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arxiv: 2604.05598 · v1 · submitted 2026-04-07 · 🧮 math-ph · math.MP· math.PR

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

T Batisse (LMBP) , A Guillin (LMBP) , B Nectoux (LMBP) , L Wu (LMBP) This is my paper

Pith reviewed 2026-05-10 19:34 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.PR
keywords kinetic Langevin processLévy noisestrong Feller propertyspectral gapquasi-stationary distributionalpha-stable processexponential ergodicity
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The pith

Kinetic Langevin processes with Lévy jumps satisfy strong Feller properties and spectral gaps even when the drift is discontinuous.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes structural and spectral properties for kinetic Langevin processes driven by pure-jump Lévy noise in both non-killed and killed settings. Even without assuming continuity of the drift, the associated semigroups are strong Feller, topologically irreducible, weakly continuous in initial conditions, and possess a spectral gap. For rotationally invariant alpha-stable driving noise with alpha between 1 and 2, weak solutions exist uniquely, transition densities appear in L^m spaces, and the semigroup is Feller on C0; stationary and quasi-stationary distributions exist and the process converges exponentially to them. These results extend to smaller alpha when the drift is smooth. The findings allow analysis of jump-driven kinetic systems under realistic low-regularity conditions that arise in physical models.

Core claim

In the low-regularity framework where the drift B need not be continuous, the non-killed and killed semigroups generated by the kinetic Langevin equation with pure-jump Lévy noise possess the strong Feller property, weak continuity of trajectories, topological irreducibility, and a spectral gap. When the noise is rotationally invariant alpha-stable with alpha in (1,2), a unique weak solution exists, densities exist in L^m spaces, the semigroup is Feller on C0(R^{2d}), and unique stationary and quasi-stationary distributions exist with exponential ergodicity for the non-killed process and exponential convergence to the quasi-stationary limit for the conditioned process. The properties hold as

What carries the argument

The Markov semigroups (non-killed and killed) generated by the kinetic Langevin SDE with pure-jump Lévy noise, together with the strong Feller property and spectral gap.

If this is right

  • A unique stationary distribution exists and the process converges exponentially fast to it.
  • The killed process admits a unique quasi-stationary distribution with exponential convergence to that distribution.
  • Transition densities exist in L^m spaces when the noise is alpha-stable.
  • The same structural and ergodic properties hold for all alpha in (0,1] provided the drift is smooth.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The jump component of the noise supplies enough regularization to overcome the loss of continuity in the drift.
  • These ergodicity results support long-time averaging arguments for observables of the process.
  • The density existence opens the possibility of likelihood-based inference from trajectory data.

Load-bearing premise

The Lévy noise must be pure-jump and sufficiently non-degenerate so that it interacts with the possibly discontinuous drift to produce the claimed regularity and mixing.

What would settle it

Exhibit a specific discontinuous drift B and rotationally invariant alpha-stable noise for which the transition kernel fails to map bounded continuous functions to continuous functions.

Figures

Figures reproduced from arXiv: 2604.05598 by A Guillin (LMBP), B Nectoux (LMBP), L Wu (LMBP), T Batisse (LMBP).

Figure 1
Figure 1. Figure 1: Schematic representation (not to scale) of the condi￾tion (3.12) and Ran4ǫ(γ0), when N = 1. s t0 t1 t2 t3 t4 t5 ∆0 ∆1 ∆2 ∆3 ∆4 L + s (δ) has exactly one jump in B(vx0→x1 (∆1) − v0, β) L + s (δ) − L + t1 (δ) has no jumps L + s (δ) − L + t2 (δ) has exactly one jump in B(vx1→x2 (∆3) − vt2 , β) L + s (δ) − L + t3 (δ) has no jumps L + s (δ) − L + t4 (δ) has exactly one jump in B(vF − vt4 , β) [PITH_FULL_IMAGE:… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic representation of the cascade of events for the increments of (L + s (δ), s ≥ 0) when N = 1 (t0 = 0 and t5 = t) and when X0 = x0. (though it is a powerful tool often used for SDEs with non-smooth coefficients which are driven by a Brownian noise, see e.g. [42, 79, 15]). The literature on the well-posedness of SDEs driven by L´evy noise is extensive. The following short review is by no means exhau… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper investigates topological and spectral properties of kinetic Langevin processes in R^{2d} given by dx_t = v_t dt, dv_t = B(x_t, v_t) dt + dL_t, where L is a pure-jump Lévy process. It considers both the non-killed process and its killed version on domains D = O × R^d. In a low-regularity setting with B merely measurable (no continuity assumed), the authors claim the strong Feller property, weak continuity of trajectories, topological irreducibility, and spectral gaps for the associated semigroups. For rotationally invariant α-stable noise with α ∈ (1,2), they assert existence and uniqueness of weak solutions, existence of densities in L^m spaces, the Feller C_0(R^{2d})-semigroup property, and exponential ergodicity to unique stationary and quasi-stationary distributions; the results are said to extend to α ∈ (0,1] when B is smooth.

Significance. If the central well-posedness and semigroup claims hold under the stated assumptions, the work would meaningfully extend ergodic theory and spectral analysis for jump-driven kinetic systems to discontinuous drifts, with relevance to non-smooth models in statistical mechanics and stochastic dynamics. The treatment of killed processes and quasi-stationary convergence is a positive addition.

major comments (1)
  1. [Abstract] Abstract (claims on weak solutions): The assertion of existence and uniqueness of weak solutions for the SDE with merely measurable drift B and rotationally invariant α-stable driving noise (α ∈ (1,2)) is load-bearing for every subsequent claim on strong Feller property, spectral gaps, densities, and exponential ergodicity. Standard results on weak uniqueness for Lévy-driven SDEs in finite dimensions typically require at least continuity of the drift or one-sided Lipschitz conditions; counter-examples to uniqueness are known for discontinuous measurable drifts. The manuscript must supply the specific argument or reference establishing this step under the low-regularity hypothesis, or the downstream semigroup and ergodicity results remain conditional.
minor comments (1)
  1. The abstract is densely packed with multiple distinct claims; separating the statements for the general Lévy case from the α-stable case, and clearly indicating which results require smoothness of B, would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comment on our manuscript. We address the point directly below and will revise the paper to strengthen the presentation of the well-posedness result.

read point-by-point responses

  1. Referee: [Abstract] Abstract (claims on weak solutions): The assertion of existence and uniqueness of weak solutions for the SDE with merely measurable drift B and rotationally invariant α-stable driving noise (α ∈ (1,2)) is load-bearing for every subsequent claim on strong Feller property, spectral gaps, densities, and exponential ergodicity. Standard results on weak uniqueness for Lévy-driven SDEs in finite dimensions typically require at least continuity of the drift or one-sided Lipschitz conditions; counter-examples to uniqueness are known for discontinuous measurable drifts. The manuscript must supply the specific argument or reference establishing this step under the low-regularity hypothesis, or the downstream semigroup and ergodicity results remain conditional.

    Authors: We agree that the existence and uniqueness of weak solutions under merely measurable B is foundational and that the manuscript must make the supporting argument fully explicit. While the abstract states that we prove this result for rotationally invariant α-stable noise with α ∈ (1,2), we acknowledge that the current presentation may not sufficiently detail or isolate the argument for the reader. We will revise the manuscript to include a dedicated subsection (or expanded paragraph in the well-posedness section) that sketches the existence proof via tightness and Skorokhod representation together with the uniqueness argument that exploits the specific Fourier properties and non-local generator of the rotationally invariant α-stable process (α > 1). We will also add relevant references to related results on Lévy-driven SDEs with irregular coefficients. This revision will ensure all subsequent claims on the strong Feller property, spectral gaps, densities, and ergodicity are clearly conditional on a rigorously established well-posedness step. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on independent existence proofs

full rationale

The paper establishes existence and uniqueness of weak solutions for the kinetic SDE under measurable drift B and rotationally invariant alpha-stable noise (alpha in (1,2)), then derives strong Feller, irreducibility, spectral gap, and ergodicity properties from that well-posedness. No self-definitional loops appear (e.g., no quantity defined in terms of itself), no fitted parameters are relabeled as predictions, and no load-bearing self-citations or imported uniqueness theorems reduce the central arguments to prior author work. The derivation chain is self-contained against external benchmarks: the low-regularity framework is explicitly stated as an assumption, and the subsequent semigroup and spectral results follow from the claimed well-posedness rather than presupposing the target conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard assumptions from stochastic analysis for Lévy-driven SDEs plus the explicit low-regularity condition on the drift.

axioms (2)
  • domain assumption Drift B is measurable (Borel) but not necessarily continuous
    Low-regularity framework invoked throughout abstract for all semigroup properties.
  • standard math Lévy process is pure-jump with suitable characteristics
    Assumed for the driving noise in the SDE definition.

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