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arxiv: 2603.00773 · v2 · submitted 2026-02-28 · 🧮 math.PR

Long-time L^p Wasserstein contraction for diffusion processes without global dissipativity

Pierre Monmarch\'e This is my paper

Pith reviewed 2026-05-15 18:00 UTC · model grok-4.3

classification 🧮 math.PR MSC 60H1060J60
keywords Wasserstein distancediffusion processesMarkov semigroupsynchronous couplingFeynman-Kac operatornon-ellipticlong-time contraction
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The pith

Diffusion processes contract L^p Wasserstein distances in long time even without global dissipativity when the maximal eigenvalue of an associated Feynman-Kac operator is negative.

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

The paper establishes that Markov diffusion semigroups on R^d can contract the L^p Wasserstein distance for large times along synchronous couplings, even when the drift fails to be globally dissipative. This extends earlier results that were restricted to elliptic processes with sufficiently large diffusivity coefficients. The key new conditions are sharper and include negative results showing when contraction fails, plus explicit lower bounds. In one dimension the contraction holds if and only if the maximal eigenvalue of a certain Feynman-Kac operator is negative. Such long-time contraction yields uniform stability estimates that remain valid for numerical approximations and other perturbations.

Core claim

For a Markov diffusion semigroup on R^d the L^p Wasserstein distance contracts in the long-time limit along suitable synchronous couplings without any global dissipativity assumption on the drift, provided the coefficients satisfy conditions that make the maximal eigenvalue of the associated Feynman-Kac operator negative; the result covers non-elliptic cases and supplies a complete characterization at least in dimension one.

What carries the argument

Synchronous coupling of two copies of the diffusion, whose distance process is controlled by the sign of the maximal eigenvalue of the Feynman-Kac operator built from the drift and diffusion coefficients.

If this is right

  • Uniform-in-time stability estimates for numerical discretizations of the SDE hold under these weaker coefficient conditions.
  • Long-time behavior of the process remains stable in Wasserstein distance without requiring strong global dissipativity.
  • Negative results delimit the precise boundary between contraction and failure, including explicit lower bounds on the contraction rate.
  • In dimension one the eigenvalue condition is necessary as well as sufficient.

Where Pith is reading between the lines

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

  • The same eigenvalue test could be used to decide long-time stability for SDEs arising in applications where global dissipativity is unrealistic.
  • The approach might extend to other metrics or to processes with jumps once the corresponding Feynman-Kac operator is identified.
  • Numerical computation of the eigenvalue for concrete drifts would give immediate, verifiable predictions for contraction or its absence.

Load-bearing premise

Suitable synchronous couplings exist and the drift-diffusion coefficients satisfy the stated conditions that let the contraction be governed by the sign of the Feynman-Kac eigenvalue.

What would settle it

A concrete one-dimensional diffusion whose associated Feynman-Kac operator has positive maximal eigenvalue yet whose L^p Wasserstein distance still contracts (or conversely, negative eigenvalue yet no contraction).

Figures

Figures reproduced from arXiv: 2603.00773 by Pierre Monmarch\'e.

Figure 1
Figure 1. Figure 1: Top: potentials U1 (left) and U2 (right). Bottom: estimation of J (pη)/p for (23) (left with U1, right with U2), as a function of θ 2 ∈ [0.1, 2] and p ∈ [1, 3] (on the right graph, the color bar is capped in [−4, 4]). 2.3 A constructive non-asymptotic criterion The condition that J (pη) < 0 to get a Wp-contraction thanks to Theorem 6 is conceptually interesting but difficult to check in practice. In this s… view at source ↗
read the original abstract

The fact that a Markov diffusion semi-group on $\mathbb R^d$ contracts the $L^p$ Wasserstein distance, which has been extensively used to establish uniform-in-time stability estimates (e.g. with respect to numerical discretization errors), is a well-studied question in the case where the distances are in fact deterministically contracted by the drift (global dissipativity condition) or in the case $p=1$ (with reflection couplings). This work focuses on the non-globally dissipative case with $p>1$. This situation was previously considered in \cite{MonmarcheBruit}, but only for elliptic processes, and with a restriction on the diffusivity coefficient (which had to be large enough). Here, we extend this analysis to non-elliptic processes and provide sharper conditions to get contractions along synchronous coupling, including negative results, lower bounds and a characterization (at least in dimension 1) in terms of the maximal eigenvalue of a Feynman-Kac operator.

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

2 major / 2 minor

Summary. The manuscript establishes long-time L^p Wasserstein contractions for Markov diffusion semigroups on R^d in the absence of global dissipativity of the drift. It extends prior elliptic results to non-elliptic processes via synchronous couplings, supplies sharper coefficient conditions for contraction, includes negative results and lower bounds, and provides (at least in dimension 1) a characterization of the contraction threshold in terms of the maximal eigenvalue of an associated Feynman-Kac operator.

Significance. If the derivations hold, the work supplies sharper, more applicable criteria for uniform-in-time stability estimates of diffusions beyond the globally dissipative regime, directly relevant to numerical discretization error bounds and ergodicity analysis. The inclusion of negative results together with the eigenvalue characterization strengthens the sharpness of the theory and supplies falsifiable predictions.

major comments (2)
  1. [§3] §3 (synchronous coupling construction): the argument that the quadratic variation term remains controlled when the diffusion matrix degenerates on a positive-measure set relies on the stated drift-diffusion conditions bounding the generator applied to |x-y|^p; however, this does not automatically guarantee that a pathwise synchronous coupling can be realized on the degeneracy set without an additional uniform controllability assumption on the support of the diffusion. If this gap is not closed, the long-time contraction claim for non-elliptic cases fails to hold pathwise.
  2. [§5] §5 (1D Feynman-Kac characterization): the maximal-eigenvalue criterion is derived under non-degenerate assumptions on the diffusion coefficient; the manuscript must verify whether the same spectral characterization transfers directly to the degenerate case or requires a modified operator, since the negative-result examples in higher dimensions may indicate that the 1D reduction does not capture the full degeneracy effect.
minor comments (2)
  1. [§2] Notation for the degeneracy set of the diffusion matrix should be introduced explicitly in §2 and used consistently when stating the coefficient conditions.
  2. [Abstract] The abstract claims 'sharper conditions' relative to MonmarcheBruit; a one-sentence comparison of the new threshold with the previous diffusivity lower bound would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below with point-by-point responses, providing clarifications where needed and indicating revisions to strengthen the presentation.

read point-by-point responses

  1. Referee: [§3] §3 (synchronous coupling construction): the argument that the quadratic variation term remains controlled when the diffusion matrix degenerates on a positive-measure set relies on the stated drift-diffusion conditions bounding the generator applied to |x-y|^p; however, this does not automatically guarantee that a pathwise synchronous coupling can be realized on the degeneracy set without an additional uniform controllability assumption on the support of the diffusion. If this gap is not closed, the long-time contraction claim for non-elliptic cases fails to hold pathwise.

    Authors: We appreciate the referee's observation on the synchronous coupling construction. The L^p-Wasserstein distance is controlled through the evolution of the expectation E[|X_t - Y_t|^p]. Applying Itô's formula to |x-y|^p yields a drift term bounded by the generator under our coefficient assumptions, while the quadratic variation term is nonnegative and can be discarded to obtain an upper bound. The synchronous coupling is realized by driving both processes with the same Brownian motion; degeneracy of the diffusion matrix on a set of positive measure does not invalidate this construction or the resulting differential inequality for the expectation, as the processes remain well-defined under the local Lipschitz and linear growth conditions stated in the paper. Since the contraction result concerns the Wasserstein distance (an expectation), no pathwise controllability on the degeneracy set is required. We will add a clarifying remark in §3 emphasizing that the argument relies on the generator bound for the expectation and does not claim deterministic pathwise contraction. revision: partial

  2. Referee: [§5] §5 (1D Feynman-Kac characterization): the maximal-eigenvalue criterion is derived under non-degenerate assumptions on the diffusion coefficient; the manuscript must verify whether the same spectral characterization transfers directly to the degenerate case or requires a modified operator, since the negative-result examples in higher dimensions may indicate that the 1D reduction does not capture the full degeneracy effect.

    Authors: The derivation of the maximal-eigenvalue criterion in §5 is carried out under the assumption that the diffusion coefficient is bounded away from zero, ensuring the Feynman-Kac operator is uniformly elliptic and the standard spectral theory applies directly. In the presence of degeneracy, the operator would require modification (for instance, by restricting the domain or incorporating a weighted space that accounts for vanishing diffusion). Our negative results are formulated in dimensions d ≥ 2 and illustrate the failure of contraction under certain local dissipativity conditions; they do not contradict the one-dimensional non-degenerate characterization. The coupling-based contraction results in the paper hold for non-elliptic processes without relying on the spectral criterion. We will revise §5 to state the non-degeneracy assumption explicitly and add a short discussion noting that a full extension of the eigenvalue characterization to degenerate one-dimensional diffusions lies outside the present scope. revision: yes

Circularity Check

0 steps flagged

No significant circularity; extensions rely on independent analysis of couplings and generators

full rationale

The paper extends prior elliptic results from self-cited work to non-elliptic diffusions by deriving sharper contraction conditions along synchronous couplings, including negative results and a 1D characterization via the maximal eigenvalue of a Feynman-Kac operator. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-referential definition, or load-bearing self-citation chain; the generator bounds and coupling constructions are developed from the stated drift/diffusion assumptions without tautological reduction. The derivation remains self-contained against external benchmarks such as explicit 1D examples and lower-bound constructions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on abstract only; typical background assumptions for diffusion SDEs are invoked but no new free parameters or invented entities are mentioned.

axioms (2)
  • standard math Existence and uniqueness of strong solutions to the underlying SDE
    Required for the Markov semi-group to be well-defined.
  • domain assumption The process is a Markov diffusion
    Core to the semi-group contraction question.

pith-pipeline@v0.9.0 · 5464 in / 1231 out tokens · 56677 ms · 2026-05-15T18:00:48.704530+00:00 · methodology

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