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

Abstract: We investigate several fundamental properties of kinetic Langevin processes in $\mathbb{R}<sup>{2d}$, 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 $\mathscr{D} = \mathscr{O} \times \mathbb{R}<sup>d$ for an arbitrary open set $\mathscr{O} \subset \mathbb{R}<sup>d$. 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 $L<sup>m$ spaces as well as the Feller $C_0(\mathbb R<sup>{2d})$-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.

• The paper establishes weak well-posedness and Feller properties for kinetic Langevin processes with low-regularity drifts and heavy-tailed Lévy noise.
• It demonstrates strong Feller and irreducibility via novel geometric skeleton constructions and decompositions of small and large jumps.
• The work proves compactness and spectral gaps, ensuring unique (quasi-)stationary distributions and exponential ergodicity under minimal conditions.

Topological and Spectral Analysis of Kinetic Langevin Processes Driven by Lévy Noise

Overview

The paper investigates topological and spectral properties of kinetic Langevin processes in $\mathbb{R}^{2d}$ of the form:

$$dx_t = v_t\, dt, \qquad dv_t = \mathbf{B}(x_t, v_t)\, dt + dL_t$$

where $L_t$ is a pure-jump Lévy process and the drift $\mathbf{B}$ is permitted to have low regularity. The primary aims are to characterize both the process and its killed variant (terminated upon exit from a domain $\mathscr{D}$ of the form $\mathscr{O} \times \mathbb{R}^d$, $\mathscr{O}$ open) with respect to properties crucial for ergodic theory and spectral analysis: strong Feller property, weak well-posedness, continuity in initial conditions, irreducibility, spectral gap, compactness, and the existence of stationary and quasi-stationary distributions.

A central focus is the setting where the driving noise is a rotationally invariant $\alpha$-stable process ($\alpha \in (0,2)$), in which the noise lacks finite variance and is thus genuinely non-Gaussian, requiring new analytic approaches compared to the classic hypoelliptic Langevin-Brownian setting.

Model and Analytical Setting

The Langevin SDE with pure-jump Lévy drivers, including the killed process within a domain $\mathscr{D} = \mathscr{O} \times \mathbb{R}^d$, is treated without smoothness assumptions on the drift, encompassing situations where $$dx_t = v_t\, dt, \qquad dv_t = \mathbf{B}(x_t, v_t)\, dt + dL_t$$0 is merely measurable and with at most linear growth or is of perturbed-gradient-type structure (superlinear growth in $$dx_t = v_t\, dt, \qquad dv_t = \mathbf{B}(x_t, v_t)\, dt + dL_t$$1 with polynomially confining potentials).

Three main classes of vector fields are considered:

• Smooth drifts: $$dx_t = v_t\, dt, \qquad dv_t = \mathbf{B}(x_t, v_t)\, dt + dL_t$$2 is $$dx_t = v_t\, dt, \qquad dv_t = \mathbf{B}(x_t, v_t)\, dt + dL_t$$3 with bounded derivatives.
• Linear Growth: $$dx_t = v_t\, dt, \qquad dv_t = \mathbf{B}(x_t, v_t)\, dt + dL_t$$4 is measurable, at most linear.
• Perturbed Gradient: $$dx_t = v_t\, dt, \qquad dv_t = \mathbf{B}(x_t, v_t)\, dt + dL_t$$5; $$dx_t = v_t\, dt, \qquad dv_t = \mathbf{B}(x_t, v_t)\, dt + dL_t$$6 satisfies certain growth and dissipativity constraints, while $$dx_t = v_t\, dt, \qquad dv_t = \mathbf{B}(x_t, v_t)\, dt + dL_t$$7 is confining.

The following properties are studied for both the non-killed Markov semigroup $$dx_t = v_t\, dt, \qquad dv_t = \mathbf{B}(x_t, v_t)\, dt + dL_t$$8 and killed semigroup $$dx_t = v_t\, dt, \qquad dv_t = \mathbf{B}(x_t, v_t)\, dt + dL_t$$9:

• Strong Feller: $L_t$0 regularizes bounded measurable functions into continuous functions.
• Topological Irreducibility: The transition kernel has positive probability to reach any open set.
• Spectral Gap & Compactness: Essential for spectral analysis and ergodic convergence.
• (Quasi-)Stationarity: Existence and uniqueness of stationary and quasi-stationary distributions with explicit rates.

Main Results

1. Weak Well-posedness and Feller Properties

• Weak Solutions: Existence and uniqueness in law (and, in the smooth case, pathwise) of solutions to the (possibly degenerate) SDE with low regularity drifts is established for all $L_t$1; extended to $L_t$2 for smooth drifts. This leverages perturbative/singular integral representations and Krylov-type estimates.
• Continuity in Initial Data: The law of solutions depends continuously on the starting point.
• Feller Property: The semigroups are proved to be $L_t$3-Feller whenever weak uniqueness holds, uniformly over the non-killed and killed processes.

2. Strong Feller and Irreducibility

• Strong Feller: For the non-killed process, shown under minimal regularity assumptions using analytic regularization properties of the degenerate Ornstein–Uhlenbeck semigroup and a perturbative Duhamel formula. For the killed process, this property is inherited using boundary exit-time estimates and localization arguments.
• Irreducibility: Achieved by explicit decomposition of the Lévy process into small and large jumps, constructing a path skeleton that enables connecting any pair of points with positive probability via a finite cascade of jumps, even in domains with non-smooth boundaries.

3. Spectral Theory and Compactness

• Essential Spectral Radius: For bounded domains $L_t$4, the essential spectral radius of $L_t$5 vanishes, and the semigroup is compact for every $L_t$6. This is shown via a measure-of-noncompactness approach, circumventing the need for Lyapunov function techniques, which are unavailable for heavy-tailed noise.
• Existence of a Spectral Gap: Compactness combined with irreducibility and Feller properties imply a strictly positive spectral gap in $L_t$7 for the killed semigroup; essential for exponential convergence in quasi-stationarity.

4. (Quasi-)Stationary and Exponential Ergodicity

• Quasi-Stationary Distributions: For bounded, irreducible killing domains, existence and uniqueness of quasi-stationary distributions are shown. Exponential convergence to the QSD is proven, with uniform rates depending on the Lyapunov structure and compactness of the semigroup.
• Stationary Distributions: Under confining, dissipative drifts (perturbed gradient with superlinear potentials), existence, uniqueness, and exponential convergence to a stationary law are established for the non-killed process, in polynomially weighted spaces.

These results are uniform in $L_t$8 for non-smooth drift, extend to $L_t$9 for smooth drift, and cover a wide range of degenerate and hypoelliptic Markovian dynamics.

Technical Contributions and Novelty

• Direct Approach to Spectral Gap: The paper bypasses the classical Lyapunov framework for the spectral gap (which fails for stable noise) by using non-compactness/essential spectrum techniques relying on the kinetic structure.
• Low Regularity Drift: A comprehensive treatment where $\mathbf{B}$0 lacks continuity, handled via perturbative analysis and integrated/probabilistic Krylov estimates, with tightness and weak convergence arguments for passage to the limit.
• Geometric Skeleton Construction: The proof of irreducibility in general domains for pure-jump, degenerate kinetic equations without regular drift, using explicit small-jump and big-jump decompositions.
• Existence of Strong Feller for both killed and non-killed processes in low-regularity, degenerate, and jump-driven regimes.

Implications and Potential Developments

Theoretical Impact

• Stochastic Kinetic Theory: The framework extends classic hypocoercivity and ergodic theory to jump-driven, degenerate kinetic equations with minimal drift regularity; this is germane for models with impulsive forcing, e.g., plasma physics, anomalous transport, or non-Gaussian thermostats.
• Markov Process Theory: Establishes a unified approach to the ergodic and quasi-ergodic analysis for degenerate, pure-jump settings — including non-Lipschitz or even discontinuous environments, thus broadening the class of well-posed Markovian SDEs with stable noise.

Application and Outlook in AI

• Numerical Sampling and MCMC: The results provide rigorous theoretical underpinnings for stochastic sampling with kinetic jump processes, pertinent for algorithms seeking non-Gaussian or robust generative sampling (e.g., in high-dimensional Bayesian inverse problems).
• Learning with Heavy-Tailed Noise: The analysis offers a basis for understanding learning dynamics, optimization, and model robustness in the presence of heavy-tailed impulsive perturbations (as seen in stochastic gradient algorithms under fat-tailed or adversarial noise).
• Potential Extensions: Next steps may include quantitative bounds for mixing times, explicit spectral gap estimates in terms of the jump measure, finer regularity properties of invariant and quasi-stationary measures, or coupling-based analysis in the presence of interactions and further degeneracies.

Conclusion

The paper delivers a rigorous foundation for the analysis of kinetic Langevin processes with pure-jump Lévy drivers, overcoming the lack of smoothness, degenerate noise structure, and unavailability of classic elliptic tools. Strong Feller, irreducibility, weak well-posedness, compactness, and (quasi-)ergodicity are established in remarkable generality. This bridges the gap between classic kinetic theory and emerging needs in stochastic analysis and AI—especially where robust, impulsive, or heavy-tailed noise is essential.

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Reference: "On some topological and spectral properties of kinetic Langevin processes driven by Lévy noises" (2604.05598)