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Ergodicity for eventually continuous Markov--Feller semigroups on Polish spaces

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Pith's one-line read A Markov–Feller semigroup on a Polish space is weakly-* mean ergodic with a unique invariant measure exactly when it is Cesàro eventually continuous and satisfies a uniform lower bound condition.

desk verdict Solid new results on Cesàro eventual continuity, but Theorem 3.2 and the main equivalence silently assume joint measurability that is not in the standing hypotheses. read the letter →

arxiv 2412.19029 v1 pith:SLTDXKD3 submitted 2024-12-26 math.PR

Fuzhou Gong , Yong Liu , Yuan Liu , Ziyu Liu This is my paper · ORCID
classification math.PR MSC 60J2537A30
keywords Markov–FellersemigroupCesàroeventualcontinuitylowerboundconditione-propertyweak-*meanergodicityergodicdecompositioniteratedfunctionsystemsstochasticLorenzsystem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Long-run behavior of Markov processes is usually studied under strong regularity assumptions such as the e-property or strong Feller condition. This paper shows that a much weaker requirement, Cesàro eventual continuity, already controls Cesàro averages: under it alone, starting from any point in the support of an ergodic invariant measure, the time averages converge weakly to that measure. The paper then proves an exact equivalence: a Cesàro eventually continuous Markov–Feller semigroup is weakly-* mean ergodic with a unique invariant measure if and only if a uniform lower bound condition holds, meaning every starting point's Cesàro averages eventually put uniformly positive mass near some fixed state. This matters because Cesàro eventual continuity is necessary for mean ergodicity and is checkable in non-equicontinuous models, including iterated function systems with jumps and noisy versions of Hopf's turbulence and Lorenz systems.

What carries the argument

The central object is the Cesàro average Q_t(x,·) = (1/t)∫_0^t P_s δ_x ds, together with the set T of states x for which {Q_t(x,·)}_{t≥0} is tight. Cesàro eventual continuity, defined by limsup_{x→z} limsup_{t→∞} |Q_t f(x) − Q_t f(z)| = 0 for every bounded Lipschitz f, expresses that nearby starting points have time averages that become close in the long run, without any uniformity over time. The lower bound conditions (C1)–(C3) are quantitative estimates requiring limsup_{t→∞} Q_t(x, B(z,ε)) > 0, with C3 making the positivity uniform in x. The argument works by showing that T is closed, that Cesàro averages starting in T converge to invariant measures, and that the assignment x ↦ ε_x is continuous; this continuity drives the ergodic decomposition and the contradiction arguments that separate invariant measures.

What would settle it

Exhibit a Cesàro eventually continuous Markov–Feller semigroup on a Polish space that satisfies condition (C3) but admits two distinct invariant measures; such an example would refute the paper's central equivalence in Theorem 3.12.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that Cesàro eventual continuity is the right weak regularity condition under which Cesàro averages converge to ergodic measures. Theorem 3.2 states: if {P_t} is Cesàro eventually continuous and admits an ergodic invariant measure μ, then for every x in supp μ the Cesàro averages Q_t(x,·) converge weakly to μ. A direct corollary is the EMDS property, that distinct ergodic measures have disjoint supports, with no e-property assumed. The sharpest result is Theorem 3.12, which characterizes weak-* mean ergodicity with a unique invariant measure as precisely Cesàro eventual continuity together with the uniform lower bound condition (C3): inf_{x∈X} limsup_{t→∞} Q_t(x, B(z,ε)) > 0 for every ε>0. Under additional stochastic continuity, the paper shows that Cesàro eventual continuity upgrades to the Cesàro e-property on the interior of the support of any ergodic measure, so the three notions coincide on such supports.

Load-bearing premise

The load-bearing premise is that the semigroup is Cesàro eventually continuous, and in the proof of the main convergence theorem the argument also quietly assumes a joint measurability regularity that is not listed among the standing definitions.

Editorial extensions

If this is right

  • For any ergodic invariant measure μ and any x in its support, the Cesàro averages Q_t(x,·) converge weakly to μ; hence the semigroup has the EMDS property under Cesàro eventual continuity alone.
  • A semigroup is weakly-* mean ergodic with a unique invariant measure if and only if it is Cesàro eventually continuous and satisfies (C3), the uniform lower bound condition.
  • If the semigroup is stochastically continuous and Cesàro eventually continuous and some ergodic measure has nonempty interior support, then the Cesàro e-property holds on that interior; on the support, weak-* mean ergodicity, the Cesàro e-property, and Cesàro eventual continuity are equivalent.
  • Cesàro eventual continuity plus the pointwise lower bound condition (C2) yields a unique invariant measure and Cesàro convergence for every initial distribution supported on T.
  • The criteria are verified on non-equicontinuous examples: an iterated function system with jumps, and stochastic versions of Hopf's turbulence model and the Lorenz system, where the e-property is known to fail or is difficult to check.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The proof of Theorem 3.2 uses a pointwise Birkhoff ergodic theorem for regular jointly measurable Markov semigroups, yet joint measurability is not among the paper's standing assumptions; if this regularity is genuinely needed, the theorem as stated holds only for that narrower class unless the argument is repaired.
  • The Baire-category step in Theorem 3.4's proof concludes eventual uniform control from a limsup bound, which is not valid as written; using a smaller threshold should repair the proof, but the Cesàro e-property conclusion currently rests on this gap.
  • The characterization suggests a practical route to mean ergodicity in high-dimensional or degenerate-noise models: verify Cesàro eventual continuity and a uniform lower bound rather than gradient estimates or coupling, a route the paper's own examples begin to map.
  • The Hopf and Lorenz examples exhibit parameter-dependent transitions in Cesàro eventual continuity; a similar phase transition may hold for other chaotic or turbulent stochastic systems, where ergodic measures multiply exactly when this weak regularity fails.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper studies ergodicity of Markov–Feller semigroups on Polish spaces under a weak regularity condition, Cesàro eventual continuity. The main theoretical results are: (a) Cesàro eventual continuity forces the support of an ergodic invariant measure to lie in T and forces convergence of the Cesàro averages from each point of that support (Theorem 3.2); (b) lower-bound conditions (C1), (C2), (C3) give existence, uniqueness, and weak-* mean ergodicity, respectively (Propositions 3.8, 3.10, Theorem 3.12); (c) under stochastic continuity, the Cesàro e-property holds on the interior of the support of an ergodic measure (Theorem 3.4); and (d) refined ergodic decomposition results are obtained in Section 3.3 for regular jointly measurable semigroups. Applications include iterated function systems with jumps, a stochastic Hopf model, and a stochastic Lorenz system; examples are given both with and without Cesàro eventual continuity.

Significance. If established, the equivalence in Theorem 3.12 is a strong and useful characterization: unique weak-* mean ergodicity is reduced to one pointwise regularity property plus a uniform lower bound. The paper also provides genuinely non-equicontinuous examples, and the contrast between Cesàro eventual continuity and the Cesàro e-property is made concrete. The main caveat is that the proof of the key convergence theorem (Theorem 3.2) invokes a pointwise ergodic theorem for jointly measurable semigroups although joint measurability is not among the standing assumptions; until this is repaired, the central equivalence is conditional. The examples in Sections 4 and 5 are worked out in enough detail to be checkable, which is a definite strength.

major comments (3)
  1. [Section 3.1, proof of Theorem 3.2] The proof invokes [58, Theorem 4.4] to obtain a full-measure set A on which lim_{t→∞} Q_t f(y) = ⟨f, μ⟩ for every f ∈ L_b(X). That theorem is stated for regular jointly measurable Markov semigroups, but joint measurability is not among the standing assumptions in Definitions 2.1–2.3; the paper itself treats 'regular jointly measurable' as an extra hypothesis at the start of Section 3.3. Moreover, the definition Q_t f(x) = (1/t)∫_0^t P_s f(x) ds already requires some measurability of s ↦ P_s f(x), which is not guaranteed by the Feller property as defined. Because Corollary 3.7, Proposition 3.10, and the (ii)⇒(i) direction of Theorem 3.12 all rely on Theorem 3.2, the headline equivalence is not established for general Markov–Feller semigroups as stated. Please add joint measurability (or a weaker sufficient measurability condition) to the standing assumptions, or prove it from the Feller property, or replace the invocation of [58, Theorem 4.4] with a pointwise ergodic theorem valid under the stated hypotheses.
  2. [Section 6.2, proof of Theorem 3.4, Step 2] The statement 'Y = ∪_n Y_n' with Y_n = {x ∈ Y : |Q_t f(x) − Q_t f(x_0)| ≤ ε/2 for all t ≥ n} does not follow from the Cesàro eventual continuity bound lim sup_{t→∞} |Q_t f(x) − Q_t f(x_0)| ≤ ε/2. A limsup bound does not imply eventual uniform boundedness at the same threshold. This is repairable by applying Cesàro eventual continuity with a smaller threshold (e.g., ε/4) so that the limsup is below ε/2 and the sets Y_n cover Y. The gap does not by itself invalidate Theorem 3.12, but Theorem 3.4 as stated needs this fix.
  3. [Section 3.2, proof of Theorem 3.12] The implication (i)⇒(ii) is delegated to [51, Corollary 5.3], but [51] is a paper on Markov semigroups with the e-property, and a weakly-* mean ergodic semigroup need not have the e-property (the paper's own Example 2.12 is weakly-* mean ergodic but fails the Cesàro e-property). Please verify the precise hypotheses of [51, Corollary 5.3] and justify its applicability, or supply a direct proof of (C3) from weak-* mean ergodicity: for z in the support of the unique invariant measure μ, weak convergence of Q_t δ_x to μ gives liminf_{t→∞} Q_t(x, B(z, ε)) ≥ μ(B(z, ε)) > 0, which yields the uniform lower bound after using Cesàro eventual continuity.
minor comments (4)
  1. [Abstract and throughout] There are numerous typos and grammatical slips, including 'serval' for 'several', 'it is showed' for 'it is shown', 'semigruop' in the keywords, 'Classifiction' for 'Classification', and 'Ces` ro' for 'Cesàro' in Proposition 2.11. These should be corrected.
  2. [Proof of Theorem 3.12] The proof refers to 'Theorem 3.10', but the intended reference appears to be Proposition 3.10; there is no Theorem 3.10 in the manuscript.
  3. [Proposition 5.5 and Example 5.7] The phrase 'does not satisfy non-Ces`aro eventual continuity' appears to be a double negative; it should read 'does not satisfy Cesàro eventual continuity' or 'is not Cesàro eventually continuous'.
  4. [Theorem 3.21] The symbol X is reused for the set ∪_{μ∈Perg} supp μ ∩ K, which clashes with the already-defined state space X; a different symbol (for example E) would improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the headline equivalence is proved in the paper; only auxiliary self-citations (one to an unpublished same-author paper) appear, and the main caveat is an unstated joint-measurability hypothesis, not a circular reduction.

full rationale

The central derivation chain is not circular. Theorem 3.12's (ii)->(i) direction is proved in the paper via Proposition 3.10, which relies on Theorem 3.2, Corollary 3.7, and Lemma 2.8; these are argued from the standing Cesaro eventual continuity assumption rather than imported from a same-author uniqueness theorem. The pointwise Birkhoff input in Theorem 3.2 is cited as [58, Theorem 4.4] (Worm-Hille, external to the authors); the paper even treats 'regular jointly measurable' semigroups as an extra hypothesis in Section 3.3, showing that the standing assumptions do not include it. That is a missing-hypothesis gap propagating to Corollary 3.7, Proposition 3.10, and Theorem 3.12 as stated, but it is a correctness issue, not a circular reduction. The same-author citations are supporting, not forcing: [15, Lemma 4.1] and [40, Lemma 3.7] are used as standard support-invariance tools in Proposition 3.5 and Theorem 3.4, and [16, Theorem 1] supplies one direction of the secondary Theorem 3.16; the latter is to appear and is a load-bearing self-citation for that side theorem, but the converse is proved here and the headline ergodicity equivalence does not depend on it. No fitted parameter is renamed as a prediction, no ansatz is smuggled in via citation, and no equation reduces to its inputs by construction. The Baire-category threshold step in Theorem 3.4's proof (passing from limsup <= epsilon/2 to eventual uniform <= epsilon/2) is a genuine but repairable gap and is again a correctness concern, not circularity. Overall, the paper's main results are self-contained against external benchmarks once the joint-measurability hypothesis is either added to Theorem 3.2 or proved for the standing assumptions.

Assumptions & free parameters 0 free parameters · 7 assumptions · 2 invented entities

The central results rest on the standing Markov-Feller assumption, on imported theorems from the e-property literature ([30, 51, 56, 57, 58]) and from the authors' own prior work ([15, 16, 40]), and on one explicitly added assumption (joint measurability in Section 3.3). No free parameters are fitted to data; all constants in the examples are model inputs. No physical entities are invented; the new objects (T, Φ, ε_x) are definitions whose properties are proven. The main fragility is that some imported theorems are applied to semigroups that only satisfy the weaker condition of Cesàro eventual continuity, with the hypothesis checks left implicit: [51, Corollary 5.3] in Theorem 3.12 and [58, Theorem 4.4] in Theorem 3.2. The unpublished companion result [16] is load-bearing for one direction of Theorem 3.16.

assumptions (7)
  • domain assumption Standing assumption: {P_t} is a Markov-Feller semigroup on a Polish space, with each P_t regular (adjoint on B_b, Definition 2.2).
    Framework for all results; see Definitions 2.1-2.3 and the paragraph after Proposition 2.6.
  • standard math Pointwise Birkhoff-type ergodic theorem [58, Theorem 4.4] for regular Markov semigroups, used in the proof of Theorem 3.2 to get Q_t f(y) → ⟨f, μ⟩ for μ-a.e. y.
    Invoked in Section 6.2 before the joint-measurability assumption of Section 3.3 is stated, so its hypotheses are not transparently satisfied.
  • standard math External lemmas from the e-process literature: [30, Lemmas 1 and 2] (tightness and Cesàro-semigroup commutation) and [15, Lemma 4.1] (invariance of ergodic supports).
    Used throughout Section 6, notably in Lemma 2.8 and Proposition 3.1; standard in this literature but not reproved here.
  • standard math External results [56, Theorem 2.3.24] (metric completeness of weak topology), [4, Proposition 3] (Poisson jump estimate (6.47)), and [7, Proposition 3.3] (control of the Lorenz system).
    Used respectively in Lemma 3.18, Proposition 4.3, and Corollary 5.10; all are published results.
  • ad hoc to paper Implication (i)⇒(ii) of Theorem 3.12 is delegated to [51, Corollary 5.3], a result for Markov semigroups with the e-property.
    The present semigroup is only assumed Cesàro eventually continuous, so the cited result's hypotheses are not verified; a direct Lipschitz-approximation argument is absent.
  • domain assumption Theorem 3.16(i)⇒(ii) rests on [16, Theorem 1], a companion paper by the same four authors that is 'to appear' and not independently checkable.
    The equivalence as stated depends on this unpublished result; the (iii)⇒(i) direction is proven in full in the present text.
  • domain assumption Joint measurability of (t, x) → P_t f(x) for f in B_b, assumed explicitly at the start of Section 3.3 rather than proved.
    The paper states 'we shall assume that {P_t} is a regular jointly measurable Markov-Feller semigroup' as the standing assumption for Section 3.3; it is an added hypothesis.
invented entities (2)
  • The set T = {x in X : {Q_t(x,·)}_{t≥0} is tight} independent evidence
    purpose: Identifies the initial states from which Cesàro averages are precompact; the main object of Theorem 1.1 and of the support theory in Section 3.1.
    Defined in (1.1). Its properties (closedness, invariance of supports, convergence of Q_t on T) are proved in Lemmas 2.7 and 2.8 and Proposition 3.1 rather than assumed.
  • The map Φ : T → P(X), x ↦ ε_x, where ε_x is the weak limit of Q_t(x,·) independent evidence
    purpose: Organizes the ergodic decomposition: equivalence classes [x], the set T_erg, and a continuous surjection from T to the ergodic measures.
    Defined in Section 3.3 after Lemma 2.8; continuity is proved in Lemma 3.18 and the fibers/classes are characterized in Proposition 3.19. These are mathematical constructions with theorems attached, not postulated physical entities.

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Pith. "Pith review of Ergodicity for eventually continuous Markov--Feller semigroups on Polish spaces." pith.science (2026). https://pith.science/paper/SLTDXKD3

@misc{pith2026241219029,
  author       = {Pith},
  title        = {Pith review of: Ergodicity for eventually continuous Markov--Feller semigroups on Polish spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SLTDXKD3}},
  note         = {Machine review of arXiv:2412.19029}
}
read the original abstract

This paper investigates the ergodicity of Markov--Feller semigroups on Polish spaces, focusing on very weak regularity conditions, particularly the Ces\`aro eventual continuity. First, it is showed that the Ces\`aro average of such semigroups weakly converges to an ergodic measure when starting from its support. This leads to a characterization of the relationship between Ces\`aro eventual continuity, Ces\`aro e-property, and weak-* mean ergodicity. Next, serval criteria are provided for the existence and uniqueness of invariant measures via Ces\`aro eventual continuity and lower bound conditions, establishing an equivalence relation between weak-* mean ergodicity and a lower bound condition. Additionally, some refined properties of ergodic decomposition are derived. Finally, the results are applied to several non-trivial examples, including iterated function systems, Hopf's turbulence model with random forces, and Lorenz system with noisy perturbations, either with or without Ces\`aro eventual continuity.

Figures

Figures reproduced from arXiv: 2412.19029 by the authors.

Figure 1
Figure 1. Ergodic decomposition of Example 3.23 4 Ergodicity for iterated function systems with jumps In this section, we present a class of iterated function systems, for which the results estab￾lished in the previous section can be applied. In particular, several specific cases are either non-equicontinuous or difficult to verify the (Ces`aro) e-property. Before going through our models in Sections 4.2 and 4.3, let us intro… view at source ↗

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