{"id":"ab0ebb60-a878-4342-b098-9398be2f2745","arxiv_id":"2504.13322","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Locally-balanced Markov jump processes are formalized on general state spaces and shown to be well-posed, reversible, ergodic, spectrally comparable to Metropolis-Hastings, uniformly ergodic for non-sub-Gaussian targets, and convergent to overdamped Langevin dynamics in the small-jump limit.","lead":"Locally-balanced Markov jump processes, continuous-time samplers for discrete and non-smooth probability distributions, are given a full mathematical foundation. The paper proves their stability and fast mixing, including a regime where they mix uniformly faster than the standard Langevin diffusion on heavy-tailed targets.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: central claims are supported, with only localized proof defects that do not change the theorems.","rationale":"The stress-test pass found no issue that would change the reader's conditional verdict. The central theory is internally consistent; the cited external theorem (Monmarche et al.) is plausible, and the moment controls in Appendix C are sufficient. The localized typos and incorrect formulas are worth correcting before final publication but do not affect the truth of the statements. I therefore keep the verdict UNCHANGED (still CONDITIONAL) rather than promoting to ACCEPT, because the paper should not be treated as final until the three flagged items are fixed.","tokens_in":28928,"tokens_out":34810,"duration_ms":307419,"concrete_test":"Re-derive the drift condition in Proposition 3.5: check whether (V=1, C=E, beta=0, b=1) is admissible in Down et al. (1995) Theorem 5.2(b); if not, verify uniform ergodicity using the standard Foster-Lyapunov condition with V(x)=min(E_x[h_C],K)+1 and confirm that the argument closes. This settles whether the flagged proof gap is repairable without changing the theorem.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I could not find a load-bearing flaw in the central argument. The main results—well-posedness and non-explosivity (Theorem 2.1), pi-reversibility and ergodicity (Theorems 2.1-2.2), spectral-gap equivalence and comparison (Propositions 3.2-3.3), uniform ergodicity for non-sub-Gaussian targets (Theorem 3.1), and the diffusion limit (Theorem 4.1)—have proofs whose essential steps check out. Assumption 2.2 is genuinely load-bearing, but the paper explicitly restricts to non-decreasing balancing functions (Remark 2.4), so this is a stated limitation rather than an oversight. The three issues noted by the reader are localized: the waiting-time scaling in Proposition 2.3 is a typo that does not invalidate the continuity argument; the 'trivial drift condition' in Proposition 3.5 can be made rigorous either by taking the whole space as the petite set with V=1 or by using a Foster-Lyapunov function based on expected hitting times; and the b''(0) formula in Appendix C is wrong in value, but only boundedness of b'' is used, so the conclusion stands. None of these undermines the central claims.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a mathematical foundation for locally-balanced Markov jump processes (LBMJPs) on Polish state spaces. For a target probability π and a Markov proposal kernel γ, the process has jump rate λ(x)=∫ g(t(x,y))γ(x,dy) and jump distribution proportional to g(t(x,y))γ(x,dy), where t(x,y) is the Radon-Nikodym ratio π(dy)γ(y,dx)/(π(dx)γ(x,dy)) and g is a balancing function. Under continuity and monotonicity of g (Assumption 2.2) plus irreducibility and support conditions, the paper proves λ finite, non-explosivity, π-reversibility, ergodic theorems, weak Feller property, and a weak generator. It then shows spectral-gap comparison with Metropolis-Hastings for bounded g, comparison theorems for unbounded g, uniform ergodicity on N for targets with exponential-type ratio tails such as π(n)∝exp(-n^a), a∈(1,2), and a diffusion limit to overdamped Langevin when Gaussian proposal variance σ_n^2→0 with time sped up by σ_n^{-2}. The proofs use standard tools from Markov chain theory and piecewise deterministic Markov processes, with several technical arguments placed in appendices.","tokens_in":29086,"tokens_out":13210,"duration_ms":119813,"significance":"If the results stand, this paper supplies a rigorous common framework for a class of continuous-time samplers that is increasingly used in Monte Carlo for discrete or non-smooth targets. The spectral-gap comparison with Metropolis-Hastings is valuable because it transfers a large body of known results; the uniform ergodicity example for non-sub-Gaussian targets is a genuinely interesting phenomenon that contrasts with Langevin diffusions; and the diffusion limit connects LBMJPs to optimal-scaling literature. The presentation is careful and the reliance on external results (Tierney, Norris, Meyn-Tweedie, Down et al., Monmarché et al.) is made explicit. The restriction to non-decreasing balancing functions is a stated limitation rather than a hidden assumption, and the authors are candid about it. I found no error that changes the main theorems; the issues below are localized and repairable.","major_comments":[{"comment":"The construction of the process in the weak-Feller proof sets τ^x_1 = λ(x)E_1 with E_1 ~ Exp(1). Since E_1/λ(x) is Exp(λ(x))-distributed and λ(x)E_1 is Exp(1/λ(x))-distributed, the construction inverts the required waiting-time scaling. Replacing λ(x) by 1/λ(x) in the displayed formulas repairs the proof without changing the conclusion, because continuity of λ and λ>0 suffice for the argument.","section":"Section 2.3, proof of Proposition 2.3"},{"comment":"The statement that the Down et al. drift condition is 'trivially satisfied' by setting V_T≡1, β(s)=0 and b=1 is not the standard form of the condition, which involves an indicator of a petite set and typically requires β(T)<1 with a non-trivial V. The argument can be made rigorous by using the uniform minorization in (24) to show that the whole space is petite, or by constructing a Lyapunov function from the uniform hitting-time bound, but the proof should spell this out.","section":"Section 3.2, proof of Proposition 3.5"}],"minor_comments":[{"comment":"Appendix C contains an incorrect numerical value: direct differentiation of b(x)=log g(e^x) gives b''(0)=g'(1)+g''(1)-g'(1)^2=1/4+g''(1), not 1/2+g''(1)-g''(1)^2 as displayed; since only boundedness of b'' is used in the proof of Theorem 4.1, the conclusion is unaffected, but the formula should be corrected.","section":"Appendix C"},{"comment":"The definitions of strong and weak Feller are conflated; as written, the sentence 'We will refer to a Markov process for which ... as strong Feller if ...' does not match the standard definitions, and the weak Feller property should be stated as P_t f∈C_b(E) for every f∈C_b(E).","section":"Section 2.3"},{"comment":"The final sentence of Remark 2.6 invokes 'Assumption 2.1' where the surrounding argument concerns Assumption 2.4; this should be corrected.","section":"Remark 2.6"},{"comment":"The kernel γ(x,·)=1/2(δ_{x-1}+δ_{x+1}) is not defined at x=0 on the state space N; a boundary convention should be stated.","section":"Section 3.2.1, Example 1"},{"comment":"The coupling of Y and \\tilde Y by 'using the same exp(1) random variables' is not explicit; since the jump rates differ, the construction should be described (e.g., via Poisson thinning or a common Poisson clock with acceptance probabilities) to justify the stochastic domination used to bound E_{n+1}[h_n].","section":"Appendix B, proof of Proposition B.1"},{"comment":"The phrase 'since T^y_n∈ N for all y' appears to be a typo: the intended meaning is that T^y_n is continuous in y, so that T^y_n=T^x_n for y sufficiently close to x.","section":"Section 2.3, proof of Proposition 2.3"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a solid contribution to applied probability and MCMC theory. The technical issues identified in the report are localized and do not affect the central theorems; I recommend acceptance after minor revision. No concerns about citation practice or overlap with prior work."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take on arXiv:2504.13322. The paper does what it says: it gives the first general-state-space treatment of locally-balanced Markov jump processes. The finite-state continuous-time version exists (Power and Goldman), and the locally balanced proposal idea is Zanella's, but the general-state-space well-posedness, the spectral gap equivalence with Metropolis-Hastings, the comparison theorem for unbounded balancing functions, the uniform ergodicity result for non-sub-Gaussian targets, and the diffusion limit are genuinely new. I checked the main lines of proof; they are coherent and use standard tools correctly. I did not find a load-bearing flaw.\n\nThe strongest piece is Section 3.2. Showing that LBMJPs can be uniformly ergodic when the target has tails like exp(-n^a) for a in (1,2), where Langevin diffusion is not uniformly ergodic, is a real qualitative result and it is proved rather than asserted. The spectral gap comparison for bounded g (Proposition 3.2) is clean and useful.\n\nThe soft spots are all localized. Assumption 2.2 (g non-decreasing, continuous) is doing a lot of work, but the paper flags that as a limitation and derives the key bound g(t) ≤ 1+t from it. That's not an oversight. The three specific issues the reader flagged are real: (i) in the proof of Proposition 2.3 the waiting time is written as λ(x)E_1 instead of E_1/λ(x); a typo, but it should be fixed because it appears in a construction that drives the continuity argument. (ii) In Proposition 3.5, the claim that the drift condition is 'trivially satisfied' with V≡1 is too quick; it becomes correct if you read the whole state space as the petite set, or use a hitting-time argument, but as written it's misleading. (iii) The formula for b''(0) in Appendix C is wrong in value. The proof only needs boundedness of b'', so the diffusion limit still stands, but the displayed formula should be corrected. None of these changes the theorems.\n\nThis is a paper for MCMC theorists and algorithm builders who work with continuous-time samplers. It deserves a serious referee; a knowledgeable referee can fix these issues in a round. I would cite it if I work in this area. Recommend sending out.","headline":"Solid, mostly rigorous foundations paper for locally-balanced Markov jump processes; central theorems hold and the localized proof errors are fixable.","tokens_in":29674,"tokens_out":2391,"would_cite":true,"duration_ms":21258,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J25","60J27","60J35","60J60","65C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes that locally-balanced Markov jump processes are well-posed, reversible and ergodic on general state spaces, and can be uniformly ergodic where overdamped Langevin diffusion is not.","keywords":["locally-balanced Markov jump process","balancing function","Markov chain Monte Carlo","ergodicity","spectral gap","diffusion limit","uniform ergodicity","non-reversible Markov processes"],"falsifier":"Exhibit a balancing function g that satisfies g(1)=1 and g(t)=t g(1/t) but is not monotone, together with a target π and proposal γ for which λ(x)=∫g(t(x,y))γ(x,dy)=∞ on a set of positive π-measure; this would show the monotonicity assumption is genuinely load-bearing and break the well-posedness theorem as stated. Alternatively, for the uniform-ergodicity claim, simulate the embedded nearest-neighbour chain of Theorem 3.1 for π(n)∝exp(−n^a), a in (1,2), with g(t)=t; if the expected hitting time of a compact set from very large n does not stay bounded as n→∞, the uniform-ergodicity conclusion would fail.","tokens_in":28662,"feed_emoji":"⚡","tokens_out":8416,"duration_ms":79032,"temperature":0.7,"pith_summary":"This paper gives locally-balanced Markov jump processes (LBMJPs) a rigorous foundation on general state spaces: continuous-time pure-jump processes whose jump kernel is built from a target measure, a proposal kernel, and a balancing function. The central message is that these processes are well-posed, non-explosive, reversible with respect to the target, and ergodic under mild conditions, and that they can mix uniformly on unbounded state spaces even when the overdamped Langevin diffusion cannot. The paper also proves that in the small-jump limit, after rescaling time by the inverse squared jump size, an LBMJP converges weakly to the overdamped Langevin diffusion with drift gradient of log target. Because LBMJPs only need local ratios of the target density and can be defined on discrete or non-smooth spaces, this supplies a theoretical foundation for a class of samplers previously studied mostly in finite state spaces.","feed_headline":"Local balancing yields uniform ergodicity where Langevin stalls","feed_subtitle":"A jump-process sampler can converge uniformly even when the target is not sub-Gaussian, and it approaches Langevin as jumps shrink.","key_machinery":"The load-bearing object is the balancing function g: R≥0 → R≥0 with g(1)=1 and g(t)=t g(1/t), combined with the Radon–Nikodym derivative t(x,y) of the time-reversed proposal against the forward one. The balancing identity makes the jump kernel J(x,dy)=g(t(x,y))γ(x,dy) reversible with respect to π; the monotonicity assumption on g supplies the two-sided bounds min(1,t)≤g(t)≤max(1,t), which yield finite jump rates, non-explosivity, integrability of the rate, and the spectral-gap comparisons; and for the diffusion limit, the identity g′(1)=1/2 (a consequence of the balancing identity and g(1)=1) is what makes the limiting drift exactly (1/2)∇log π.","core_discovery":"The central claim is that the locally-balanced Markov jump process, defined through a balancing function g, a base kernel γ, and target π, is a genuine Markov jump process: under Assumptions 2.1–2.4 it is non-explosive, π-reversible, and ergodic from π-almost every starting state, and from every starting state in the support of π under Assumption 2.4. The key structural fact is that the jump kernel J(x, dy) = g(t(x, y))γ(x, dy), with t(x, y) the Radon–Nikodym derivative of the time-reversed proposal against the forward one, satisfies detailed balance because g(t) = t g(1/t), so π(dx)J(x, dy) is symmetric. With g non-decreasing and continuous, the paper proves min(1,t) ≤ g(t) ≤ max(1,t), which makes the jump rate λ integrable under π and the process non-explosive. It further claims: a spectral-gap equivalence between LBMJPs with bounded g and Metropolis–Hastings algorithms with the same proposal; comparison theorems for unbounded g; uniform ergodicity on unbounded spaces for targets like π(n) ∝ exp(−n^a) with a in (1,2) when g grows, a regime where overdamped Langevin diffusion is not uniformly ergodic; and a weak diffusion limit to the overdamped Langevin diffusion as the proposal variance goes to zero with time rescaled by $σ_n^{{−2}}$.","pith_inferences":["Combining the uniform-ergodicity result on a fixed grid with the diffusion limit on the same grid suggests a genuine order-of-limits phenomenon: a fixed fine grid can yield uniform ergodicity for a heavy-tailed target, while the continuum limit loses it, so the choice of grid spacing relative to the time horizon is a substantive modelling decision rather than a numerical artefact.","The monotonicity of g is used heavily, but the reversibility property itself follows from the balance identity alone; a plausible extension is that some non-monotone balancing functions also yield well-posed processes, and the paper explicitly leaves this case open in Remark 2.4.","The spectral-gap comparison with Metropolis–Hastings suggests a practical recipe: take any geometrically ergodic Metropolis–Hastings sampler, replace its accept-reject step by the continuous-time locally-balanced jump kernel with the same proposal, and retain exponential ergodicity without extra dimension-dependent tuning.","The variance reduction of importance tempering over continuous-time averaging, proved in the paper's setting, likely also applies to the non-reversible extensions sketched in Section 5.2, since the embedded-chain structure is preserved there, although the paper does not prove this."],"forward_implications":["If a Metropolis–Hastings chain with proposal γ and target π has a positive spectral gap, then the corresponding LBMJP with any non-decreasing balancing function and the same γ also has a positive spectral gap, so LBMJPs inherit a large body of geometric-ergodicity results.","With unbounded g, LBMJPs can be uniformly ergodic on unbounded state spaces for targets with tails that are not sub-Gaussian, such as π(n) ∝ exp(−n^a) with a in (1,2), a regime where the overdamped Langevin diffusion is not uniformly ergodic.","In the small-jump limit with time rescaled, a LBMJP converges weakly to the overdamped Langevin diffusion, so the behaviour of a finely tuned LBMJP approaches that of a Langevin sampler.","The embedded discrete-time chain of a LBMJP can be used for Monte Carlo estimation either through continuous-time averages or through importance-tempered averages, with the latter having lower asymptotic variance.","The same balancing construction extends to non-reversible processes through skewed or modified detailed balance, giving a route to non-reversible locally-balanced samplers."],"supporting_citations":[{"why":"Defines the Radon–Nikodym derivative t(x,y) and the symmetry condition on which the balancing ratio is built.","marker":"Tierney [1998]"},{"why":"Introduced the continuous-time locally-balanced Markov process in finite state spaces, which this paper generalises to arbitrary Polish spaces.","marker":"Power and Goldman [2019]"},{"why":"Supplies the ergodic theorem used to prove non-explosivity and the time-average law of large numbers for the process.","marker":"Asmussen and Glynn [2011]"},{"why":"Provides the criterion linking almost-sure finiteness of the sum of inverse holding rates to non-explosivity of a jump process.","marker":"Norris [1997]"},{"why":"Supplies the piecewise-deterministic Markov process martingale and generator theory used to characterise the weak generator.","marker":"Davis [1984]"},{"why":"Gives the petite-set criteria used to show that compact sets are small for skeleton chains of the process.","marker":"Meyn and Tweedie [1992]"},{"why":"Provides the drift and minorisation criterion used to conclude uniform ergodicity from bounded hitting times of petite sets.","marker":"Down et al. [1995]"},{"why":"Documents the failure of uniform ergodicity for overdamped Langevin diffusions with non-sub-Gaussian targets, the contrast that motivates and calibrates Theorem 3.1.","marker":"Roberts and Tweedie [1996b]"},{"why":"Supplies the generator-expansion and scaling template for random-walk Metropolis whose Langevin limit the diffusion proof of Theorem 4.1 follows.","marker":"Gelman et al. [1997]"},{"why":"Provides the abstract weak-convergence theorem for Markov processes with converging generators used to conclude the diffusion limit.","marker":"Monmarché et al. [2022]"}],"fun_headline_variants":["Jump process samplers reach uniform ergodicity where Langevin stalls","Uniform ergodicity for non-sub-Gaussian targets: a jump process win","LBMJP: uniform mixing on unbounded spaces, Langevin can't","Local balance beats Langevin: uniform convergence on heavy tails","Jump process achieves uniform ergodicity without sub-Gaussian tails"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Assumption 2.2: the balancing function is non-decreasing and continuous; without monotonicity, the paper's two-sided bounds on g fail, and the jump rate can in principle be infinite, which would make the process itself undefined.","fun_headline_variants_meta":{"raw":{"variants":["Jump process samplers reach uniform ergodicity where Langevin stalls","Uniform ergodicity for non-sub-Gaussian targets: a jump process win","LBMJP: uniform mixing on unbounded spaces, Langevin can't","Local balance beats Langevin: uniform convergence on heavy tails","Jump process achieves uniform ergodicity without sub-Gaussian tails"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000423,"raw_usage":{"total_tokens":2232,"prompt_tokens":1064,"completion_tokens":1168,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":680,"completion_tokens_details":{"reasoning_tokens":1075}},"tokens_in":680,"tokens_out":1168,"duration_ms":11199,"temperature":1.0,"reasoning_tokens":1075,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T12:12:41.706767+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a balancing function g that satisfies g(1)=1 and g(t)=t g(1/t) but is not monotone, together with a target π and proposal γ for which λ(x)=∫g(t(x,y))γ(x,dy)=∞ on a set of positive π-measure; this would show the monotonicity assumption is genuinely load-bearing and break the well-posedness theorem as stated. Alternatively, for the uniform-ergodicity claim, simulate the embedded nearest-neighbour chain of Theorem 3.1 for π(n)∝exp(−n^a), a in (1,2), with g(t)=t; if the expected hitting time of a compact set from very large n does not stay bounded as n→∞, the uniform-ergodicity conclusion would fail.","supporting_citations":[],"review_version":1}