class of applicable potentials and coefficients governing these equations completes our analysis. |
| title | The essential m-dissipativity for degenerate infinite dimensional stochastic Hamiltonian systems and applications |
| topic | Probability Functional Analysis 47D07, 37K45, 35K57, 35J70 |
| url | https://arxiv.org/abs/2410.15993 |