{"id":"42b52f04-edb7-4dd9-bf3e-942f440d357e","arxiv_id":"2412.15626","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Partial resetting makes any strictly α-stable Lévy process ergodic with an explicit stationary density, and for Brownian motion the relaxation to that density changes type across |y|=2t.","lead":"The paper proves that a stable Lévy process with partial resetting has a stationary state, gives explicit formulas for its density and moments, and shows that for Brownian motion the large-time density changes behavior across a phase transition curve. It matters because partial resetting models search, TCP congestion control, and growth-collapse processes, and this is a rigorous treatment beyond the one-dimensional cases studied before.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.16's embedded-chain recursion misses a factor c; the cited Borovkov–Foss/Harris arguments are applied to the wrong AR(1) dynamics, leaving the uniform ergodicity Theorem 3.7 with a gap.","rationale":"The reader correctly identified Lemma 2.16 as the weakest point, but the more specific issue is not merely reliance on external theorems: the embedded-chain recursion stated in the proof is algebraically wrong, missing the factor c multiplying the innovation. This is load-bearing for the uniform ergodicity theorem (3.12) and the resulting L1 convergence, because those are the only results that use the total-variation step in an essential way. The central claims of Theorem A (pointwise limit, moments, stationary density) and the Brownian phase transition Theorem D do not depend on Lemma 2.16: the pointwise limit follows from weak convergence of ν_t, the moment computations are self-contained, and Section 4.4 uses steepest descent on the series representation. The paper's own Remark 2.17 states that an analytic proof is deferred, so the gap is acknowledged but not filled. The error is concrete and fixable, so the appropriate action is conditional acceptance: replace or repair the proof of Lemma 2.16, or state Theorem 3.7 with the total-variation step as an assumption. I also credit the paper's independent support: the moment recursion and Carleman-based weak convergence are explicit and verifiable, and the asymptotic analysis in Section 4 is detailed with matched expansions. My recommendation is therefore CONDITIONAL rather than REJECT, and I partially agree with the reader's weakest-assumption identification because it points at the right lemma but not at the incorrect recurrence within its proof. No ad hominem is intended; this is a technical proof gap that can be settled by the proposed re-derivation.","tokens_in":104442,"tokens_out":15093,"duration_ms":133711,"concrete_test":"Re-derive the embedded chain for Y_t=t directly from (1.1): show that Z_{n+1}=cZ_n+c(τ_{n+1}−τ_n) with τ_{n+1}−τ_n iid Exp(1). Then verify whether this corrected AR(1) chain satisfies conditions I–III of [15, p.18] and the hypotheses of [2, Theorem 2.1 and Remark B] (aperiodicity, positive Harris recurrence, absolute continuity on (0,∞)). If the conditions hold, replace the erroneous line in Lemma 2.16 with the corrected recursion and confirm that sc-convergence of this chain implies total-variation convergence of ν_t to ν; if they fail, Theorem 3.7 needs a different proof or a modified hypothesis.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Lemma 2.16, which supplies the total-variation convergence used in Theorem 3.7 and Corollary 3.8, contains a concrete algebraic error. For the deterministic drift case Y_t=t used there, (1.1) gives X_t = c X_{τ_n-} + (t−τ_n) on [τ_n,τ_{n+1}), so just before the next reset X_{τ_{n+1}-} = X_{τ_n} + (τ_{n+1}−τ_n), and after the reset Z_{n+1}=X_{τ_{n+1}} = c X_{τ_{n+1}-} = c Z_n + c(τ_{n+1}−τ_n). The paper instead writes Z_{n+1}=c Z_n + (τ_{n+1}−τ_n). The driver is thus c·Exp(1), not Exp(1), and the stationary law of the embedded chain is not the limiting measure ν of (2.7); its mean is c/(1−c), whereas ν has mean 1/(1−c). While the corrected chain is still an AR(1) with iid exponential innovations, the external criteria [15, Theorem 1(3), Theorem 8] and [2, Theorem 2.1] are not verified for this corrected dynamics, and the identification with ν via sc-convergence is not immediate. This does not threaten the pointwise limit (weak convergence via Theorem 2.14 would suffice) or the moment formulas, but it does undermine the uniform ergodicity statement (3.12) and the L1 convergence in Corollary 3.8 as written. Remark 2.17 explicitly defers an analytic proof, so the gap is real and currently unresolved in the manuscript.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a d-dimensional stochastic process X obtained from a strictly α-stable Lévy process Y (α∈(0,2]) by partial resetting: at independent Poisson epochs the position is multiplied by c∈(0,1), and between epochs it evolves as Y. The main results are: (Theorem A) pointwise ergodicity of the transition density p(t;x,y) to an explicit stationary density ρ_Y(y), expressed as an integral of the stable transition density against a limiting measure μ, with closed-form moment formulas involving the q-Gamma function; (Theorem B) uniform convergence of p/ρ_Y to 1 for isotropic α-stable processes away from the origin; (Theorem C, D) for Brownian motion, a precise dichotomy in the space-time region |y|≈2t, with p staying of order ρ_Y inside the band and reverting to a Gaussian-type asymptotic above |y|=2t. The proofs are built on a series representation of p in terms of recursively defined splines, a complete computation of their moments via q-series, weak and total-variation convergence of auxiliary measures μ_t, and a Laplace/steepest-descent analysis for the Brownian case.","tokens_in":104707,"tokens_out":26010,"duration_ms":210794,"significance":"If the results hold, this is a substantial contribution to the rigorous theory of stochastic resetting. The paper gives the first systematic treatment of partial (multiplicative) resetting for multidimensional stable processes, with explicit stationary densities, moment formulas, and a fully characterized phase transition for Brownian motion. The moment machinery via q-series is original and appears correct. The Brownian asymptotics in Theorems C and D are concrete, falsifiable predictions and go well beyond previous formal results. The NESS verification via non-self-adjointness is also a useful rigorous check. The main caveat is the total-variation step (Lemma 2.16), which is load-bearing for the uniform ergodicity theorem and is justified by external probabilistic results rather than by a self-contained argument.","major_comments":[{"comment":"The embedded chain is defined by Z_n = X_{τ_n}, and the recursion is written as Z_{n+1} = c Z_n + (τ_{n+1} − τ_n). Under the paper's own convention (1.1), X_{τ_n} is the post-reset value, so the correct recursion for the post-reset chain is Z_{n+1} = c Z_n + c(τ_{n+1} − τ_n). The stationary law of that chain is the law of cZ (with Z∼ν), not ν. If the intended chain is the pre-reset chain X_{τ_n-}, the recursion is correct but the definition Z_n = X_{τ_n} is misleading and must be changed. As written, the chain whose total-variation convergence is cited from [15] and [2] is not the chain that demonstrably has stationary law ν, so the conclusion ‖ν_t − ν‖_TV → 0 is not established. Since Theorem 3.7 and Corollary 3.8 rely on Lemma 2.16 through Lemmas 2.20 and 2.21, this is a load-bearing gap; Remark 2.17 explicitly defers an analytic proof, leaving the gap unresolved in the manuscript.","section":"Lemma 2.16 (proof), Section 2.3"},{"comment":"The proof asserts that proving ergodicity of the continuous-time process X_TCP (with Y_t≡t) is equivalent to the total-variation convergence of ν_t, and then invokes [15, Theorem 1(3)] to pass from the embedded chain to the continuous-time process. This passage is not explained: the reader is not told how the Poisson structure and the residual times are handled, nor how the sc-convergence of the chain implies TV convergence of the law of X_t at arbitrary times t. The authors should either give a direct argument or state precisely which theorem in [15] covers this equivalence and why its conditions apply. Without this, the uniform result (3.12) and the L^1 convergence (3.14) remain insufficiently supported.","section":"Lemma 2.16 (proof), Section 2.3"}],"minor_comments":[{"comment":"The notation C_0^\\infty(R^d) is nonstandard if it is intended to mean smooth functions that vanish at infinity together with all derivatives; usually C_0^\\infty denotes compact support. Please clarify the function space used.","section":"Section 2.4, Proposition 2.22"},{"comment":"The sentence 'The measure μ has finite moments of all orders β∈R' can mislead readers, since for negative integers the moments are defined through the limiting procedure (2.23) and are not ordinary integrals for β ≤ −1. The statement is true because the density (2.35) is flat at 0, but this should be stated explicitly.","section":"Theorem 2.14"},{"comment":"The identity 'the probability distribution of X_t equals ν_t' for the drift process relies on [64, Theorem 3], which is not stated. Since this identification is load-bearing, please include the exact theorem or a short proof of the moment identity.","section":"Lemma 2.16 (proof), Section 2.3"},{"comment":"The manuscript contains a number of typographical errors and OCR-style artifacts (e.g., 'resett ing' in the title, stray 'u1D451' symbols throughout). A careful proofreading pass is recommended.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The main technical contribution is strong and the moment computations are convincing. The proof of Lemma 2.16, however, contains a genuine inconsistency in the embedded-chain recursion and relies on a terse appeal to external references, with Remark 2.17 acknowledging that an analytic proof is deferred. Since Theorem 3.7 and Corollary 3.8 depend on this lemma, the paper needs repair in this section. The issue appears fixable without changing the main results, but it currently prevents me from recommending acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, it delivers what it promises: a spline-based series representation for the transition density of a stable process under partial resetting, closed-form stationary moments via the q-gamma function, pointwise ergodicity with an explicit stationary density, a NESS proof, and sharp space-time asymptotics that include a genuine Brownian phase transition across |y|=2t. The moment computations are first-principles, detailed, and I found no error in the spline/algebraic core. The paper is also honest about its own limits: Remark 2.17 defers a purely analytic proof, and the region below q^2 for the Brownian asymptotics is left open. That is credit where it is due.\n\nSecond, the stress-test is right about Lemma 2.16, and this is a real soft spot. For the deterministic drift case Y_t=t used there, the embedded chain at Poisson epochs satisfies Z_{n+1}=c Z_n + c E_n, not c Z_n + E_n as the paper writes. The driver is c times an exponential, not an exponential. So the paper applies the Borovkov–Foss and Harris criteria to the wrong AR(1) dynamics. The consequence is that the total-variation convergence of ν_t to ν, which the proof of Lemma 2.16 is meant to establish, is not proven as written. That feeds into Lemma 2.20/2.21 and therefore into the uniform part of Theorem 3.7 and Corollary 3.8. The pointwise limit does not collapse: weak convergence via Theorem 2.14 plus the representation (3.7) gives (1.3), and the Section 4 asymptotics do not depend on Lemma 2.16. The stress-test overreaches when it says the corrected dynamics fail the external criteria: an AR(1) with iid exponential innovations of rate 1/c is still positive Harris recurrent and absolutely continuous, so the same theorems should apply. But the paper must fix the recursion and re-verify the chain, and the uniform ergodicity statement currently sits on a false equation.\n\nWho gets value: people working on stochastic resetting, AIMD/queueing models, and stable process theory. The paper deserves a serious referee. My recommendation is to send it out, with the referee asked to check Lemma 2.16 and require the authors to correct the embedded-chain argument or supply the deferred analytic proof. If that is cleaned up, the paper should be accepted.","headline":"A serious and mostly rigorous paper on partial resetting for stable processes, with a real but likely fixable algebraic error in the embedded-chain proof used for uniform ergodicity.","tokens_in":105317,"tokens_out":12083,"would_cite":true,"duration_ms":95650,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G10","60J35","60K40","82C05","82C31","35K08","60J65","60G51"],"pacs":[],"model":"deepseek-v4-flash","headline":"Partial resetting forces stable processes into explicit stationary states, with a Brownian phase transition exactly at distance 2t.","keywords":["partial resetting","transition density","ergodic measure","non-equilibrium stationary state","phase transition","q-Gamma function","heat kernel","asymptotic behavior"],"falsifier":"Simulate the embedded stationary chain $Z_{n+1} = cZ_n + E_n$ with unit-rate exponential increments — the resetting process observed at its reset epochs — and histogram the stationary law: the paper predicts $\\rho_{\\mathbf{Y}}(y)/(|y|^{-(d-1)/2}e^{-|y|}) \\to (1/2)(q;q)_\\infty^{-1}(2\\pi)^{-(d-1)/2}$ for Brownian motion, together with the closed moment formula for every real $\\beta$, including the negative-integer values defined by the limiting $\\varepsilon$-trick; a mismatch in either the tail prefactor or any moment would falsify the identification of $\\rho_{\\mathbf{Y}}$. A second, independent check is the phase transition itself: evaluate $p(t;0,y)$ from the series representation with $|y|/(2t)$ held fixed and compare the logarithm of the density as $t$ grows — the paper predicts an abrupt change of functional form as the ratio crosses $1$, with the $O(t^{-1})$ approach to $\\rho_{\\mathbf{Y}}$ below the threshold and the explicit heat-kernel expansion above it.","tokens_in":104158,"feed_emoji":"🌀","tokens_out":21628,"duration_ms":156205,"temperature":0.7,"pith_summary":"Partial resetting — scaling the position of a particle by a fixed factor $c \\in (0,1)$ at independent exponential epochs — is a standard model for TCP congestion control, growth–collapse systems, and intermittent search. This paper proves that when the underlying free process $\\mathbf{Y}$ is a strictly $\\alpha$-stable L\\'evy process with a transition density, the resetting process $\\mathbf{X}$ always settles into a stationary state, and that the stationary density $\\rho_{\\mathbf{Y}}$ is explicit: it is an average of the free process's density from the origin against a universal measure $\\nu$ whose moments are $k!/(q;q)_k$, $q = c^\\alpha$. All moments of the stationary law follow in closed form, $\\int_{\\mathbb{R}^d} |y|^\\beta \\rho_{\\mathbf{Y}}(y)\\,dy = [\\Gamma(\\beta/\\alpha+1)/\\Gamma_q(\\beta/\\alpha+1)](1-q)^{-\\beta/\\alpha} \\mathbb{E}|Y_1|^\\beta$ for every real $\\beta$. The same spline machinery yields sharp space-time asymptotics: for isotropic stable processes $p(t;x,y)/\\rho_{\\mathbf{Y}}(y) \\to 1$ uniformly for $|x| \\le \\kappa|y|$, while for Brownian motion a phase transition occurs at $|y| = 2t$ — inside the cone the density approaches $\\rho_{\\mathbf{Y}}(y)$, outside it keeps a heat-kernel-like exponential form. A careful reader would care because these theorems turn the resetting literature's mostly numerical claims, and the physicists' predicted change of behavior near $|y| \\approx 2t$, into statements with computable constants.","feed_headline":"Resetting Brownian motion flips its asymptotics at distance 2t","feed_subtitle":"New spline series give exact stationary densities; the crossover sits exactly at distance 2t.","key_machinery":"The engine is a sequence of splines $\\{s_j\\}$ on $[0,1]$: recursively defined, homogeneous piecewise-polynomial densities whose moments satisfy the two-term recursion $(j+1+\\beta)A(\\beta,j+1) = A(\\beta,j) + \\beta q^{j+1}A(\\beta-1,j+1)$. Solving this recursion for all real $\\beta$ — negative integers require a limiting $\\varepsilon$-trick because the recursion breaks down at $\\beta = 0$ — and summing the resulting series with the $q$-binomial theorem produces the explicit moments in terms of the $q$-Gamma function $\\Gamma_q$. The splines assemble into probability measures $\\nu_t$ on $[0,t]$ via $\\nu_t(ds) = e^{-t}\\delta_t(ds) + e^{-t}\\sum_{j\\ge1} t^j s_j(s/t)\\,ds/t$, and the load-bearing identity is the representation $p(t;0,y) = \\int_0^\\infty p_0(s;0,y)\\,\\nu_t(ds)$. Ergodicity, moment formulas, and the uniform asymptotics are then read off from the convergence of $\\nu_t$ to the limiting measure $\\nu$ with moments $k!/(q;q)_k$, together with the classical heat-kernel estimates for stable densities. For the Brownian phase transition the decisive object is the phase function $\\vartheta(u) = -(d/2)\\log u - \\Theta/u + \\log\\Phi(t,u)$ on $(0,1]$, with $\\Theta = |y|^2/4t$ and $\\Phi$ the spline series; whether its saddle point lies inside or outside $(q,1)$ selects the stationary regime or the heat-kernel regime, with the crossover at $\\Theta/t = 1$, i.e. $|y| = 2t$.","core_discovery":"The central claim is that a strictly $\\alpha$-stable process $\\mathbf{Y}$ with density $p_0$, run with multiplicative resets at rate one, has a transition density $p$ that converges as $t \\to \\infty$ to the smooth density $\\rho_{\\mathbf{Y}}(y) = (1/(q;q)_\\infty)\\sum_{k\\ge 0} (-1)^k q^{k(k-1)/2} (q;q)_k^{-1} \\int_0^\\infty e^{-q^{-k}s} p_0(s;0,y)\\,ds$, with explicit, uniform asymptotics attached to the convergence. All steady-state moments are $\\int_{\\mathbb{R}^d}|y|^\\beta \\rho_{\\mathbf{Y}}(y)\\,dy = [\\Gamma(\\beta/\\alpha+1)/\\Gamma_q(\\beta/\\alpha+1)](1-q)^{-\\beta/\\alpha}\\mathbb{E}|Y_1|^\\beta$, with the quotient of Gamma functions continued to negative integers by a limiting $\\varepsilon$-trick. For isotropic $\\alpha$-stable laws the ratio $p(t;x,y)/\\rho_{\\mathbf{Y}}(y)$ converges to $1$ uniformly in the region $|x| \\le \\kappa|y|$, uniformly also in the resetting factor below any $\\kappa_1 < 1$. For Brownian motion the paper identifies exactly where uniform convergence to $\\rho_{\\mathbf{Y}}$ holds: in the band $q^2+\\delta \\le |y|^2/(4t^2) \\le 1-\\delta$ one has $p(t;0,y) = \\rho_{\\mathbf{Y}}(y)(1+O(t^{-1}))$, whereas in $|y|^2/(4t^2) \\ge 1+\\delta$ one has $p(t;0,y) = e^{-t}(4\\pi t)^{-d/2}e^{-|y|^2/4t}\\{1+(4t^2/|y|^2)\\psi(4t^2/|y|^2)+O(t/|y|^2)\\}$ for an explicit $q$-series $\\psi$, so the asymptotic regime changes discontinuously across the curve $|y| = 2t$. The same representation yields the Fokker–Planck equation for $p$, the harmonicity $\\mathcal{A}^*\\rho_{\\mathbf{Y}} = 0$, and a proof that the generator is not self-adjoint on $L^2(\\mathbb{R}^d, \\rho_{\\mathbf{Y}}\\,dy)$ — a non-equilibrium stationary state (NESS).","pith_inferences":["The crossover curve $|y|=2t$ has a deterministic reading: a Brownian particle cannot travel farther than about $2t$ without a reset, so multiplicative resets that shrink positions should leave no stationary mass beyond the no-reset light cone; the thin bands around $|y|\\approx 2t$ left open by the paper's theorems are the natural place to look for an interpolating intermediate asymptotic.","The moment formula is a $q$-deformation of the stable scaling identity $\\mathbb{E}|Y_s|^\\beta = s^{\\beta/\\alpha}\\mathbb{E}|Y_1|^\\beta$ and suggests viewing $\\rho_{\\mathbf{Y}}$ as a $q$-analogue of the stable law; a testable extension would be to check whether $\\rho_{\\mathbf{Y}}$ obeys a $q$-analogue of self-decomposability, which would yield recurrence relations for its orthogonal polynomials.","The NESS proof exhibits one bump function witnessing non-self-adjointness but does not quantify the departure from reversibility; the explicit density makes a quantitative version accessible, such as the operator norm of $\\mathcal{A} - \\mathcal{A}^*$ on the stationary $L^2$ space or the entropy production rate, which is the quantity stochastic-thermodynamics applications actually need.","Because the total-variation step is the only piece with a deferred analytic proof, a purely analytic replacement would likely extend uniform ergodicity to the currently excluded extremes ($q\\to1$, cylindrical processes), where the pointwise limit is expected to hold already by the moment-based weak-convergence argument."],"forward_implications":["The stationary law of the resetting process is explicit: all steady-state moments are computable in closed form from $\\mathbb{E}|Y_1|^\\beta$ and the $q$-Gamma ratio, so mean displacement, energy, and fluctuation measures need no simulation.","Uniform ratio convergence $p/\\rho_{\\mathbf{Y}}\\to 1$ for $|x|\\le\\kappa|y|$ means that for isotropic stable laws the stationary density governs the transition density's behaviour on the natural scale of the L\\'evy measure, including the power-law tail $|y|^{-(d+\\alpha)}$.","For Brownian motion, the transition density converges to $\\rho_{\\mathbf{Y}}$ uniformly (at rate $O(t^{-1})$) only inside the cone $|y|<2t$; outside the cone it is carried by the no-reset Gaussian term, so long excursions follow the large-deviation factor $e^{-|y|^2/4t - t}$ rather than the stationary $e^{-|y|}$ tail.","The stationary state is provably non-equilibrium: the process generator is not self-adjoint on $L^2(\\mathbb{R}^d,\\rho_{\\mathbf{Y}}dy)$, giving a rigorous NESS certificate of the kind the resetting literature usually argues heuristically.","The density solves the Fokker–Planck equation and $\\rho_{\\mathbf{Y}}$ solves the adjoint harmonicity equation $\\mathcal{A}^*\\rho_{\\mathbf{Y}}=0$, so the stationary measure is analytically characterised, not just numerically observed."],"supporting_citations":[{"why":"Supplies the sc-convergence framework for stochastic recursive sequences used in Lemma 2.16 to convert convergence of the embedded chain into total-variation convergence of the auxiliary measures $\\nu_t$, the step behind uniform ergodicity.","marker":"[15]"},{"why":"Provides the Harris-recurrence criterion for the embedded AR(1) chain that Lemma 2.16 must verify before the sc-convergence theorem applies.","marker":"[2]"},{"why":"Gives the absolute continuity of the chain's distribution, completing the three conditions needed in Lemma 2.16's total-variation argument.","marker":"[6]"},{"why":"Underlies Theorem 2.14: tightness from uniform moment bounds and determinacy of the moment problem identify the limiting measure $\\nu$ through its moments $k!/(q;q)_k$.","marker":"[12]"},{"why":"Supplies the explicit AIMD steady-state density from which the paper reads off the series form of the limiting measure $\\nu$ and hence of $\\rho_{\\mathbf{Y}}$.","marker":"[65]"},{"why":"Mittag-Leffler asymptotics justify the uniform moment limits (2.26)-(2.29) from which the explicit moments of $\\nu$ and of $\\rho_{\\mathbf{Y}}$ are derived.","marker":"[38]"},{"why":"The heat-kernel estimates $p_0(s;0,y) \\approx \\min\\{s^{-d/\\alpha}, s|y|^{-d-\\alpha}\\}$ drive the uniform far-field asymptotics for isotropic stable processes and stable subordinators.","marker":"[13]"},{"why":"Earlier Fourier-transform computation of the partial-resetting density under the a priori assumption that the stationary density exists; the spline method removes the assumption and works in all dimensions.","marker":"[25]"}],"fun_headline_variants":["Exact stationary densities for stable processes with partial resets","Resetting Brownian motion: asymptotics flip at |y|=2t","Partial resetting yields non-equilibrium steady states exactly","Stable processes under partial reset: ergodic density in closed form","Brownian resets show phase transition in tail asymptotics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the auxiliary measures $\\nu_t$ converge in total variation, which the paper establishes by identifying them with the law of an embedded first-order autoregressive chain and applying an external ergodicity theorem for stochastic recursive sequences; the paper notes (Remark 2.17) that a purely analytic proof is deferred, so if those external criteria do not apply exactly at some parameter values the uniform ergodicity statement would need replacement, although the pointwise limit is expected to survive.","fun_headline_variants_meta":{"raw":{"variants":["Exact stationary densities for stable processes with partial resets","Resetting Brownian motion: asymptotics flip at |y|=2t","Partial resetting yields non-equilibrium steady states exactly","Stable processes under partial reset: ergodic density in closed form","Brownian resets show phase transition in tail asymptotics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000801,"raw_usage":{"total_tokens":3771,"prompt_tokens":1444,"completion_tokens":2327,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":1060,"completion_tokens_details":{"reasoning_tokens":2255}},"tokens_in":1060,"tokens_out":2327,"duration_ms":13674,"temperature":1.0,"reasoning_tokens":2255,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:14:59.344204+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the embedded stationary chain $Z_{n+1} = cZ_n + E_n$ with unit-rate exponential increments — the resetting process observed at its reset epochs — and histogram the stationary law: the paper predicts $\\rho_{\\mathbf{Y}}(y)/(|y|^{-(d-1)/2}e^{-|y|}) \\to (1/2)(q;q)_\\infty^{-1}(2\\pi)^{-(d-1)/2}$ for Brownian motion, together with the closed moment formula for every real $\\beta$, including the negative-integer values defined by the limiting $\\varepsilon$-trick; a mismatch in either the tail prefactor or any moment would falsify the identification of $\\rho_{\\mathbf{Y}}$. A second, independent check is the phase transition itself: evaluate $p(t;0,y)$ from the series representation with $|y|/(2t)$ held fixed and compare the logarithm of the density as $t$ grows — the paper predicts an abrupt change of functional form as the ratio crosses $1$, with the $O(t^{-1})$ approach to $\\rho_{\\mathbf{Y}}$ below the threshold and the explicit heat-kernel expansion above it.","supporting_citations":[{"cited_title":"Bogdan, A","cited_arxiv_id":null,"evidence_quote":"Supplies the sc-convergence framework for stochastic recursive sequences used in Lemma 2.16 to convert convergence of the embedded chain into total-variation convergence of the auxiliary measures $\\nu_t$, the step behind uniform ergodicity."},{"cited_title":"Avrachenkov, A","cited_arxiv_id":null,"evidence_quote":"Gives the absolute continuity of the chain's distribution, completing the three conditions needed in Lemma 2.16's total-variation argument."},{"cited_title":"Berezhkovskii, A","cited_arxiv_id":null,"evidence_quote":"Underlies Theorem 2.14: tightness from uniform moment bounds and determinacy of the moment problem identify the limiting measure $\\nu$ through its moments $k!/(q;q)_k$."},{"cited_title":"Ott and J.H.B","cited_arxiv_id":null,"evidence_quote":"Supplies the explicit AIMD steady-state density from which the paper reads off the series form of the limiting measure $\\nu$ and hence of $\\rho_{\\mathbf{Y}}$."},{"cited_title":"Gerber and R","cited_arxiv_id":null,"evidence_quote":"Mittag-Leffler asymptotics justify the uniform moment limits (2.26)-(2.29) from which the explicit moments of $\\nu$ and of $\\rho_{\\mathbf{Y}}$ are derived."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The heat-kernel estimates $p_0(s;0,y) \\approx \\min\\{s^{-d/\\alpha}, s|y|^{-d-\\alpha}\\}$ drive the uniform far-field asymptotics for isotropic stable processes and stable subordinators."},{"cited_title":"Derrida, Non equilibrium steady states: fluctuations and large devia tions of the density and of the current , J","cited_arxiv_id":null,"evidence_quote":"Earlier Fourier-transform computation of the partial-resetting density under the a priori assumption that the stationary density exists; the spline method removes the assumption and works in all dimensions."}],"review_version":1}