On some topological and spectral properties of kinetic Langevin processes driven by L{é}vy noises
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914451579469824 |
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| author | Batisse, T Guillin, A Nectoux, B Wu, L |
| author_facet | Batisse, T Guillin, A Nectoux, B Wu, L |
| contents | We investigate several fundamental properties of kinetic Langevin processes in $\mathbb{R}^{2d}$, defined as solutions to the following system: $$dx\_t = v\_t \, dt, \qquad dv\_t = \mathbf{B}(x\_t, v\_t) \, dt + dL\_t$$ where $(L\_t, t \ge 0)$ is a pure-jump L{é}vy process. Our analysis covers both the original process and its killed counterpart, where killing occurs upon exiting domains of the form $\mathscr{D} = \mathscr{O} \times \mathbb{R}^d$ for an arbitrary open set $\mathscr{O} \subset \mathbb{R}^d$. Operating within a low-regularity framework - where the drift $\mathbf{B}$ is not assumed to be continuous - we establish key structural and spectral properties for both the associated non-killed and killed semigroups. These include: the strong Feller property, weak continuity of trajectories with respect to initial conditions, topological irreducibility and the existence of a spectral gap. Furthermore, we prove, in this low-regularity framework, the existence and uniqueness of a weak solution when the driving noise is a rotationally invariant $α$-stable process, when $α\in (1,2)$. For this specific case, we show that the aforementioned properties hold and further establish the existence of densities within certain $L^m$ spaces as well as the Feller $C\_0(\mathbb R^{2d})$-semigroup property. Finally, we address the existence and uniqueness of stationary and quasi-stationary distributions, proving exponential ergodicity for the non-killed process and exponential convergence to the quasi-stationary limit for the conditioned process. We show that these results extend to every $α\in (0,1]$ when the drift is smooth. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05598 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On some topological and spectral properties of kinetic Langevin processes driven by L{é}vy noises Batisse, T Guillin, A Nectoux, B Wu, L Mathematical Physics Probability We investigate several fundamental properties of kinetic Langevin processes in $\mathbb{R}^{2d}$, defined as solutions to the following system: $$dx\_t = v\_t \, dt, \qquad dv\_t = \mathbf{B}(x\_t, v\_t) \, dt + dL\_t$$ where $(L\_t, t \ge 0)$ is a pure-jump L{é}vy process. Our analysis covers both the original process and its killed counterpart, where killing occurs upon exiting domains of the form $\mathscr{D} = \mathscr{O} \times \mathbb{R}^d$ for an arbitrary open set $\mathscr{O} \subset \mathbb{R}^d$. Operating within a low-regularity framework - where the drift $\mathbf{B}$ is not assumed to be continuous - we establish key structural and spectral properties for both the associated non-killed and killed semigroups. These include: the strong Feller property, weak continuity of trajectories with respect to initial conditions, topological irreducibility and the existence of a spectral gap. Furthermore, we prove, in this low-regularity framework, the existence and uniqueness of a weak solution when the driving noise is a rotationally invariant $α$-stable process, when $α\in (1,2)$. For this specific case, we show that the aforementioned properties hold and further establish the existence of densities within certain $L^m$ spaces as well as the Feller $C\_0(\mathbb R^{2d})$-semigroup property. Finally, we address the existence and uniqueness of stationary and quasi-stationary distributions, proving exponential ergodicity for the non-killed process and exponential convergence to the quasi-stationary limit for the conditioned process. We show that these results extend to every $α\in (0,1]$ when the drift is smooth. |
| title | On some topological and spectral properties of kinetic Langevin processes driven by L{é}vy noises |
| topic | Mathematical Physics Probability |
| url | https://arxiv.org/abs/2604.05598 |