First-passage statistics of random walks: a general approach via Riemann-Hilbert problems

Giampaolo Cristadoro; Mattia Radice

REVIEW 3 major objections 4 minor 74 references

First-passage statistics of random walks: a general approach via Riemann-Hilbert problems

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper claims that the first-passage statistics of any one-dimensional random walk with independent jumps can be written exactly in terms of the jump characteristic function alone, for continuous and discrete, symmetric and asymmetric…

desk verdict Genuinely new b>0 and asymmetric first-passage formulas, with a proof that needs tightening: 'universal' overstates the verified assumptions, and the H_0=1 contour argument is wrong as written. read the letter →

arxiv 2507.11196 v1 pith:2EA6IXB5 submitted 2025-07-15 cond-mat.stat-mech

Mattia Radice , Giampaolo Cristadoro This is my paper · ORCID

classification cond-mat.stat-mech PACS 05.40.Fb02.50.-r

keywords randomwalksfirst-passagestatisticsRiemann-HilbertproblemWiener-HopffactorizationsurvivalprobabilityLévyflightsleap-overcharacteristicfunction

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

The paper aims to show that the complete first-passage statistics of a one-dimensional random walk with independent, identically distributed jumps can be obtained from a small set of explicit formulae that depend only on the characteristic function of the jumps. This would settle, in one framework, problems that previously required separate treatments for continuous versus discrete jumps and for symmetric versus asymmetric jump distributions. If the claim holds, first-passage times, first-passage positions (leap-over), and survival probabilities become directly computable, exactly or asymptotically, for essentially any jump law, including Lévy flights with drift and skewness. The payoff is a unified tool for extreme-value statistics, search processes, record theory, and resetting models.

What carries the argument

The load-bearing object is the canonical solution ψ0(z,ζ) of the homogeneous Riemann-Hilbert problem, ψ0(z,ζ)=exp(1/(2πi)∫_{−∞}^{∞} ln[1−ζφ̂(t)]/(t−z) dt) for continuous jumps, with a unit-circle analogue for discrete jumps. It factorizes the coefficient 1−ζφ̂(k) into boundary values of a sectionally analytic function, and the Sokhotski-Plemelj formulae turn that factorization into the explicit phase factors and the kernel H_b(k,ζ) that appear in the final representations. The same machinery yields the exact asymptotic exponents and prefactors for survival probabilities of Lévy flights in Section 4.

What would settle it

For an asymmetric discrete jump distribution whose characteristic function makes 1−ζφ̂(z) have a zero inside the unit circle for some ζ in (0,1), the residue formula (15) for H_b(k,ζ) would omit the contribution of that zero, so direct numerical evaluation of Q_b(k,ζ) from simulating millions of walks would disagree with (10b); agreement for all ζ would confirm the factorization assumption.

Watch

Extended reading notes

Core claim

The paper's central claim is that for any 0<ζ<1 and b≥0, the generating functions F_b(k,ζ) and Q_b(k,ζ) — which encode the joint distribution of the first-passage time and first-passage position beyond b, and the distribution of the walk that has not crossed b by step n — admit the exact representations (6a)-(6b) for continuous jumps and (10a)-(10b) for discrete jumps, written solely in terms of the jump characteristic function φ̂(k). The same formulae apply to symmetric and asymmetric jump distributions, which previous Pollaczek-Spitzer and Hopf-Ivanov-type identities did not. This rests on reading the basic renewal relation 1 − F_b = (1 − ζ φ̂) Q_b as a non-homogeneous Riemann-Hilbert problem and solving it through the canonical solution of the homogeneous problem.

Load-bearing premise

Everything rests on the assumption that 1 − ζ φ̂(z) admits a canonical factorization with no zeros in the relevant half-planes (or inside the unit circle for discrete jumps) and with the factors approaching 1 at infinity; the paper does not verify this for the full class of jump distributions it claims, in particular heavy-tailed asymmetric ones.

Editorial extensions

If this is right

For symmetric continuous jump distributions, formulae (6a)-(6b) reproduce the Sparre-Andersen universality Q0(0,ζ)=1/√(1−ζ), and for b=0 they recover the known b=0 result from [54].
For asymmetric continuous jumps, the survival probability's large-n decay exponent changes: table 1 gives the exponent as a function of α, skewness β, and drift μ, with exact prefactors; in the region 1<α<2 with negative drift the survival probability decays to a positive constant.
For discrete jumps, formulae (10a)-(10b) generalize the symmetric b=0 result of [48] to asymmetric jumps and to any threshold b, with the Bernoulli walk worked out in closed form.
The leap-over distribution for symmetric Lévy flights is predicted to decay as ℓ^{−1−α/2}, always with diverging mean for 0<α<2, and with the b-dependence given by H_b(0,1) whose small-b and large-b asymptotics are derived.
First-passage duality appears in the explicit examples: conditioned on eventual escape, the first-passage time distribution is independent of the sign of the drift for the skewed Laplace and Bernoulli walks.

Reading between the lines

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

Because the formulae are written entirely in terms of φ̂(k), one could in principle tabulate F_b and Q_b numerically from φ̂ for any jump law, turning first-passage calculations into a quadrature problem rather than a simulation problem; the paper does not explore this computational route.
The Riemann-Hilbert formulation suggests the same technique applies to the cost-process generalization mentioned in Section 6, and possibly to walks with correlated increments, where the coefficient G would carry extra structure; that is an extension the authors gesture at but do not develop.
The exact prefactors for Lévy-flight survival probabilities in (B.14), (B.25) and (B.28) are directly testable in Monte Carlo experiments beyond the paper's own checks; a mismatch in the constants would pinpoint the limits of the canonical-factorization assumption.
The paper's universality claim is conditional on a factorization property of 1−ζφ̂(z) that is verified in the examples but not proven for the full class; checking the zero-location of 1−ζφ̂(z) in the complex plane for a given jump law is a practical precondition for using the formulae.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

This paper advances the toolkit for first-passage statistics of 1D random walks. The joint generating functions for crossing time/position and for non-crossing walks are written solely in terms of the jump characteristic function, for b≥0 and for asymmetric jumps. The b>0 formulas and the asymmetric treatment are new; the exact decay coefficients for skewed Lévy flights go beyond known exponents. The Bernoulli example reproduces the known generating function, and simulations back the main asymptotics. This is real, useful work.

Two soft spots, both in the derivation rather than the answers. First, the 'universally valid' claim in the abstract is too strong. Section 5 states RH conditions—Hölder continuity and a power-law decay condition on G(k)=1−ζφ̂(k)—but never verifies them for the full claimed class. For slowly decaying characteristic functions, the phase integral (7) need not converge, and the canonical solution argument requires assumptions that are absent. The worked examples satisfy the conditions, so they don't establish the universal claim.

Second, the proof that H_0=1 in Section 5.1 is flawed as written. With the contour B passing to the right of u=0, the integral ∫_B du/u is iπ, not 2πi, at b=0. So the first term in (119) is 1/2, not 1; the claimed H_0=1 follows only by taking the limit b→0+, not by evaluating the representation directly. The b>0 formulas likely survive—the checks point that way—but the text at the end of Section 5.1 is inconsistent with its own contour convention.

Both issues are fixable. State the precise conditions under which the RH factorization holds, and either define H_0 as a limit or correct the contour calculation. I would trust the explicit examples and the b>0 forms, but I would not cite this for the 'general' claim without the repair.

Recommendation: send it to a serious referee. It deserves referee time, but it needs revision before acceptance.

Best

Referee Report

3 major / 4 minor

Summary. The paper studies first-passage statistics for one-dimensional random walks with iid jumps, focusing on the joint generating function F_b(k,ζ) of the first-passage time and position beyond a threshold b≥0, and on the generating function Q_b(k,ζ) of the position of walks that have not crossed b by time n. The authors propose Riemann-Hilbert solutions of the Pollaczek-Spitzer-type relation (5), obtaining the explicit representations (6a)-(6b) for continuous jumps and (10a)-(10b) for discrete jumps, written solely in terms of the jump characteristic function. They illustrate the formalism with skewed Laplace, Erlang-mixture, and Bernoulli walk examples, and derive asymptotic results for Lévy flights and leap-over distributions, with supporting numerical simulations. The paper claims the approach is universally valid for symmetric and asymmetric, continuous and discrete jump distributions.

Significance. The approach is attractive and potentially unifying: it generalizes known fluctuation identities to asymmetric and discrete cases and provides explicit formulae from which asymptotics can be extracted. The worked examples are careful and pass nontrivial checks, including recovery of E[ζ^τ]=(1-√(1-ζ²))/ζ for the symmetric Bernoulli walk, matching of the b=0 limits with known results, and good agreement with simulations. If the general formulae can be established under precisely stated hypotheses, this would be a valuable contribution to the statistical physics of first-passage processes. However, the proof as written contains a concrete error in the b=0 contour evaluation and does not justify the full universality claim; these issues are fixable but require revision.

major comments (3)

[Section 5.1, Eq. (119)] The statement immediately below Eq. (119), that 'The first integral at the rhs is equal to 1 for any b ≥ 0', is incorrect at b=0. With B defined in Eq. (9) as the vertical line along the imaginary axis indented so that u=0 remains on the left, and with the principal-value convention of Eq. (117), one computes ∫_B du/u = πi. Hence the first term (1/2πi)∫_B e^{ub}/u du equals 1/2 at b=0; it equals 1 only for b>0, where the closing semicircle in the left half-plane has vanishing contribution. Consequently, the proof of H_0(k,ζ)=1 given after Eq. (119) is invalid, and the assertion that (6a)-(6b) with H_b from (9) reduce at b=0 to the known results is not established. The b=0 formulae themselves are correct when obtained from the homogeneous solution (105)-(108), but the unified representation (9) is discontinuous at b=0 as written. The authors should repair this by defining H_0 as the b→0+ limit or by correcting the contour evaluation.
[Section 5, conditions (i)-(iii); Section 5.1, Eqs. (105)-(117)] The RH derivation assumes that G(k)=1-ζφ̂(k) is Hölder continuous, nonzero, decays to its limit as |k|^{-μ} with μ>0, and that the canonical solution Ψ±0(z) exists with Ψ±0(z)→1 at infinity, with the phase integral (7) convergent and identity (111) justified. These hypotheses are not verified for the full class claimed in the abstract ('universally valid ... continuous and discrete, symmetric and asymmetric jump distributions'). For example, if φ̂(k) decays as 1/ln|k|, then ln[1-ζφ̂(k)] decays only logarithmically, the phase integral (7) diverges, and the Cauchy integral in (111) is not meaningful. All examples in the paper (Laplace, Erlang mixtures, stable laws, Bernoulli) satisfy much stronger decay and analyticity conditions, so they do not establish universality. The authors should either state precise conditions on φ̂ under which (6a)-(6b) and (10a)-(10b) are proved, or restrict the universality claim accordingly.
[Section 5.2, Eq. (15)] The residue representation for H_b(k,ζ) involves w^{-b-1} and is well-defined only when b is a nonnegative integer (or, more generally, a multiple of the lattice spacing). The paper presents formulae (10a)-(10b) for all b≥0 without stating this restriction. For lattice walks, any real threshold is equivalent to an integer threshold, so the restriction is benign, but it should be stated explicitly in the statement of the main formulae.

minor comments (4)

[Eq. (48)] The expression 'Σ n qn(n)(κ²+1)^{n-1}η^{n-1}' appears to contain a typographical error: qn(n) should likely be q_b(n), and the parameter κ used in this equation is not defined.
[Table 1] The α=1 row of Table 1 is garbled: the second and third rows appear to give the same exponent, and the role of µ=0 in that row is unclear.
[Section 2.1, Eq. (9)] The description of the contour B as 'a vertical line along the imaginary axis, deformed in such a way that u=0 remains on the left' is ambiguous; the orientation of the contour and the side of the singularity on which it passes should be specified explicitly.
[Section 4.1] The phrase 'Such kinds of random walks are called Lévy flights' should be 'This kind of random walk is called a Lévy flight'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main formulae are derived from the jump characteristic function through a Riemann-Hilbert solution of an exact fluctuation identity, with no fitted parameter or target result used as input.

full rationale

The derivation chain starts from relation (5), an exact identity linking F_b(k,ζ) and Q_b(k,ζ) in terms of 1−ζφ̂(k). The only input throughout is the jump characteristic function φ̂(k). The canonical solution Ψ0(z,ζ) is constructed directly from ln[1−ζφ̂(t)] via the Cauchy integral (105), and the boundary-value problem is solved using Sokhotski-Plemelj formulae, decay conditions at infinity, and the constant A_b=1/2 fixed by (96). The factor H_b(k,ζ) is then an explicit contour integral, with no fitting step and no appeal to the target F_b or Q_b as an input. The worked examples proceed by factorizing 1−ζφ̂ and evaluating residues, which is algebra rather than circular reasoning. The self-citations [12] and [57] are not load-bearing: [57] supplies an example jump density and [12] supplies an integral evaluation used in an asymptotic prefactor, both peripheral to the main theorem. The paper's universality claim is stronger than what is verified: Section 5 requires Hölder and decay conditions on G(k)=1−ζφ̂, and the b=0 contour convention around (119) warrants technical scrutiny. Those are validity concerns, not circularity, since they concern whether the stated assumptions cover the claimed domain, not whether the outputs are equivalent to the inputs by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted constants or invented entities. The formulas are derived from the jump characteristic function, a standard input. The main non-standard premise is the canonical RH factorization with the required analyticity and decay, which is reasonable because 1-ζφ has positive real part, but which is asserted rather than proven for the general class of distributions claimed. All other ingredients are standard.

assumptions (4)

standard math Sokhotski-Plemelj formulae and Cauchy principal value calculus for singular integrals
Used throughout Section 5 and Appendix A to take boundary values of sectionally analytic functions.
ad hoc to paper Canonical factorization of 1-ζφ̂(k) with ψ±(z) analytic, nonzero, and tending to 1 in the respective half-planes
Needed for the canonical solution (105) and the identity (111); not explicitly verified for the full class of jump distributions claimed in the abstract.
domain assumption The random walk has iid real jumps, starts at 0, and the threshold satisfies b≥0; in the discrete case the lattice spacing is set to 1 and b is integer
Model definition in Section 2; discrete derivation in Section 5.2 assumes integer b.
standard math Tauberian theorems connecting ζ→1 singular behavior of generating functions to large-n asymptotics
Used in Section 4 and Appendices B-C to extract q0(n) asymptotics from (B.1) and similar.

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Pith. "Pith review of First-passage statistics of random walks: a general approach via Riemann-Hilbert problems." pith.science (2026). https://pith.science/paper/2EA6IXB5

@misc{pith2026250711196,
  author       = {Pith},
  title        = {Pith review of: First-passage statistics of random walks: a general approach via Riemann-Hilbert problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2EA6IXB5}},
  note         = {Machine review of arXiv:2507.11196}
}

abstract

We study first-passage statistics for one-dimensional random walks $S_n$ with independent and identically distributed jumps starting from the origin. We focus on the joint distribution of the first-passage time $\tau_b$ and first-passage position $S_{\tau_b}$ beyond a threshold $b\geq0$, as well as the distribution of $S_n$ for the walks that do not cross $b$ up to step $n$. By solving suitable Riemann-Hilbert problems, we are able to obtain exact and semi-explicit general formulae for the quantities of interest. Notably, such formulae are written solely in terms of the characteristic function of the jumps. In contrast with previous results, our approach is universally valid, applicable to both continuous and discrete, symmetric and asymmetric jump distributions. We complement our theoretical findings with explicit examples.

Figures

Figures reproduced from arXiv: 2507.11196 by the authors.

**Figure 2.** Conditional leap-over PDF for the case m = 2, given by the ratio (65) over (67). Panel (a) shows distributions with different µ and b fixed, while panel (b) displays different b with µ fixed. The plots are in logarithmic scale on the ℓ-axis to show that major differences are visible for small ℓ [see, in particular, panel (b)]. The data (markers) are the results of numerical simulations, obtained by evolving N random… view at source ↗

**Figure 3.** Asymptotic decay of q0(n) for L´evy flights with jumps drawn from skewed L´evy stable laws. The asymptotic behaviour of the data is compared with our analytical predictions, see below. Panel (a): case 0 < α < 1. The data (markers) are obtained by evolving 106 random walks and the analytical prediction (solid lines) are given by (B.14). Panel (b)-(c): case 1 < α < 2, with µ > 0. The number of simulations is 108 . The… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Exact survival probability q0(n) for L´evy flights, with jumps drawn from skewed L´evy stable laws with µ = 0. The exact theoretical result of (86) is compared with our numerical simulations (markers). Each value of the survival probability is obtained by evolving 107 …

**Figure 5.** Figure 5: Asymptotic power-law decay of the leap-over PDF for symmetric L´evy flights, [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]

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Works this paper leans on

74 extracted references · 74 canonical work pages

[1]

van Kampen N G 2007 Stochastic Processes in Physics and Chemistry (Amsterdam: Elsevier)

work page 2007
[2]

Sethna J P 2021 Statistical mechanics: entropy, order parameters, and complexity (Oxford: Oxford University Press)

work page 2021
[3]

Gardiner C W 2003 Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Berlin: Springer)

work page 2003
[4]

Altan-Bonnet G, Mora T and Walczak A M 2020 Phys. Rep. 849 1

work page 2020
[5]

Port S C 1963 J. Math. Anal. Appl. 6 109

work page 1963
[6]

Feller W 1971 An introduction to probability theory and its applications vol 2 (New York: Wiley)

work page 1971
[7]

Redner S 2001 A guide to first-passage processes (Cambridge: Cambridge University Press)

work page 2001
[8]

Metzler R, Oshanin G and Redner S (eds) 2014 First-passage phenomena and their applications (Singapore: World Scientific)

work page 2014

Show all 74 references

[9]

Koren T, Lomholt H A, Chechkin A V, Klafter J and Metzler R 2007 Phys. Rev. Lett. 99 160602

2007
[10]

Palyulin V V, Chechkin A V and Metzler R 2014 Proc. Natl. Acad. Sci. USA 111 2931

2014
[11]

Barkai E 2003 Phys. Rev. Lett. 90 104101

2003
[12]

Radice M and Cristadoro G 2024 Phys. Rev. E 110 L022103

2024
[13]

Sinai Y G 1957 Theory Prob. Appl. 2 122

1957
[14]

Ray D 1958 Trans. Am. Math. Soc. 89 16

1958
[15]

Rogozin B A 1964 Theory Prob. Appl. 9 450

1964
[16]

Theory Relat

Doney R A and Greenwood P E 1993 Probab. Theory Relat. Fields 94 457

1993
[17]

Theory Relat

Doney R A 1995 Probab. Theory Relat. Fields 101 577

1995
[18]

Bianchi A, Cristadoro G and Pozzoli G 2025 Stocastic Process. Appl. 188 104666

2025
[19]

Majumdar S N, Mounaix P and Schehr G 2013 Phys. Rev. Lett. 111 070601

2013
[20]

Majumdar S N, Mounaix P and Schehr G 2014 J. Stat. Mech. 2014 P09013

2014
[21]

Godr` eche C, Majumdar S N and Schehr G 2016 Phys. Rev. Lett. 117 010601

2016
[22]

Godr` eche C, Majumdar S N and Schehr G 2017 J. Phys. A: Math. Theor. 50 333001

2017
[23]

Godr` eche C and Luck J M 2025arXiv:2501.17268

arXiv
[24]

Majumdar S N 2005 Curr. Sci. 89 2076

2005
[25]

Majumdar S N 2010 Physica A 389 4299

2010
[26]

Artuso R, Onofri M, Pozzoli G and Radice M 2022 J. Stat. Mech. 2022 103209

2022
[27]

Majumdar S N, Comtet A and Ziff R M 2005 J. Stat. Phys. 122 833

2005
[28]

Comtet A and Majumdar S N 2005 J. Stat. Mech. 2005 P06013 First-passage statistics of random walks 39

2005
[29]

Artuso R, Cristadoro G, Degli Esposti M and Knight G 2014 Phys. Rev. E 89 052111

2014
[30]

Majumdar S N, Mounaix P and Schehr G 2017 J. Phys. A: Math. Theor. 50 465002

2017
[31]

Majumdar S N and Sire C 1996 Phys. Rev. Lett. 77 1420

1996
[32]

Oerding K, Cornell S J and Bray A J 1997 Phys. Rev. E 56 R25

1997
[33]

Majumdar S N, Rosso A and Zoia A 2010 Phys. Rev. Lett. 104 020602

2010
[34]

Bray A J, Majumdar S N and Schehr G 2013 Adv. Phys. 62 225

2013
[35]

Hopf E 1934 Mathematical problems of radiative equilibrium (Cambridge: Cambridge University Press)

1934
[36]

Pollaczek F 1952 C. R. Acad. Sci. Paris 234 2334

1952
[37]

Sparre-Andersen E 1954 Math. Scand. 2 194

1954
[38]

Baxter G 1958 Pacific J. Math. 8 649

1958
[39]

Spitzer F 1956 Trans. Amer. Math. Soc. 82 323

1956
[40]

Spitzer F 1957 Duke Math. J. 24 327

1957
[41]

Spitzer F 1960 Trans. Amer. Math. Soc. 94 150

1960
[42]

Finance 20 259

Green R, Fusai G and Abrahams I D 2010 Math. Finance 20 259

2010
[43]

Fusai G, Germano G and Marazzina D 2016 Eur. J. Oper. Res. 251 124

2016
[44]

Astrophys

Ivanov V V 1994 Astron. Astrophys. 286 328

1994
[45]

Le Doussal P and Wiese K J 2009 Phys. Rev. E 79 051105

2009
[46]

Majumdar S N, Schehr G and Wergen G 2012 J. Phys. A: Math. Theor. 45 355002

2012
[47]

Mounaix P, Majumdar S N and Schehr G 2018 J. Stat. Mech. 2018 083201

2018
[48]

Mounaix P, Majumdar S N and Schehr G 2020 J. Phys. A: Math. Theor. 53 415003

2020
[49]

Burenev I N and Majumdar S N 2025 arxiv:2504.04409

2025 arXiv
[50]

Noble B 1958 Methods based on the Wiener-Hopf technique (Oxford: Pergamon press)

1958
[51]

Gakhov F D 1990 Boundary value problems (New York: Dover)

1990
[52]

Kisil A V 2015 IMA J. Appl. Math. 80 1569

2015
[53]

Kisil A V, Abrahams I D, Mishuris G and Rogosin S V 2021 Proc. R. Soc. A 477 20210533

2021
[54]

Klafter J and Sokolov I M 2011 First steps in random walks (Oxford: Oxford university press)

2011
[55]

Luck J M, Funke M and Nieuwenhuizen T M 1991 J. Phys. A: Math. Theor. 24 4155

1991
[56]

Battilana M, Majumdar S N and Schehr G 2020 Markov Proc. Rel. Fields 26 57

2020
[57]

Radice M, Cristadoro G and Thapa S 2025 Chaos 35 023131

2025
[58]

Krapivsky P L and Redner S 2018 J. Stat. Mech. 2018 093208

2018
[59]

Padash A, Capa la K, Kantz H, Dybiec B, Shokri B, Metzler R and Chechkin A V 2025 J. Phys. A: Math. Theor. 58 185002

2025
[60]

Padash A, Chechkin A V, Dybiec B, Pavlyukevich I, Shokri B and Metzler R 2019 J. Phys. A: Math. Theor. 52 454004

2019
[61]

Padash A, Chechkin A V, Dybiec B, Magdziarz M, Shokri B and Metzler R 2020 J. Phys. A: Math. Theor. 53 275002

2020
[62]

Sato K 2013 L´ evy processes and infinitely divisible distributions(Cambridge: Cambridge University Press)

2013
[63]

Ablowitz M J and Fokas A S 1997 Complex variables: introduction and applications (Cambridge: Cambridge University Press)

1997
[64]

Its A R 2003 Not. Am. Math. Soc. 50 1389

2003
[65]

Mori F, Majumdar S N and Vivo P 2024 Phys. Rev. Res. 6 043053

2024
[66]

Majumdar S N, Mori F and Vivo P 2023 Phys. Rev. Lett. 130 237102

2023
[67]

Majumdar S N, Mori F and Vivo P 2023 Phys. Rev. E 108 064122

2023
[68]

Fuchs J, Goldt S and Seifert U 2016 Europhys. Lett. 113 60009

2016
[69]

Mori F, Olsen K S and Krishnamurthy S 2023 Phys. Rev. Res. 5 023103

2023
[70]

Sunil J C, Blythe R A, Evans M R and Majumdar S N 2023 J. Phys. A: Math. Theor. 56 395001

2023
[71]

Pal P S, Pal A, Park H and Lee J S 2024 Phys. Rev. E 108 044117

2024
[72]

Sunil J C, Blythe R A, Evans M R and Majumdar S N 2025 arXiv: 2503.03697

2025
[73]

Gradshteyn I S and Ryzhik I M 2007 Table of integrals, series, and products (Amsterdam: Elsevier First-passage statistics of random walks 40 academic press)

2007
[74]

Bateman H 1954 Tables of integral transforms vol 1 (New York: McGraw-Hill Book Company)

1954

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

van Kampen N G 2007 Stochastic Processes in Physics and Chemistry (Amsterdam: Elsevier)

work page 2007

[2] [2]

Sethna J P 2021 Statistical mechanics: entropy, order parameters, and complexity (Oxford: Oxford University Press)

work page 2021

[3] [3]

Gardiner C W 2003 Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Berlin: Springer)

work page 2003

[4] [4]

Altan-Bonnet G, Mora T and Walczak A M 2020 Phys. Rep. 849 1

work page 2020

[5] [5]

Port S C 1963 J. Math. Anal. Appl. 6 109

work page 1963

[6] [6]

Feller W 1971 An introduction to probability theory and its applications vol 2 (New York: Wiley)

work page 1971

[7] [7]

Redner S 2001 A guide to first-passage processes (Cambridge: Cambridge University Press)

work page 2001

[8] [8]

Metzler R, Oshanin G and Redner S (eds) 2014 First-passage phenomena and their applications (Singapore: World Scientific)

work page 2014