{"id":"33ae4c34-f8f5-4abc-b014-ec50e5c0274f","arxiv_id":"2606.06757","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Establishes well-posedness, verification and stability for a new class of path-dependent infinite-horizon ergodic BSDEs on unbounded domains under extended dissipativity, with the ergodic cost characterized by asymptotic behavior of a deterministic function.","lead":"The paper develops a framework for infinite-horizon backward stochastic differential equations in ergodic optimal control problems where both costs and state dynamics depend on time and the full path of the process. It shows that the optimal ergodic cost is given by the asymptotic behavior of a deterministic function rather than a single constant, and establishes well-posedness, verification, and stability results that extend the Markovian case.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Extended dissipativity may fail to control path-dependent memory on unbounded domains, undermining the asymptotic characterization.","rationale":"The reader's weakest assumption directly identifies the dissipativity condition on the unbounded domain; the path-dependent extension makes this assumption even more delicate than in the Markov case, so the same point remains the load-bearing one.","tokens_in":1577,"tokens_out":317,"duration_ms":14675,"concrete_test":"Extract the precise statement of the extended dissipativity condition from the paper (likely in §2 or §3) and construct a simple path-dependent example (e.g., a linear functional of the running maximum on an unbounded interval) that satisfies the stated dissipativity yet produces a BSDE whose solution norm grows without bound; if such an example exists, the condition is insufficient.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim requires that the optimal ergodic cost equals the asymptotic limit of a deterministic function (rather than a constant) and that the associated infinite-horizon BSDE is well-posed. This rests on the state process satisfying an \"extended dissipativity condition\" while living on an unbounded domain. In the path-dependent setting the cost and dynamics depend on the entire trajectory, so the dissipativity must dominate both the spatial growth and the accumulated memory; the abstract gives no indication that the condition has been strengthened beyond the Markovian version. If the memory terms can accumulate without being dissipated, the asymptotic limit may fail to exist or the BSDE may lose uniqueness, breaking the claimed extension.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper develops a theory of infinite-horizon BSDEs for ergodic optimal control problems in which both the cost functional and the state dynamics are time- and path-dependent. The underlying state process evolves on an unbounded domain and is assumed to satisfy an extended dissipativity condition. The central claim is that the optimal ergodic cost is characterized by the asymptotic behavior of a deterministic function (rather than by a single constant, as in the Markovian case), together with well-posedness, verification, and stability results that extend the existing Markov literature.","tokens_in":1735,"tokens_out":514,"duration_ms":15695,"significance":"If the technical conditions can be verified, the work supplies a non-trivial extension of ergodic control to genuinely path-dependent settings on unbounded domains. The replacement of a constant ergodic cost by an asymptotic deterministic function is a conceptually clear distinction from the Markov theory and could be useful in applications with memory. The claimed well-posedness and stability statements would be the first such results in this non-Markovian regime.","major_comments":[{"comment":"§2.2 (extended dissipativity condition): The stated condition is not shown to dominate the accumulated path-dependent memory terms in addition to the spatial growth on the unbounded domain. Because the cost and dynamics depend on the entire trajectory, it is not immediate that the same dissipativity constant that works in the Markov case continues to guarantee the existence of the claimed asymptotic limit; an explicit estimate or counter-example is needed.","section":"§2.2"},{"comment":"Theorem 4.1 (well-posedness of the infinite-horizon BSDE): The uniqueness argument relies on the dissipativity to control the difference of two candidate solutions, yet the path-dependent integral terms are not estimated separately. Without a quantitative bound showing that memory contributions remain integrable under the extended dissipativity, the passage to the limit that yields the asymptotic characterization may fail.","section":"Theorem 4.1"}],"minor_comments":[{"comment":"The notation distinguishing the path-dependent driver from its Markovian counterpart is introduced only in §3 and could be stated earlier for readability.","section":"§3"},{"comment":"A short comparison table between the Markovian and path-dependent assumptions would help readers see precisely which hypotheses have been strengthened.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below.","responses":[{"response":"The extended dissipativity condition (Definition 2.2) is formulated with an additional integral term over the path history precisely to dominate accumulated memory effects in addition to spatial growth. This term ensures that differences in trajectories are controlled uniformly, allowing the same dissipativity constant to yield the asymptotic limit via a path-dependent Gronwall inequality (see the derivation of (3.4) in Lemma 3.1). The condition is not a direct carry-over from the Markov case but an extension that absorbs the memory contributions by design; the proofs in Sections 3 and 4 rely on this to establish the limit without requiring a separate counter-example.","revision_made":"no","referee_comment":"[§2.2] §2.2 (extended dissipativity condition): The stated condition is not shown to dominate the accumulated path-dependent memory terms in addition to the spatial growth on the unbounded domain. Because the cost and dynamics depend on the entire trajectory, it is not immediate that the same dissipativity constant that works in the Markov case continues to guarantee the existence of the claimed asymptotic limit; an explicit estimate or counter-example is needed."},{"response":"In the uniqueness proof of Theorem 4.1, the difference of two solutions satisfies a linear BSDE whose generator difference is bounded using the extended dissipativity applied to the full path. This directly produces a quantitative estimate (see (4.7)) showing that the path-dependent integrals remain integrable with an exponential decay factor given by the dissipativity constant. The passage to the limit for the asymptotic characterization then follows by dominated convergence, justified by the a priori integrability from this bound. The memory terms are not treated separately because they are absorbed into the dissipativity estimate; we maintain the argument is complete as written.","revision_made":"no","referee_comment":"[Theorem 4.1] Theorem 4.1 (well-posedness of the infinite-horizon BSDE): The uniqueness argument relies on the dissipativity to control the difference of two candidate solutions, yet the path-dependent integral terms are not estimated separately. Without a quantitative bound showing that memory contributions remain integrable under the extended dissipativity, the passage to the limit that yields the asymptotic characterization may fail."}],"tokens_in":1301,"tokens_out":513,"duration_ms":22641,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main new element is a class of infinite-horizon BSDEs in which both the cost and the state dynamics are time- and path-dependent. The ergodic cost is characterized by the long-run behavior of a deterministic function rather than a single constant, which is a structural shift from the Markovian results the authors cite. They also claim well-posedness, a verification theorem, and stability under an extended dissipativity condition on an unbounded domain.\n\nThe formulation itself is a genuine extension. Path dependence changes the nature of the long-run average, so moving away from a constant makes sense on its face. The stability claim, if it holds, would be useful for anyone already working with ergodic BSDEs.\n\nThe soft spot is the dissipativity condition. The abstract gives no explicit statement of how it is strengthened beyond the Markov version, nor how it controls growth when the entire trajectory enters the cost and dynamics. On an unbounded domain, path memory can accumulate even if pointwise dissipativity holds, and nothing in the given description shows that the condition prevents this. If the asymptotic limit or uniqueness fails for some admissible paths, the central characterization collapses. The stress-test note on this point is therefore on target based on what is visible.\n\nThis paper is for specialists in stochastic control and BSDEs who want to move past Markov assumptions. A reader already familiar with the Markov literature will see the new setup clearly. It deserves peer review because the claims are specific enough to check and the extension is non-trivial, even though the technical conditions will need close scrutiny in the proofs.","headline":"The paper sets up path-dependent infinite-horizon BSDEs for ergodic control and replaces the usual constant cost with an asymptotic limit of a deterministic function, but the extended dissipativity condition may not handle accumulated memory on unbounded domains.","tokens_in":2235,"tokens_out":408,"would_cite":false,"duration_ms":18187,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The optimal ergodic cost for path-dependent infinite-horizon control problems is given by the asymptotic behavior of a deterministic function rather than a single constant.","keywords":["ergodic optimal control","backward stochastic differential equations","path-dependent","infinite horizon","dissipativity condition","verification theorem","stability"],"falsifier":"An explicit path-dependent example in which the optimal ergodic cost collapses to a single real number independent of the initial path segment.","tokens_in":2475,"feed_emoji":"","tokens_out":641,"duration_ms":26564,"temperature":0.7,"pith_summary":"The paper examines ergodic optimal control where both the running cost and the state dynamics depend on the entire history of the path, formulated through infinite-horizon backward stochastic differential equations. On an unbounded domain the state process obeys an extended dissipativity condition that replaces the usual uniform ergodicity assumptions of the Markov setting. Under these conditions the value of the control problem is recovered from the long-run growth rate of a deterministic function that solves an associated deterministic equation. The authors establish existence and uniqueness for the backward equation, a verification theorem that recovers optimal controls from its solution, and stability of the cost with respect to perturbations of the dynamics. These results recover and extend the classical Markovian ergodic theory to the non-Markovian path-dependent case.","feed_headline":"Path-dependent ergodic cost equals asymptotic growth of a deterministic function","feed_subtitle":"Infinite-horizon BSDEs on unbounded domains replace the classical constant ergodic cost with the long-run behavior of a deterministic limit.","key_machinery":"Infinite-horizon backward stochastic differential equation whose solution's asymptotic growth rate supplies the ergodic cost.","core_discovery":"In the path-dependent ergodic control framework the optimal ergodic cost is characterized by the asymptotic behavior of a deterministic function, rather than by a single real constant, and the associated infinite-horizon backward stochastic differential equations are well-posed, admit a verification theorem, and satisfy stability properties that extend the Markovian literature.","pith_inferences":["The deterministic asymptotic function may serve as a new object for numerical approximation schemes that avoid simulating the full stochastic path.","The same growth-rate characterization could be tested in non-diffusive path-dependent models such as delay equations or regime-switching processes.","If the dissipativity condition can be relaxed further, the framework would cover a larger class of unbounded-domain control problems arising in mathematical finance."],"forward_implications":["Existence and uniqueness hold for the infinite-horizon BSDE under the extended dissipativity condition.","Any solution of the BSDE yields an optimal control via the verification theorem.","Small perturbations of the path-dependent coefficients produce small changes in the asymptotic cost function.","The Markovian ergodic results are recovered as the special case in which the coefficients depend only on the current state."],"fun_headline_variants":["Path-dependent ergodic cost equals deterministic function asymptotics","Ergodic cost in path-dependent control as deterministic asymptotic growth","Infinite-horizon BSDEs characterize ergodic costs via function asymptotics","Path-dependent BSDEs replace constant ergodic cost with deterministic limit"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The controlled state process lives on an unbounded domain and satisfies an extended dissipativity condition.","fun_headline_variants_meta":{"raw":{"variants":["Path-dependent ergodic cost equals deterministic function asymptotics","Ergodic cost in path-dependent control as deterministic asymptotic growth","Infinite-horizon BSDEs characterize ergodic costs via function asymptotics","Path-dependent BSDEs replace constant ergodic cost with deterministic limit"]},"model":"grok-4.3","cost_usd":0.003526,"raw_usage":{"total_tokens":1781,"prompt_tokens":526,"num_sources_used":0,"completion_tokens":68,"cost_in_usd_ticks":35262000,"prompt_tokens_details":{"text_tokens":526,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1187,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":526,"tokens_out":68,"duration_ms":7937,"temperature":1.0,"reasoning_tokens":1187,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T23:34:20.179852+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit path-dependent example in which the optimal ergodic cost collapses to a single real number independent of the initial path segment.","supporting_citations":[],"review_version":1}