{"id":"b02876cb-1481-46ab-9bc4-6b89cc1d0bf9","arxiv_id":"2412.16374","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":2.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A self-contained lecture-note introduction to diffusion with stochastic resetting, covering stationary states, first-passage properties, large deviations, and the cost of resetting.","lead":"These lecture notes introduce the mathematics of stochastic resetting, a process in which a random walker is repeatedly returned to a starting point. They show how to derive stationary states, first-passage times, and large deviation functions, and they include recent results on the cost of resetting.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the derivations are internally consistent, and the finite-mean waiting-time condition is explicitly stated rather than hidden.","rationale":"The reader identified the finite-mean waiting-time assumption as the weakest point, and that is indeed the main mathematical condition in the non-Poissonian section. However, it is not a hidden or unstated assumption: the notes explicitly derive Eq. (117) as the condition for a non-equilibrium stationary state to exist, and every concrete example considered in the notes uses exponential or deterministic resetting, both of which have finite mean. The internal derivations I checked are consistent: the renewal equations have the correct convolution structure, the Laplace inversions are valid, the saddle-point equations for the cost large deviation function are correct, and the small-s expansions that yield the stationary states are standard. The typos flagged by the reader and confirmed here are cosmetic and localized. Therefore the central claim — that the notes are self-contained and allow the reader to reproduce the calculations — is not threatened. I would not adjust the ACCEPT verdict, though a copyedit pass for Eq. (70) and Eq. (119) would improve the pedagogical value.","tokens_in":20358,"tokens_out":23032,"duration_ms":206684,"concrete_test":"Independently re-derive Eq. (120) from Eq. (119) by setting s=0 and applying Fubini to the numerator integral, verifying that ∫ Ψ(t) q0(t) dt equals ∫ ψ(t) ∫_0^t q0(τ) dτ dt; if this identity fails, the non-Poissonian MFPT formula would need revision, but it should hold for any integrable waiting-time density.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is pedagogical: a self-contained introduction that lets a reader execute the calculations using Laplace transforms and renewal equations. I checked the main derivation chains — Poissonian propagator and steady state (§4), survival probability and MFPT (§5), the renewal construction for additive functionals (§6), the cost generating function and its saddle-point inversion (§7), and the non-Poissonian renewal equations (§8). The algebra is consistent: Eq. (48) correctly follows from Eq. (46); the residue in Eq. (72) matches a direct derivative of the denominator of Eq. (64); Eq. (97) follows from the first-renewal cost equation; and Eq. (120) is the correct s=0 limit of Eq. (119), with the numerator identity obtained by Fubini. The finite-mean waiting-time condition in Eq. (117) is a genuine restriction, but the notes state it explicitly and all worked examples (exponential and deterministic resetting) satisfy it. The only issues I found are minor typos already noted: the spurious 'vt' in Eq. (70), the wrong Laplace variable in Eq. (119), and a harmless use of x_r where x_0=x_r in Eq. (72). None of these changes any result or undermines the self-contained exposition.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"These lecture notes provide a self-contained introduction to diffusion with stochastic resetting, developed from elementary Laplace-transform and renewal-equation techniques. The paper derives the free diffusion propagator and the survival probability in an absorbing half-line, introduces Poissonian resetting through master and renewal equations, obtains the Laplace stationary state and its relaxation front, computes the mean first-passage time and its optimal resetting rate, extends the renewal construction to additive functionals and to the cost of resetting (including an explicit large-deviation function for linear cost), and finally treats general non-Poissonian resetting subject to a finite mean inter-reset time. The presentation is aimed at readers who wish to reproduce every calculation.","tokens_in":20472,"tokens_out":20621,"duration_ms":173186,"significance":"The paper's claim is pedagogical rather than a claim of new scientific results. Its strength is that the derivations are standard, internally consistent, and complete enough for a student to follow: the renewal equations, the Laplace transforms, and the saddle-point inversions are all shown, with appendices supplying the contour and asymptotic tools. There are no fitted parameters or hidden numerical inputs, and the main restriction, namely the finite mean waiting time needed for a non-equilibrium stationary state in Eq. (117), is stated explicitly and is satisfied by all worked examples. The explicit large-deviation function for linear resetting cost in Eqs. (111)-(112) provides a concrete checkable result. I found no circularity: the notes re-derive known results from first principles. Once the equation typos listed below are fixed, these notes would be a valuable entry point to the field.","major_comments":[],"minor_comments":[{"comment":"With the scaling variable defined in Eq. (10) as z=(x-x0)/(Dt)^{1/2}, the normalized solution of Eq. (12) is f(z)=(4π)^{-1/2} e^{-z^2/4}, not (4π)^{-1/2} e^{-z^2/2}; the latter has norm 1/√2 and does not reproduce the propagator in Eq. (9).","section":"Section 2.1, Eq. (12)"},{"comment":"The pole condition contains a spurious 'vt' and should read s0 + r exp(-x0 sqrt((r+s0)/D))=0; in the residue formula (72), x_r should be replaced by x0 under the assumption x_r=x0 stated just before Eq. (63).","section":"Section 5.2, Eqs. (70) and (72)"},{"comment":"The displayed large-deviation form has the wrong sign; it should be P_r(A_t,t) ~ e^{-t I(a)} to agree with Eq. (79) and with the definition of I(a) in Eq. (90).","section":"Section 6.1, Eq. (89)"},{"comment":"These renewal equations implicitly assume that the initial position equals the resetting position, x0=x_r; please state this at the beginning of Section 6.1, as is done for the cost model in Section 7.","section":"Section 6.1, Eqs. (80)-(84)"},{"comment":"The denominator is written with e^{-s t} inside an integral over τ; it should be e^{-s τ} ψ(τ) q0(τ|x0), consistently with the numerator and with Eq. (120).","section":"Section 8.1, Eq. (119)"},{"comment":"Please proofread for small typos: 'Theses notes' in Section 1 should be 'These notes', and the caption of Fig. 4 abbreviates 'Mean FTP' where 'MFPT' is meant.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is well suited to SciPost Physics Lecture Notes and the heavy self-citation is understandable given the authors' role in developing the subject. The Section 2.1 scaling slip and the equation typos in Sections 5, 6, and 8 should be corrected before publication, but they do not affect any of the subsequent results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"These lecture notes do exactly what they claim: a self-contained walk through diffusion with stochastic resetting, from the renewal equations for the propagator and survival probability to the large deviation function for the cost of resetting and a short chapter on non-Poissonian resetting. The novelty is zero — the key results are from the cited literature, and the recent-material sections (cost, non-Poissonian optimization) are taken from the authors' own published work — but for a lecture-note venue that is not a flaw. The pedagogical value is real. The authors are careful to show the Laplace-transform and saddle-point steps, and the appendices fill in the nontrivial integral identities. I checked the main derivation chains (the renewal argument in Section 4, the pole location in Section 5.2, the cost generating function in Section 7, and the non-Poissonian stationary state in Section 8) and the algebra is consistent. The stress-test note matches my reading; I found no hidden circularity or fitting, and the finite-mean waiting-time condition in Eq. (117) is explicitly flagged by the authors rather than glossed over.\n\nThe soft spots are minor. Eq. (70) has a spurious 'vt' inside the exponential, and Eq. (119) has the wrong variable in the Laplace transform of the survival term — the stress-test note is right that Eq. (120) follows once you correct it. Eq. (72) uses x_r where the text has already set x_0 = x_r; it is a notational shortcut, not an error. Readers should also know that the restriction to finite mean reset times in Section 8 is real: the stationary-state analysis genuinely fails for infinite-mean waiting time distributions, though the notes are honest about it. On citation practice: the paper cites the primary literature and the authors' own work is a large part of that literature. It is heavy self-citation only because Evans is one of the field's founders; the derivations stand alone, so I do not read it as a problem.\n\nWho is this for? A graduate student or a researcher from another area who wants to sit down and actually re-derive the standard resetting results. That audience will get a lot out of it. It is not a research contribution and should not be judged as one. I would send it to a referee, with the request to check the typos and the saddle-point steps in Section 7 and Appendix D. After a light revision fixing the typos, I would accept.","headline":"Self-contained lecture notes that correctly re-derive the standard resetting results; zero new science by design, minor typos only, and genuinely useful for students.","tokens_in":21114,"tokens_out":2279,"would_cite":true,"duration_ms":19507,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Resetting turns infinite search times into finite, optimizable ones","keywords":["stochastic resetting","diffusion","first passage time","large deviations","renewal equations","nonequilibrium steady state","cost of resetting","non-Poissonian resetting"],"falsifier":"Take a waiting-time distribution with infinite mean, such as $\\psi(t)\\sim t^{-(1+\\alpha)}$ with $0<\\alpha<1$, run the resetting diffusion to long times, and measure the position distribution: a localized stationary distribution would contradict the paper's claim that a finite mean reset time is required for a non-equilibrium stationary state.","tokens_in":20051,"feed_emoji":"🎯","tokens_out":9744,"duration_ms":76856,"temperature":0.7,"pith_summary":"These lectures aim to make diffusion with stochastic resetting fully tractable from a standing start: a reader who knows basic probability and Laplace transforms can carry out every central calculation in full. The payoff is a clear view of how a simple resetting rule converts the infinite mean first-passage time of ordinary diffusion into a finite, tunable quantity, and how it generates a non-equilibrium stationary state with a cusp at the resetting point. The same renewal-equation machinery extends to additive functionals of the trajectory, yielding large-deviation rate functions for quantities such as the total cost of resetting, and to non-Poissonian resetting with general waiting-time distributions. If the notes succeed, they provide an entry point to a field whose core results normally sit behind a long bibliography.","feed_headline":"Resetting turns infinite search times into finite, optimizable ones","feed_subtitle":"Lecture notes make every key calculation reproducible from scratch, including large-deviation rate functions for resetting cost.","key_machinery":"The machinery is the pair of renewal equations — one integrating over the time of the last reset and one over the first reset — together with Laplace transforms in time. In the Laplace domain, the no-resetting propagator $\\tilde P_0(x,s)=(4sD)^{-1/2}e^{-\\sqrt{s/D}|x-x_0|}$ and the no-resetting survival probability $\\tilde q_0(s|x_0)=(1-e^{-x_0\\sqrt{s/D}})/s$ are the basic building blocks; every resetting quantity is expressed as a rational combination of these ingredients, with a pole that dominates long-time behaviour. The same structure produces the generating function $\\tilde G_r(k,s)=\\tilde G_0(k,s+r)/(1-r\\tilde G_0(k,s+r))$, whose pole $s_0(k,r)$ is converted into the large-deviation rate function $I(a)=-\\sup_k(ka+s_0(k,r))$ by a saddle-point/Legendre-Fenchel step. This pipeline — renewal equation, Laplace transform, locate the pole, invert by saddle point — is what carries every calculation in the notes.","core_discovery":"The paper's central claim is that the entire phenomenology of diffusion with stochastic resetting — the stationary state, the finite optimal mean first-passage time, the exponential long-time survival probability, and the large deviations of additive functionals — follows from one renewal-equation formalism plus Laplace transforms. For Poissonian resetting at rate $r$, the propagator obeys a last-renewal equation whose long-time limit is the Laplace distribution $P^*_r(x)=\\frac{\\alpha_0}{2}e^{-\\alpha_0|x-x_r|}$, $\\alpha_0=\\sqrt{r/D}$, and the mean first-passage time to an absorbing target at known distance $x_0$ becomes $\\langle T_r\\rangle=(e^{y}-1)/r$ with $y=x_0\\sqrt{r/D}$, minimized at $y=1.5936\\ldots$. The survival probability decays exponentially rather than as a power law, with a Gumbel-like form in the large-$y$ regime. The notes then derive an exact Laplace-domain expression for the generating function of any additive functional, whose pole $s_0(k,r)$ generates the large-deviation rate function by Legendre-Fenchel transform, and they apply it to the cost of resetting. Finally, for non-Poissonian resetting, the same renewal equations show that a stationary non-equilibrium state exists if and only if the mean time between resets is finite.","pith_inferences":["The notes leave implicit that the same pole-and-saddle-point scheme should yield large-deviation rate functions for other additive functionals of the resetting process, such as the area swept or the local time at the resetting site, without new ideas.","A natural extension the notes do not develop is optimizing the total cost of a search: the linear-cost rate function could be used to minimize the cost required to reach a target, rather than minimizing the mean first-passage time alone.","The finite-mean condition in Section 8 implies that waiting-time distributions with infinite mean would produce an ageing regime with no stationary state; the notes state the condition but do not explore that regime.","Because the large-deviation calculation in Section 7.3 keeps only the saddle-point exponential, the sub-exponential prefactors for $P_r(C,t)$ remain uncomputed; a reader could extract them from the full Bromwich integral."],"forward_implications":["A reader can reproduce the stationary state of Poissonian resetting as a Laplace distribution with decay length $\\sqrt{D/r}$, and see the probability current that makes it a non-equilibrium steady state.","The mean first-passage time to a target at known distance is finite for every finite resetting rate $r$, diverges as $r\\to0$ and $r\\to\\infty$, and has a unique optimum at dimensionless rate $y=1.5936\\ldots$.","Long-time survival under resetting becomes exponential, $q_r(t|x_0)\\sim e^{-r t e^{-y}}$ in the large-$y$ regime, replacing the diffusive power-law tail with a Gumbel-type decay.","For any additive functional with $f\\ge0$, the rate function is obtained from the pole of $\\tilde G_0(k,s+r)$ by a Legendre-Fenchel transform; the notes work out the linear-cost case explicitly.","For non-Poissonian resetting, a stationary state exists only when the mean waiting time is finite, and when the target distance is known the optimal waiting-time distribution is deterministic resetting."],"supporting_citations":[{"why":"Introduces diffusion with stochastic resetting and the renewal-equation treatment of the propagator and first-passage properties that the notes reproduce.","marker":"[1]"},{"why":"The detailed review that the notes complement; supplies the broader bibliography and the standard results the notes re-derive.","marker":"[2]"},{"why":"Gives the dynamical transition in relaxation to the stationary state, used in the notes for the equilibration front and the first large-deviation function.","marker":"[9]"},{"why":"Establishes the large-deviation formalism for Markov processes with resetting that underlies Section 6's generating-function derivation.","marker":"[14]"},{"why":"Supplies the theory of additive functionals of Brownian motion with resetting, the setting for the large-deviation claims in Section 6.","marker":"[15]"},{"why":"Defines and studies the cost of stochastic resetting, the example worked out in Section 7.","marker":"[17]"},{"why":"Computes cost statistics and optimisation for searches with reset, supporting the mean-cost and large-deviation calculations.","marker":"[18]"},{"why":"First-passage under restart, the basis for the non-Poissonian and deterministic-resetting optimality discussion in Section 8.","marker":"[24]"},{"why":"Diffusion with optimal resetting, the reference for optimising resetting when target distance may be random.","marker":"[27]"},{"why":"Provides the result that exponential waiting-time resetting is optimal for an exponential target distribution, used in the final optimisation discussion.","marker":"[28]"}],"fun_headline_variants":["Reset to find faster: optimal search via stochastic resetting","Large deviations reveal optimal resetting rates for search","Diffusion with resetting: from Poissonian to non-Poissonian","How resetting turns infinite search times finite"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that for non-Poissonian resetting the mean time between resets is finite; if a waiting-time distribution with an infinite mean is used, the stationary state disappears and the renewal-equation analysis no longer applies.","fun_headline_variants_meta":{"raw":{"variants":["Reset to find faster: optimal search via stochastic resetting","Large deviations reveal optimal resetting rates for search","Diffusion with resetting: from Poissonian to non-Poissonian","How resetting turns infinite search times finite"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000385,"raw_usage":{"total_tokens":2018,"prompt_tokens":912,"completion_tokens":1106,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":528,"completion_tokens_details":{"reasoning_tokens":1040}},"tokens_in":528,"tokens_out":1106,"duration_ms":8103,"temperature":1.0,"reasoning_tokens":1040,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:37:58.688231+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a waiting-time distribution with infinite mean, such as $\\psi(t)\\sim t^{-(1+\\alpha)}$ with $0<\\alpha<1$, run the resetting diffusion to long times, and measure the position distribution: a localized stationary distribution would contradict the paper's claim that a finite mean reset time is required for a non-equilibrium stationary state.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the large-deviation formalism for Markov processes with resetting that underlies Section 6's generating-function derivation."},{"cited_title":"Den Hollander, S","cited_arxiv_id":null,"evidence_quote":"Supplies the theory of additive functionals of Brownian motion with resetting, the setting for the large-deviation claims in Section 6."}],"review_version":1}