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Stationary states for stable processes with partial resetting

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Pith's one-line read Partial resetting forces stable processes into explicit stationary states, with a Brownian phase transition exactly at distance 2t.

desk verdict 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. read the letter →

arxiv 2412.15626 v1 pith:O4LKKNTA submitted 2024-12-20 math.PR

Tomasz Grzywny , Karol Szczypkowski , Zbigniew Palmowski , Bartosz Trojan This is my paper · ORCID
classification math.PR MSC 60G1060J3560K4082C0582C3135K0860J6560G51
keywords partialresettingtransitiondensityergodicmeasurenon-equilibriumstationarystatephaseq-Gammafunctionheatkernelasymptoticbehavior
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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. $|y| = 2t$.

What would settle it

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.

Watch

Extended reading notes

Core claim

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).

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

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.

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 (2)
  1. [Lemma 2.16 (proof), Section 2.3] 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.
  2. [Lemma 2.16 (proof), Section 2.3] 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.
minor comments (4)
  1. [Section 2.4, Proposition 2.22] 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.
  2. [Theorem 2.14] 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.
  3. [Lemma 2.16 (proof), Section 2.3] 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.
  4. [General] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the stationary density and its moments are derived from explicit spline/moment computations, while the only flagged limitation is a potential non-circular correctness gap.

full rationale

The central derivation chain is self-contained. The paper starts from the renewal equation (3.2)-(3.4), derives the spline representation in Theorem 3.4 and the measure representation (3.9), and then computes the moments of the splines and of the measures nu_t directly in Theorem 2.9, Corollary 2.10, Corollary 2.11, Proposition 2.13 and Theorem 2.14. There is no fitted parameter renamed as a prediction: the limiting measure nu is characterized by its moments through Carleman's condition, an independent determinacy criterion, and the stationary density rho_Y is defined by convolution in (2.42), not by assuming the target formula. The moment identity (1.4) follows from the computed moments of nu together with the self-similarity of Y, so it is a consequence rather than an input. The explicit q-series density (2.35) is recognized after the fact from the known TCP/AIMD law in [65] as matching the independently derived moments; this identification is not used to prove convergence or the moment formulas. Theorems C and D are obtained by steepest-descent analysis of the representation (3.10), with no ansatz smuggled in by citation. Self-citations such as [25], [42] and [43] occur as background references for prior Fourier-transform approaches or heat-kernel regularity estimates and are not load-bearing for the main claims. The manuscript itself flags in Remark 2.17 that a purely analytic proof of the total-variation lemma is deferred; the supplied probabilistic argument via [15], [2] and [6] may contain a correctness issue (the embedded-chain recursion appears to drop a factor c in the innovation term), but this is a potential error or gap, not a circular reduction of the theorem to its own assumptions. Accordingly, no circular step is identified.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters appear: c, q=c^α, α, d, and the Lévy measure parameters are model inputs, and all derived moment formulas are parameter-free in terms of those inputs. The splines g_k and measures ν_t are auxiliary mathematical constructions used to represent the transition density, not new physical degrees of freedom. The axioms listed are standard stable-process background and external criteria; none of them encodes the target stationary density formula as an assumed input.

assumptions (7)
  • domain assumption Y is a strictly α-stable Lévy process in R^d, α∈(0,2], with an absolutely continuous transition density p0 satisfying the scaling and derivative estimates of Lemma A.4.
    This is the standing hypothesis of Theorems A, 3.4, 3.7, 3.14, 3.15 and the asymptotic sections. It excludes degenerate stable processes such as deterministic drift, and supplies the smoothness and moment inputs used in Lemmas 2.20, 2.21 and 3.9.
  • standard math Heat-kernel asymptotics p0(s;0,y) ≈ min{s^{-d/α}, s/|y|^{d+α}} and the Blumenthal-Getoor limit for isotropic stable processes, cited [13], with subordinator variants from [14,26,85].
    Used in Sections 4.1 through 4.3 to convert integrals against ν_t into ratios involving the Lévy measure; these asymptotics are imported from the literature and not re-derived.
  • standard math Total variation convergence criteria for autoregressive chains: Borovkov-Foss sc-convergence [15], Harris recurrence of iterated random Lipschitz functions [2], and absolute continuity of GARCH-type distributions [6].
    These external results carry Lemma 2.16, which upgrades weak convergence of ν_t to total variation convergence. Remark 2.17 notes that a purely analytic proof is deferred.
  • standard math Mittag-Leffler asymptotic lim_{t→∞} t^{β-1} e^{-t} E_{α,β}(t^α)=1/α from [38, Theorem 4.3].
    Used at the end of Proposition 2.13 to evaluate the limiting scaled moments of ν_t.
  • standard math Carleman's condition and determinacy of the Stieltjes moment problem, plus Billingsley's subsequence and moment-convergence theorems [12].
    Used in Theorem 2.14 to pass from explicit moment formulas to weak convergence of ν_t and to identify the limit measure uniquely.
  • standard math The q-binomial theorem, q-Pochhammer identities, and the limiting interpretation (2.23) at negative integer moments.
    Central to the moment computations in Theorem 2.9 and Corollary 2.10; the paper uses these q-series identities as background machinery.
  • standard math The Lévy measure of a non-degenerate strictly stable process is atomless, as given in Appendix A, Eq. (A.1).
    Used in Theorem 3.15 to show that the two generator evaluations at a test function have different limits as the support radius shrinks, proving non-self-adjointness.

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Pith. "Pith review of Stationary states for stable processes with partial resetting." pith.science (2026). https://pith.science/paper/O4LKKNTA

@misc{pith2026241215626,
  author       = {Pith},
  title        = {Pith review of: Stationary states for stable processes with partial resetting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O4LKKNTA}},
  note         = {Machine review of arXiv:2412.15626}
}
abstract

We study a $d$-dimensional stochastic process $\mathbf{X}$ which arises from a L\'evy process $\mathbf{Y}$ by partial resetting, that is the position of the process $\mathbf{X}$ at a Poisson moment equals $c$ times its position right before the moment, and it develops as $\mathbf{Y}$ between these two consecutive moments, $c \in (0, 1)$. We focus on $\mathbf{Y}$ being a strictly $\alpha$-stable process with $\alpha\in (0,2]$ having a transition density: We analyze properties of the transition density $p$ of the process $\mathbf{X}$. We establish a series representation of $p$. We prove its convergence as time goes to infinity (ergodicity), and we show that the limit $\rho_{\mathbf{Y}}$ (density of the ergodic measure) can be expressed by means of the transition density of the process $\mathbf{Y}$ starting from zero, which results in closed concise formulae for its moments. We show that the process $\mathbf{X}$ reaches a non-equilibrium stationary state. Furthermore, we check that $p$ satisfies the Fokker--Planck equation, and we confirm the harmonicity of $\rho_{\mathbf{Y}}$ with respect to the adjoint generator. In detail, we discuss the following cases: Brownian motion, isotropic and $d$-cylindrical $\alpha$-stable processes for $\alpha \in (0,2)$, and $\alpha$-stable subordinator for $\alpha\in (0,1)$. We find the asymptotic behavior of $p(t;x,y)$ as $t\to +\infty$ while $(t,y)$ stays in a certain space-time region. For Brownian motion, we discover a phase transition, that is a change of the asymptotic behavior of $p(t;0,y)$ with respect to $\rho_{\mathbf{Y}}(y)$.

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Reference graph

Works this paper leans on

85 extracted references · 79 canonical work pages · cited by 1 Pith paper

  1. [15]

    Bogdan, A

    K. Bogdan, A. Stós, and P. Sztonyk, Harnack inequality for stable processes on/u1D451-sets, Studia Math. 158 (2003), no. 2, 163–198

    work page 2003

  2. [2]

    Akahori, C

    J. Akahori, C. Imamura Y . Constantinescu, and H. Pham, An application of risk theory to mortgage lending , Scand. Actuar. J. 5 (2022), 447–469

    2022

  3. [1]

    We can assume that /u1D461(1 −/u1D4620) converges to/u1D454∈ [ 0, ∞]. Since /u1D4620 satisfies (4.46), we have 1 /u1D4622 0 − 1 = /u1D451 2/u1D43F 1 /u1D4620 + /u1D461 /u1D43F (/u1D719′(/u1D461(1 −/u1D4620)) /u1D719(/u1D461(1 −/u1D4620)) − /u1D719′ (/u1D454) /u1D719(/u1D454) ) + /u1D461 /u1D43F (/u1D719′ (/u1D454) /u1D719(/u1D454) − /u1D43F /u1D461 ) . Hen...

    2021

  4. [3]

    Alsmeyer, On the Harris recurrence of iterated random Lipschitz functions and related convergence rate results, J

    G. Alsmeyer, On the Harris recurrence of iterated random Lipschitz functions and related convergence rate results, J. Theoret. Probab. 16 (2003), no. 1, 217–247

    2003

  5. [4]

    Economou, and M.J

    J.R Artalejo, A. Economou, and M.J. Lopez-Herrero, Evaluating growth measures in populations subject to binom ial and geometric catastrophes, Math. Biosci. Eng. 4 (2006), no. 4, 573–94

    2006

  6. [5]

    Assaf, A

    M. Assaf, A. Kamenev, and B. Meerson, Population extinction risk in the aftermath of a catastroph ic event, Phys. Rev. E 79 (2009), 011127

    2009

  7. [6]

    Avrachenkov, A

    K. Avrachenkov, A. Piunovskiy, and Y . Zhang, Markov processes with restart, J. Appl. Probab. 50 (2013), no. 4, 960–968

    work page 2013

  8. [7]

    Basrak, R.A

    B. Basrak, R.A. Davis, and T. Mikosch, Regular variation of GARCH processes , Stochastic Process. Appl. 99 (2002), no. 1, 95–115

    work page 2002

Show all 85 references
  1. [8]

    thesis, Braunschweig Universite, 2011

    Anita Behme, Generalized Ornstein–Uhlenbeck processes and extensions , Ph.D. thesis, Braunschweig Universite, 2011

    2011

  2. [9]

    Bell, Searching behaviour: The behavioural ecology of finding res ources, Chapman and Hall, 1991

    W.J. Bell, Searching behaviour: The behavioural ecology of finding res ources, Chapman and Hall, 1991

    1991

  3. [10]

    Bénichou, M

    O. Bénichou, M. Coppey, M. Moreau, P-H. Suet, and R. Voitu riez, Optimal search strategies for hidden targets , Phys. Rev. Lett. 94 (2005), 198101

    2005

  4. [11]

    Bénichou, C

    O. Bénichou, C. Loverdo, Moreau M., and R. Voituriez, Intermittent search strategies, Rev. Mod. Phys. 83 (2011), 81

    2011

  5. [12]

    Berezhkovskii, A

    A. Berezhkovskii, A. Szabo, R. Urbakh, and A. Kolomeisk y, Dependence of the enzymatic velocity on the substrate dissociation rate, J. Phys. Chem. B 121 (2016), 3437. STATIONARY STATES FOR STABLE PROCESSES WITH PARTIAL RESETT ING 59

    2016

  6. [13]

    P. Billingsley, Probability and measure, third ed., Wiley Series in Probability and Mathematical St atistics, John Wiley & Sons, Inc., New Y ork, 1995, A Wiley-Interscience Publication

    1995

  7. [14]

    Blumenthal and R.K

    R.M. Blumenthal and R.K. Getoor, Some theorems on stable processes , Trans. Amer. Math.Soc. 95 (1960), 263–273

    1960

  8. [16]

    Borovkov and S.G

    A.A. Borovkov and S.G. Foss, Stochastically recursive sequences and their generalizat ions, Siberian Adv. Math. 2 (1992), no. 1, 16–81

    1992

  9. [17]

    Borovkov and D

    K. Borovkov and D. Vere-Jones, Explicit formulae for stationary distributions of stress r elease processes, J. Appl. Probab. 37 (2000), no. 2, 315–321

    2000

  10. [18]

    Böttcher, R

    B. Böttcher, R. Schilling, and J. Wang, Lévy matters. III, Lecture Notes in Mathematics, vol. 2099, Springer, Cham, 2 013

  11. [19]

    Boxma, D

    O. Boxma, D. Perry, W. Stadje, and Sh. Zacks, A Markovian growth-collapse model , Adv. in Appl. Probab. 38 (2006), no. 1, 221–243

    2006

  12. [20]

    Brilliantov and T

    N.V . Brilliantov and T. Poschel, Kinetic theory of granular gases , Oxford University Press, 2004

    2004

  13. [21]

    Calabrese, A

    S. Calabrese, A. Porporato, F. Laio, P.D. Odorico, and L . Ridolfi, Age distribution dynamics with stochastic jumps in mortality, Proc. R. Soc. A: Math. Phys. Eng. Sci. 473 (2017), 20170451

    2017

  14. [22]

    Clark, Ecological disturbance as a renewal process: Theory and app lication to fire history , Oikos 56 (1989), 17

    J.S. Clark, Ecological disturbance as a renewal process: Theory and app lication to fire history , Oikos 56 (1989), 17

    1989

  15. [23]

    Cofré, L

    R. Cofré, L. Videla, and F. Rosas, An introduction to the non-equilibrium steady states of max imum entropy spike trains , Entropy 21 (2019), no. 9

    2019

  16. [24]

    Dahlenburg, A

    M. Dahlenburg, A. Chechkin, R. Schumer, and R. Metzler, Stochastic resetting by a random amplitude , Phys. Rev. E 103 (2021), 052123

    2021

  17. [25]

    Derrida, Non equilibrium steady states: fluctuations and large devia tions of the density and of the current , J

    B. Derrida, Non equilibrium steady states: fluctuations and large devia tions of the density and of the current , J. Stat. Mech. Theory Exp. 10 (2007), no. 7, 7023

    2007

  18. [26]

    Di Bello, A

    C. Di Bello, A. Chechkin, A. Hartmann, Z. Palmowski, and R. Metzler, Time-dependent probability density function for partial resetting dynamics, New J. Phys. 25 (2023), 082002

    2023

  19. [27]

    Doetsch, Introduction to the theory and application of the Laplace tr ansformation, Springer-Verlag, New Y ork-Heidelberg, 1974

    G. Doetsch, Introduction to the theory and application of the Laplace tr ansformation, Springer-Verlag, New Y ork-Heidelberg, 1974

    1974

  20. [28]

    Dumas, F

    V . Dumas, F. Guillemin, and Ph. Robert, A Markovian analysis of additive-increase multiplicative -decrease algorithms, Adv. in Appl. Probab. 34 (2002), no. 1, 85–111

    2002

  21. [29]

    Eliazar and J

    I. Eliazar and J. Klafter, A growth-collapse model: Lévy inflow, geometric crashes, an d generalized Ornstein-Uhlenbeck dynamics, Phys. A 334 (2004), no. 1-2, 1–21

    2004

  22. [30]

    Eule and J

    S. Eule and J. Metzger, Non-equilibrium steady states of stochastic processes with intermittent resetting, New J. Phys.18 (2016), 033006

    2016

  23. [31]

    Evans and S.N

    M.R. Evans and S.N. Majumdar, Diffusion with stochastic resetting , Phys. Rev. Lett. 106 (2011), 160601

    2011

  24. [32]

    , Diffusion with resetting in arbitrary spatial dimension , J. Phys. A 47 (2014), no. 28, 285001, 19

    2014

  25. [33]

    Evans, S.N

    M.R. Evans, S.N. Majumdar, and K. Mallick, Optimal diffusive search: nonequilibrium resetting versus equilibrium dynamics, J. Phys. A 46 (2013), no. 18, 185001, 13

    2013

  26. [34]

    Evans, S.N

    M.R. Evans, S.N. Majumdar, and G. Schehr, Stochastic resetting and applications, J. Phys. A 53 (2020), no. 19, 193001, 67

    2020

  27. [36]

    Floreani and A

    S. Floreani and A. González Casanova, Non-equilibrium steady state of the symmetric exclusion process with reservoirs, ArXiv: 2307.02481, 2023

    2023 arXiv

  28. [37]

    Fuchs, S

    J. Fuchs, S. Goldt, and U. Seifert, Stochastic thermodynamics of resetting, EPL-Europhys Lett. 113 (2016), no. 6, 60009

    2016

  29. [38]

    Gerber and R

    L.R. Gerber and R. Hilborn, Catastrophic events and recovery from low densities in popu lations of otariids: implications for risk of extinction, Mammal Rev. 31 (2001), no. 9, 131

    2001

  30. [39]

    Gorenflo, A.A

    R. Gorenflo, A.A. Kilbas, F. Mainardi, and S. Rogosin, Mittag-Leffler functions, related topics and applications , Springer Monographs in Mathematics, Springer, Berlin, 2020

    2020

  31. [40]

    Grier and Y

    D.G. Grier and Y . Roichman, Holographic optical trapping, Appl. Opt. 45 (2006), no. 5, 880–887

    2006

  32. [41]

    Grincevičius, Random difference equations and renewal theory for products of random matrices, Lithuanian Math

    A.R. Grincevičius, Random difference equations and renewal theory for products of random matrices, Lithuanian Math. J. 15 (1975), 580–589

    1975

  33. [42]

    Grünwald and R.H

    D. Grünwald and R.H. Singer, In vivo imaging of labelled endogenous /u1D6FD-actin mRNA during nucleocytoplasmic transport , Nature 467 (2010), no. 7315, 604

    2010

  34. [43]

    Grzywny and K

    T. Grzywny and K. Szczypkowski, Heat kernels of non-symmetric Lévy-type operators, J. Differ. Equations 267 (2019), no. 10, 6004–6064

    2019

  35. [44]

    4, 3191–3223

    , Lévy processes: concentration function and heat kernel bou nds, Bernoulli 26 (2020), no. 4, 3191–3223

    2020

  36. [45]

    Guillemin, Ph

    F. Guillemin, Ph. Robert, and B. Zwart, AIMD algorithms and exponential functionals , Ann. Appl. Probab. 14 (2004), no. 1, 90–117

    2004

  37. [46]

    Gupta and A.M

    Sh. Gupta and A.M. Jayannavar, Stochastic resetting: A (very) brief review , Front. Phys. 10 (2022), 789097

    2022

  38. [47]

    Hardin, Skewed stable variables and processes, Tech

    D. Hardin, Skewed stable variables and processes, Tech. report, North Carolina University, 1984

    1984

  39. [48]

    Hartman and A

    P. Hartman and A. Wintner, On the infinitesimal generators of integral convolutions , Am. J. Math. 64 (1942), 273–298

    1942

  40. [49]

    Iserles, On the generalized pantograph functional-differential equ ation, Eur

    A. Iserles, On the generalized pantograph functional-differential equ ation, Eur. J. Appl. Math. 4 (1993), no. 1, 1–38. 60 TOMASZ GRZYWNY, ZBIGNIEW PALMOWSKI, KAROL SZCZYPKOWSKI , AND BARTOSZ TROJAN

    1993

  41. [50]

    Kella, D

    O. Kella, D. Perry, and W. Stadje, A stochastic clearing model with a Brownian and a compound Po isson component,, Probab. Eng. Inf. Sci. 17 (2003), 1

    2003

  42. [51]

    Knopova and R

    V . Knopova and R. Schilling, A note on the existence of transition probability densities of Lévy processes , Forum Math. 25 (2013), no. 1, 125–149

    2013

  43. [52]

    Kou, A jump-diffusion model for option pricing , Manag

    S. Kou, A jump-diffusion model for option pricing , Manag. Sci 48 (2002), 1086

    2002

  44. [53]

    Kusolitsch, Why the theorem of Scheffé should be rather called a theorem of Riesz, Period

    N. Kusolitsch, Why the theorem of Scheffé should be rather called a theorem of Riesz, Period. Math. Hungar. 61 (2010), no. 1-2, 225–229

    2010

  45. [54]

    Löpker and W

    A. Löpker and W. Stadje, Hitting times and the running maximum of Markovian growth-c ollapse processes, J. Appl. Probab. 48 (2011), no. 2, 295–312

    2011

  46. [55]

    Löpker, J.S.H

    A. Löpker, J.S.H. van Leeuwaarden, and T.J. Ott, TCP and iso-stationary transformations, Queueing Syst. 63 (2009), 459–475

    2009

  47. [56]

    Majumdar, S

    S.N. Majumdar, S. Sabhapandit, and G. Schehr, Dynamical transition in the temporal relaxation of stochas tic processes under resetting, Phys. Rev. E 91 (2015), no. 5, 052131, 8

    2015

  48. [57]

    Malakar, V

    K. Malakar, V . Jemseena, A. Kundu, K. Kumar, S. Sabhapan dit, S. Majumdar, S. Redner, and A. Dhar, Steady state, relaxation and first-passage properties of a run-and-tumble particle i n one-dimension, J. Stat. Mech. 93 (2018), 043215

    2018

  49. [58]

    X.Y . Mau, X. Feng, and A. Porporato, Multiplicative jump processes and applications to leachin g of salt and contaminants in the soil, Phys. Rev. E 90 (2014), 1

    2014

  50. [59]

    Mendoza, Sudden stops, financial crises, and leverage , Am

    E.G. Mendoza, Sudden stops, financial crises, and leverage , Am. Econ. Rev. 100 (2010), 1941

    2010

  51. [60]

    Merton, Option pricing when underlying stock returns are discontin uous, J

    R.C. Merton, Option pricing when underlying stock returns are discontin uous, J. Financ. Econ. 3 (1976), 125

    1976

  52. [61]

    S. P. Meyn and R.L. T weedie, Markov chains and stochastic stability , Communications and Control Engineering Series, Springer-Verlag London, 1993

    1993

  53. [62]

    Mikosch and G

    T. Mikosch and G. Samorodnitsky, The supremum of a negative drift random walk with dependent h eavy-tailed steps, Ann. Appl. Probab. 10 (2000), no. 3, 1025–1064

    2000

  54. [63]

    Mukherjee, K

    B. Mukherjee, K. Sengupta, and S.N. Majumdar, Quantum dynamics with stochastic reset, Phys. Rev. B 98 (2018), 104309

    2018

  55. [64]

    Odorico, F

    P.D. Odorico, F. Laio, L. Ridolfi, P.D. Odorico, F. Laio, and L. Ridolfi, A probabilistic analysis of fire-induced tree-grass coexistence in savannas, Am. Nat. 167 (2006), 78

    2006

  56. [65]

    Ott and J.H.B

    T.J. Ott and J.H.B. Kemperman, Transient behavior of processes in the TCP paradigm, Probab. Engrg. Inform. Sci. 22 (2008), no. 3, 431–471

    2008

  57. [66]

    Ott, J.H.B

    T.J. Ott, J.H.B. Kemperman, and M. Mathis, The stationary behavior of ideal TCP congestion avoidance , http://www.teunisott.com/Papers/TCP_Paradigm/TCPwindow.pdf, 1996

    1996

  58. [67]

    Pal, Diffusion in a potential landscape with stochastic resettin g, Phys

    A. Pal, Diffusion in a potential landscape with stochastic resettin g, Phys. Rev. E 91 (2015), 012113

    2015

  59. [68]

    Plata, D

    C.A. Plata, D. Gupta, and S. Azaele, Asymmetric stochastic resetting: modeling catastrophic e vents, Phys. Rev. E 102 (2020), no. 5, 052116, 9

    2020

  60. [69]

    Reuveni S., Urbakh and J

    M. Reuveni S., Urbakh and J. Klafter, Role of substrate unbinding in Michaelis–Menten enzymatic reactions, Proc. Natl. Acad. Sci. USA 111 (2014), 4391

    2014

  61. [70]

    Robin, L

    T. Robin, L. Hadany, and M. Urbakh, Random search with resetting as a strategy for optimal polli nation, Phys. Rev. E 99 (2019), 052119

    2019

  62. [71]

    Robin, S

    T. Robin, S. Reuveni, and M. Urbakh, Single-molecule theory of enzymatic inhibition , Nat. Commun. 9 (2018), no. 1, 779

    2018

  63. [72]

    Roldan, A

    E. Roldan, A. Lisica, Sanchez-Taltavull, D., and S.W. G rill, Stochastic resetting in backtrack recovery by RNA polymera ses, Phys. Rev. E 93 (2016), no. 6, 062411

    2016

  64. [73]

    Sato, Lévy processes and infinitely divisible distributions , Cambridge Studies in Advanced Mathematics, vol

    K.-I. Sato, Lévy processes and infinitely divisible distributions , Cambridge Studies in Advanced Mathematics, vol. 68, Cam- bridge University Press, Cambridge, 1999, Translated from the 1990 Japanese original, Revised by the author

    1999

  65. [74]

    Shanbhag and M

    D.N. Shanbhag and M. Sreehari, On certain self-decomposable distributions , Z. Wahrscheinlichkeitstheorie verw Gebiete 38 (1977), 217–222

    1977

  66. [75]

    Sornette, Why stock markets crash: critical events in complex financia l systems, Princeton University Press, 2017

    D. Sornette, Why stock markets crash: critical events in complex financia l systems, Princeton University Press, 2017

    2017

  67. [76]

    Suweis, A

    S. Suweis, A. Porporato, A. Rinaldo, and A. Maritan, Prescription-induced jump distributions in multiplicati ve Poisson processes, Phys. Rev. E 83 (2011), 1

    2011

  68. [77]

    Suweis, A

    S. Suweis, A. Rinaldo, S. E. Van Der Zee, E. Daly, A. Marit an, and A. Porporato, Stochastic modeling of soil salinity, Geophys. Res. Lett. 37 (2010), 1

    2010

  69. [78]

    Tal-Friedman, A

    O. Tal-Friedman, A. Pal, A. Sekhon, Sh. Reuveni, and Y . R oichman, Experimental realization of diffusion with stochastic resetting, J. Phys. Chem. Lett. 11 (2020), no. 17, 7350–7355

    2020

  70. [79]

    Tal-Friedman, Y

    O. Tal-Friedman, Y . Roichman, and Sh. Reuveni, Diffusion with partial resetting , Phys. Rev. E 106 (2022), no. 5, Paper No. 054116, 13

    2022

  71. [80]

    van der Hofstad, S

    R. van der Hofstad, S. Kapodistria, Z. Palmowski, and S.Shneer, Unified approach for solving exit problems for additive-increase and multiplicative-decrease processes, J. Appl. Probab. 60 (2023), no. 1, 85–105

    2023

  72. [81]

    van Leeuwaarden, A.H

    J.S.H. van Leeuwaarden, A.H. Löpker, and T.J. Ott, TCP and iso-stationary transformations, Queueing Syst. 63 (2009), no. 1-4, 459–475

    2009

  73. [82]

    Whitehouse, M.R

    J. Whitehouse, M.R. Evans, and S.N. Majumdar, Effect of partial absorption on diffusion with resetting , Phys. Rev. E87 (2013), 022118. STATIONARY STATES FOR STABLE PROCESSES WITH PARTIAL RESETT ING 61

    2013

  74. [83]

    Wiśniewolski, On the probabilistic representations of solutions of panto graph equations and triangle coefficients , J

    M. Wiśniewolski, On the probabilistic representations of solutions of panto graph equations and triangle coefficients , J. Differ. Equations 379 (2024), 600–625

    2024

  75. [84]

    Wu Y . and W.Q. Zhu, Stochastic analysis of a pulse-type prey-predator model , Phys. Rev. E 77 (2008), 1

    2008

  76. [85]

    Zheng, Ergodic theorems for stress release processes, Stochastic Process

    X.G. Zheng, Ergodic theorems for stress release processes, Stochastic Process. Appl. 37 (1991), no. 2, 239–258

    1991

  77. [86]

    Zolotarev, One-dimensional stable distributions , Translations of Mathematical Monographs, vol

    V .M. Zolotarev, One-dimensional stable distributions , Translations of Mathematical Monographs, vol. 65, Americ an Mathe- matical Society, Providence, RI, 1986. T/o.pc/m.pc/a.pc/s.pc/z.pc G/r.pc/z.pc/y.pc/w.pc/n.pc/y.pc, W/y.pc/d.pc/z.pc/i.pc/a.pc/lslash.pc M/a.pc/t.pc/e.pc/m...

    1986

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