{"id":"8cf2b671-98dc-4777-b5c3-e69026ffa62a","arxiv_id":"2412.19029","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Cesàro eventual continuity together with a uniform lower-bound condition is shown to be exactly equivalent to weak-* mean ergodicity for Markov-Feller semigroups on Polish spaces.","lead":"This mathematics paper shows that a very weak regularity condition, called Cesàro eventual continuity, is enough to force the time-averaged behavior of a Markov-Feller semigroup to converge to an ergodic steady state from every point in that state's support. It also gives an exact two-condition characterization of unique mean ergodicity and applies the theory to jump processes, a stochastic turbulence model, and a noisy Lorenz system.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.2 invokes a pointwise Birkhoff theorem for jointly measurable semigroups, a hypothesis absent from the standing assumptions; this propagates to Corollary 3.7, Proposition 3.10, and the central Theorem 3.12.","rationale":"The reader's weakest_assumption identifies exactly the load-bearing gap I find: the use of a joint-measurability-dependent Birkhoff theorem in Theorem 3.2. This is the most serious issue because Theorem 3.2 is the engine behind the EMDS property (Corollary 3.7), the uniqueness criterion (Proposition 3.10), and therefore the (ii)⇒(i) part of the central equivalence Theorem 3.12. Without pointwise Cesàro convergence on a full-measure set for ergodic measures, the argument that every x in supp μ has Q_t(x, ·) converging to μ cannot go through. The gap is not a counterexample, and the paper's examples and computations appear coherent, so the appropriate verdict remains CONDITIONAL: the claims are plausible and the proofs are mostly complete, but the standing assumptions must be augmented (or a new proof supplied) before Theorem 3.2 and its consequences are fully rigorous. The Baire-category issue in Theorem 3.4 is secondary because Theorem 3.4 contributes the Cesàro e-property on interiors, which is not used to prove Theorem 3.12. I therefore agree with the reader's verdict and do not call for a change.","tokens_in":45407,"tokens_out":7794,"duration_ms":187763,"concrete_test":"Open [58, Theorem 4.4] and confirm whether its hypotheses include 'regular jointly measurable Markov semigroup'. Then check whether the paper anywhere proves that a Cesàro eventually continuous Markov–Feller semigroup (Definitions 2.1–2.3) is necessarily regular jointly measurable, or at least that the map (t,x) ↦ P_t(x,A) is jointly measurable for Borel A. As a practical check, verify joint measurability explicitly for the IFS Example 4.1 and the Hopf model Example 5.1; if these examples satisfy it, adding the hypothesis is mild, but it must be stated. If the citation indeed requires joint measurability and no derivation is supplied, Theorem 3.2 is incomplete as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 3.2 uses [58, Theorem 4.4] to obtain a full-measure set A on which the Cesàro averages Q_t f(y) converge to ⟨f, μ⟩ for every f ∈ L_b(X). However, [58] is the Worm–Hille ergodic decomposition paper for 'regular jointly measurable Markov semigroups', and the paper's standing assumptions (Definitions 2.1–2.3) only define Markov–Feller semigroups without any joint measurability or stochastic continuity condition. The paper itself treats joint measurability as an extra hypothesis when it introduces 'regular jointly measurable' semigroups at the start of Section 3.3, so it cannot be assumed silently in Theorem 3.2. Moreover, the very 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 stated Feller property. Because Corollary 3.7 (EMDS) and Proposition 3.10 (unique ergodicity under (C2)) both rely on Theorem 3.2, the (ii)⇒(i) direction of Theorem 3.12 — the paper's headline equivalence — is not established for general Markov–Feller semigroups as stated. This is a missing hypothesis rather than a numerical or logical contradiction: the examples in Sections 4 and 5 may well be jointly measurable, but the paper needs to say so and to add the hypothesis or prove it. The Baire-category threshold issue in Theorem 3.4 is real but repairable and less central, since Theorem 3.4 is not needed for the main uniqueness equivalence.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":45704,"tokens_out":8051,"duration_ms":80731,"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":[{"comment":"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.","section":"Section 3.1, proof of Theorem 3.2"},{"comment":"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.","section":"Section 6.2, proof of Theorem 3.4, Step 2"},{"comment":"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.","section":"Section 3.2, proof of Theorem 3.12"}],"minor_comments":[{"comment":"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.","section":"Abstract and throughout"},{"comment":"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.","section":"Proof of Theorem 3.12"},{"comment":"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'.","section":"Proposition 5.5 and Example 5.7"},{"comment":"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.","section":"Theorem 3.21"}],"recommendation":"major_revision","confidential_remarks":"The main issue is concentration of hypotheses: Theorem 3.2 and its consequences silently assume joint measurability that is not part of Definitions 2.1–2.3. This is repairable by adding the assumption or proving it for the examples, but it is load-bearing, so a major revision is appropriate. The authors should also double-check the applicability of [51, Corollary 5.3] before resubmission. If the repairs are made, the paper would be a solid contribution to the ergodic theory of non-equicontinuous Markov semigroups."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Overall: worth a serious referee, but the central line has a fixable missing hypothesis. The paper really does extend the regularity ladder: Theorem 3.12 gives a clean necessary-and-sufficient condition for weak-* mean ergodicity, Theorem 3.2 gives Cesàro convergence from the support of an ergodic measure, and the IFS example with internal randomness is genuinely non-equicontinuous. The proofs are mostly careful, and the examples are worked out in enough detail to check.\n\nThe soft spot: Theorem 3.2 uses [58, Theorem 4.4], a pointwise ergodic theorem for regular jointly measurable Markov semigroups. Joint measurability in time is not among Definitions 2.1–2.3, and the paper itself treats it as an extra hypothesis when it defines 'regular jointly measurable' semigroups in Section 3.3. Without it, the full-measure convergence set A does not come from the standing assumptions. That missing hypothesis propagates: Corollary 3.7, Proposition 3.10, and Theorem 3.12(ii)⇒(i) all lean on it. The examples likely satisfy joint measurability, but the paper needs to say so, and the main theorems should state the assumption explicitly or prove it from the Feller property, which is not generally possible. This is the main thing I would ask the authors to fix.\n\nThere is also a small gap in Theorem 3.4, Step 2: Y = ∪ Y_n does not follow from a limsup ≤ ε/2 bound; you need to shrink the threshold. That is a standard repair. Also, the (i)⇒(ii) direction of Theorem 3.12 cites [51, Corollary 5.3] from the e-property world; as written the hypotheses don't obviously match, and the dependency on the unpublished companion [16] for Theorem 3.16 is fragile. These are less serious than the joint-measurability point.\n\nIf I were the editor I would send it to a referee, and the referee should ask for the joint-measurability fix and the Baire repair. The core ideas are sound and the paper is worth having in the literature once the hypotheses are aligned.","headline":"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.","tokens_in":46388,"tokens_out":3565,"would_cite":false,"duration_ms":34659,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J25","37A30"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Markov–Feller semigroup","Cesàro eventual continuity","lower bound condition","e-property","weak-* mean ergodicity","ergodic decomposition","iterated function systems","stochastic Lorenz system"],"falsifier":"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.","tokens_in":45030,"feed_emoji":"📈","tokens_out":8445,"duration_ms":72808,"temperature":0.7,"pith_summary":"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.","feed_headline":"Weak Cesaro continuity characterizes unique mean ergodicity","feed_subtitle":"A Markov semigroup averages to a unique invariant measure exactly when this weak regularity and a uniform lower bound hold.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the pointwise Birkhoff-type ergodic theorem for regular jointly measurable Markov semigroups used to prove Theorem 3.2.","marker":"[58, Theorem 4.4]"},{"why":"Introduces eventual continuity and gives the existence-of-invariant-measure criterion by lower bound that Proposition 3.8 extends to continuous time.","marker":"[15, Theorem 1.6]"},{"why":"Establishes the equivalence between asymptotic stability and eventual continuity plus a uniform lower bound, used in Theorem 3.16.","marker":"[16, Theorem 1]"},{"why":"Provides the e-property unique-ergodicity criterion with lower bound (C2) that Proposition 3.10 generalizes to Cesàro eventual continuity.","marker":"[30, Theorem 1]"},{"why":"Supplies the lower bound technique and contradiction construction used in the proof of Proposition 3.8.","marker":"[37, Theorem 3.1]"},{"why":"Gives the necessity direction of Theorem 3.12, that weak-* mean ergodicity implies Cesàro eventual continuity and the lower bound condition (C3).","marker":"[51, Corollary 5.3]"},{"why":"Provides the ergodic-decomposition-with-weak-concentration template used in Theorem 3.21.","marker":"[51, Theorem 3.8]"},{"why":"Yields the stochastic Lorenz ergodic measures and controllability estimate used to show failure of EMDS and Cesàro eventual continuity.","marker":"[7, Theorem 1.1 and Proposition 3.3]"},{"why":"Provides the iterated function system with jumps whose non-equicontinuous modifications are treated in Section 4.","marker":"[4]"}],"fun_headline_variants":["Cesàro continuity with lower bound determines unique ergodicity","Unique ergodicity from Cesàro continuity and lower bound","Weak Cesàro regularity pinpoints unique invariant measure","Mean ergodicity characterized by weak Cesàro condition","Cesàro averages lock onto unique measure under mild bound"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Cesàro continuity with lower bound determines unique ergodicity","Unique ergodicity from Cesàro continuity and lower bound","Weak Cesàro regularity pinpoints unique invariant measure","Mean ergodicity characterized by weak Cesàro condition","Cesàro averages lock onto unique measure under mild bound"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000852,"raw_usage":{"total_tokens":3700,"prompt_tokens":941,"completion_tokens":2759,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":557,"completion_tokens_details":{"reasoning_tokens":2677}},"tokens_in":557,"tokens_out":2759,"duration_ms":21352,"temperature":1.0,"reasoning_tokens":2677,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T01:00:47.352833+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Bessaih, R","cited_arxiv_id":null,"evidence_quote":"Provides the iterated function system with jumps whose non-equicontinuous modifications are treated in Section 4."}],"review_version":1}