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

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.