Exponential stability and hypoelliptic regularization for the kinetic Fokker-Planck equation with confining potential
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866907771192999936 |
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| author | Arnold, Anton Toshpulatov, Gayrat |
| author_facet | Arnold, Anton Toshpulatov, Gayrat |
| contents | This paper is concerned with a modified entropy method to establish the large-time convergence towards the (unique) steady state, for kinetic Fokker-Planck equations with non-quadratic confinement potentials in whole space. We extend previous approaches by analyzing Lyapunov functionals with non-constant weight matrices in the dissipation functional (a generalized Fisher information). We establish exponential convergence in a weighted $H^1$-norm with rates that become sharp in the case of quadratic potentials. In the defective case for quadratic potentials, i.e. when the drift matrix has non-trivial Jordan blocks, the weighted $L^2$-distance between a Fokker-Planck-solution and the steady state has always a sharp decay estimate of the order $\mathcal O\big( (1+t)e^{-tν/2}\big)$, with $ν$ the friction parameter. The presented method also gives new hypoelliptic regularization results for kinetic Fokker-Planck equations (from a weighted $L^2$-space to a weighted $H^1$-space). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_06410 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Exponential stability and hypoelliptic regularization for the kinetic Fokker-Planck equation with confining potential Arnold, Anton Toshpulatov, Gayrat Analysis of PDEs This paper is concerned with a modified entropy method to establish the large-time convergence towards the (unique) steady state, for kinetic Fokker-Planck equations with non-quadratic confinement potentials in whole space. We extend previous approaches by analyzing Lyapunov functionals with non-constant weight matrices in the dissipation functional (a generalized Fisher information). We establish exponential convergence in a weighted $H^1$-norm with rates that become sharp in the case of quadratic potentials. In the defective case for quadratic potentials, i.e. when the drift matrix has non-trivial Jordan blocks, the weighted $L^2$-distance between a Fokker-Planck-solution and the steady state has always a sharp decay estimate of the order $\mathcal O\big( (1+t)e^{-tν/2}\big)$, with $ν$ the friction parameter. The presented method also gives new hypoelliptic regularization results for kinetic Fokker-Planck equations (from a weighted $L^2$-space to a weighted $H^1$-space). |
| title | Exponential stability and hypoelliptic regularization for the kinetic Fokker-Planck equation with confining potential |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2310.06410 |