On some topological and spectral properties of kinetic Langevin processes driven by L{\'e}vy noises

TL;DR

This paper studies fundamental topological and spectral properties of kinetic Langevin processes driven by Lévy noises, including existence, uniqueness, ergodicity, and spectral gaps, in low-regularity settings.

Contribution

It establishes key structural and spectral properties for kinetic Langevin processes with Lévy noise under low regularity, including existence of densities and ergodic behavior.

Findings

Proved strong Feller property and irreducibility for the processes.

Established existence and uniqueness of solutions driven by rotationally invariant α-stable Lévy processes.

Proved exponential ergodicity and convergence to quasi-stationary distributions.

Abstract

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{\'e}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 $\alpha$-stable process, when $\alpha \in (1,2)$. 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 $\alpha \in (0,1]$ when the drift is smooth.