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arxiv: 2605.05986 · v1 · submitted 2026-05-07 · 🧮 math.PR

Convergence rate of the occupation measure of classes of ergodic processes toward their invariant distribution in mean Wasserstein distance

Gilles Pag\`es , Fabien Panloup This is my paper

Pith reviewed 2026-05-08 06:21 UTC · model grok-4.3

classification 🧮 math.PR
keywords occupation measuresWasserstein distanceergodic processesconvergence ratesfractional Brownian motiondiffusionsinvariant distributionconditional convergence
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The pith

Occupation measures of ergodic processes converge to invariant distributions at the same mean Wasserstein rates as stationary mixing processes under conditional equilibrium convergence.

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

The paper extends the L^p-mean Wasserstein convergence rate bounds obtained by Fournier and Guillin for stationary mixing processes to a larger class of ergodic processes. It achieves this by supplying general criteria that rely on an assumption of conditional convergence to equilibrium in total variation or Wasserstein distance, without requiring regularization. The results apply directly to occupation measures of processes that need not be stationary or Markovian, including Brownian diffusions and additive SDEs driven by fractional Brownian motion or other Gaussian processes with stationary increments. Readers care because the extension makes quantitative ergodic control available to a broader set of models that start from arbitrary initial conditions.

Core claim

Assuming conditional convergence to equilibrium in Total Variation or Wasserstein distance recovers the same L^p-mean rates of convergence in Wasserstein distance for the occupation measures of a larger class of ergodic processes, including non-stationary and non-Markovian ones, and yields explicit conditions for Brownian diffusions and additive SDEs driven by fractional Brownian motions or Gaussian processes with stationary increments.

What carries the argument

Conditional convergence to equilibrium in Total Variation or Wasserstein distance, which controls forgetting of the initial condition and extends the mixing-based proof strategy to non-stationary settings.

If this is right

  • The same rates hold for non-stationary ergodic processes.
  • Explicit conditions produce the rates for Brownian diffusions.
  • Additive SDEs driven by fractional Brownian motion or stationary-increment Gaussian processes satisfy the rates.
  • The proofs avoid regularization techniques.

Where Pith is reading between the lines

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

  • Many non-Markovian models with memory can have their long-term empirical statistics approximated at known rates starting from arbitrary initial conditions.
  • The criteria may extend to other ergodic theorems once conditional convergence can be verified for the driving noise.
  • Quantitative bounds could support error estimates in numerical simulations of processes initialized away from stationarity.

Load-bearing premise

The processes must satisfy conditional convergence to equilibrium in total variation or Wasserstein distance.

What would settle it

An ergodic process that meets the conditional convergence assumption yet whose occupation measure fails to achieve the predicted L^p-mean Wasserstein rate, or a process lacking the conditional assumption that nonetheless attains the rate.

read the original abstract

N. Fournier and A. Guillin obtained in their 2015 PTRF paper some bounds of the L^p-mean rate of convergence in Wasserstein distance of empirical distributions for a class of stationary mixing processes. In this paper, we propose to extend their strategy of proof and provide general criterions which allow to keep similar rates for a larger class of processes. These results (which do not require regularization techniques) lead to various applications to occupation measures of ergodic processes which may be not stationary or not Markovian under an assumption of {\em conditional} convergence to equilibrium in Total Variation or Wasserstein distance. We then provide explicit conditions which lead to these rates for Brownian diffusions and additive SDEs driven by fractional Brownian Motions {or by Gaussian processes with stationary increments}.

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 extends the 2015 results of Fournier and Guillin on L^p-mean rates of convergence in Wasserstein distance for empirical (occupation) measures of stationary mixing processes. It introduces general criteria based on an assumption of conditional convergence to equilibrium in total variation or Wasserstein distance (allowed to depend on the past sigma-field) that permit the same rates to hold for a larger class of ergodic processes, including non-stationary and non-Markovian ones. Explicit conditions yielding these rates are then derived for Brownian diffusions and for additive SDEs driven by fractional Brownian motion or by Gaussian processes with stationary increments.

Significance. If the central extension is correct, the work broadens the scope of mean-Wasserstein convergence rates for occupation measures beyond the stationary Markov setting without relying on regularization. The provision of explicit conditions for the applications to fBM-driven SDEs is a concrete strength that could be useful for analyzing long-time behavior of processes with memory. The paper ships a general criterion rather than case-by-case arguments, which is a positive feature.

major comments (2)
  1. [§2] §2 (general criterion, presumably the main theorem extending Fournier-Guillin): the statement that the conditional convergence assumption transfers the mean-Wasserstein rate to the occupation measure (1/n)∑_{k=1}^n δ_{X_k} must be accompanied by an explicit uniformity requirement on the conditional rate with respect to the conditioning time k (or with respect to the random initial condition). Without such uniformity, the interchange between conditional expectation and Cesàro averaging can lose the optimal rate; the current formulation appears to allow pointwise-in-k decay.
  2. [Applications section] Applications to additive SDEs driven by fBM (final section): the explicit conditions given for conditional convergence in Wasserstein distance must be checked to imply uniformity in the starting time. If the decay rate depends on the random initial sigma-field in a non-uniform way, the claimed rate for the occupation measure does not automatically follow from the general criterion.
minor comments (2)
  1. [Introduction] The abstract states that the results 'do not require regularization techniques'; this should be contrasted explicitly with the original Fournier-Guillin approach in the introduction to clarify the technical gain.
  2. [§2] Notation for the conditional convergence assumption (e.g., the random variable E[ W_p(μ_k, π) | F_0 ] or its TV analogue) should be introduced once and used consistently; several passages in the general criterion section use slightly varying phrasing.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the precise identification of a technical point regarding uniformity in the conditional rates. The comments are well-taken and highlight a subtlety in passing from conditional convergence to averaged rates. We address both major comments below and will revise the manuscript to incorporate the required uniformity assumptions explicitly.

read point-by-point responses

  1. Referee: §2 (general criterion, presumably the main theorem extending Fournier-Guillin): the statement that the conditional convergence assumption transfers the mean-Wasserstein rate to the occupation measure (1/n)∑_{k=1}^n δ_{X_k} must be accompanied by an explicit uniformity requirement on the conditional rate with respect to the conditioning time k (or with respect to the random initial condition). Without such uniformity, the interchange between conditional expectation and Cesàro averaging can lose the optimal rate; the current formulation appears to allow pointwise-in-k decay.

    Authors: We agree that an explicit uniformity condition is necessary to justify the interchange of conditional expectation and Cesàro averaging while preserving the optimal rate. Our current Assumption 2.1 formulates the conditional convergence in total variation or Wasserstein distance but does not state uniformity over the starting time k. In the revised manuscript we will strengthen the assumption to require that there exists a deterministic rate function r(n) (independent of k and of the conditioning sigma-field) such that the conditional expectation of the distance is bounded by r(n) uniformly in k. The proof of the main theorem will be updated to invoke this uniformity when bounding the averaged term, and we will add a short remark explaining why the uniformity is indispensable. revision: yes

  2. Referee: [Applications section] Applications to additive SDEs driven by fBM (final section): the explicit conditions given for conditional convergence in Wasserstein distance must be checked to imply uniformity in the starting time. If the decay rate depends on the random initial sigma-field in a non-uniform way, the claimed rate for the occupation measure does not automatically follow from the general criterion.

    Authors: We concur that the applications require verification that the derived conditional rates are uniform in the starting time. For the additive SDEs driven by fractional Brownian motion (or Gaussian processes with stationary increments), the time-homogeneity of the driving noise and the Lipschitz assumptions on the drift ensure that the Wasserstein contraction bounds obtained via the coupling argument are independent of the initial time k. Nevertheless, this uniformity is currently implicit rather than stated. In the revision we will insert a short lemma (or remark) after the derivation of the conditional rates, confirming that the bounds satisfy the uniform version of Assumption 2.1 introduced in §2. Should any parameter regime fail uniformity, we will restrict the stated conditions accordingly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; extends external 2015 result under added assumption

full rationale

The derivation extends the 2015 Fournier-Guillin PTRF result (external authors) via a new proof strategy and general criteria under an explicit added assumption of conditional convergence in TV or Wasserstein distance. No self-citations are load-bearing for the central claim, no parameters are fitted then renamed as predictions, and no definitional reductions or ansatz smuggling occur. The chain remains independent of its own outputs, with applications derived from the stated assumptions rather than by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on one domain assumption (conditional convergence) and the extension of an existing proof strategy; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption Conditional convergence to equilibrium in Total Variation or Wasserstein distance
    Invoked to extend rates to non-stationary/non-Markovian ergodic processes.

pith-pipeline@v0.9.0 · 5434 in / 1134 out tokens · 29956 ms · 2026-05-08T06:21:09.588618+00:00 · methodology

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