In 1995, Zhan Shi and Wendelin Werner converted fixed-time generalized arcsine laws into sharp almost-sure integral tests for the time spent in the positive half-line by strictly stable Lévy processes and by perturbed reflecting Brownian motion (PRBM), Xt=∣Bt∣−μℓtX_t=|B_t|-\mu\ell_t. The paper itself acknowledged three unfinished directions: (1) extending the stable-process test beyond strict self-similarity, to wider classes of Lévy processes and to discrete-time analogues such as random walks; (2) writing out the small-time (t→0+t\to 0+) analogue of the stable-process test, which the authors left to the reader; and (3) replacing their polynomial lower bound P(T1>t)≥c3 t−1/(2μ)\mathbb{P}(T_1>t)\ge c_3\,t^{-1/(2\mu)} for the PRBM hitting time of level 1 by the exact tail asymptotic with explicit constant, which they indicated could be extracted from Carmona–Petit–Yor Corollary 3.4.1 plus a Tauberian argument but found technically troublesome. Later literature supplies fixed-time occupation-time distributions for special or general Lévy classes (Landriault–Renaud–Zhou 2011; Pitters 2025), integral tests for the extrema of doubly perturbed Brownian motion (Chaumont–Doney–Hu 2000), and path decompositions of PRBM (Aïdékon–Hu–Shi 2020), none of which is an almost-sure occupation-time liminf test beyond the strictly stable class or a sharp PRBM hitting-time asymptotic. The problem is open.

Background

A Lévy process is a stochastic process with stationary independent increments and càdlàg paths (paths that are right-continuous with left limits), starting at 0; Brownian motion and Poisson-type jump processes are the basic examples, and the general one combines a Gaussian part, a drift, and jumps. For a real-valued process ZZ, the occupation time (or sojourn time) of the positive half-line is Ht=∫0t1{Zs>0} dsH_t=\int_0^t \mathbf{1}_{\{Z_s>0\}}\,ds, the time spent positive up to tt. For Brownian motion, Lévy's arcsine law gives the distribution of Ht/tH_t/t at fixed times; Carmona, Petit and Yor extended this to generalized arcsine laws, and for a strictly stable Lévy process ZZ of index α∈(0,2)\alpha\in(0,2) — a self-similar Lévy process whose jumps scale so that Zct=dc1/αZtZ_{ct}\overset{d}{=}c^{1/\alpha}Z_t — with positivity parameter ρ=P(Z1>0)∈(0,1)\rho=\mathbb{P}(Z_1>0)\in(0,1), the fixed-time law of Ht/tH_t/t is the generalized arcsine law with density proportional to uρ−1(1−u)−ρu^{\rho-1}(1-u)^{-\rho}. A classical companion question, going back to Erdős, Feller, Kolmogorov and Petrowsky's integral tests for Brownian motion, concerns the almost-sure behavior: for which functions ff does Ht/(tf(t))H_t/(t f(t)) tend to 0 or to +∞+\infty along the path? Hobson (1994) proved that for Brownian motion lim inf⁡t→∞Ht/(tf(t))\liminf_{t\to\infty} H_t/(t f(t)) equals 0 or +∞+\infty according as ∫∞f(x)1/2 dx/x\int^\infty f(x)^{1/2}\,dx/x diverges or converges. Shi and Werner (1995) established the analogous dichotomies for strictly stable Lévy processes, with exponent ρ\rho in place of 1/21/2, and for perturbed reflecting Brownian motion. PRBM is the process Xt=∣Bt∣−μℓtX_t=|B_t|-\mu\ell_t, where BB is a linear Brownian motion, ℓt\ell_t is its local time at 0 (a continuous measure of time spent at 0 that increases only when Bt=0B_t=0), and μ>0\mu>0 is a fixed constant: reflecting Brownian motion ∣Bt∣|B_t| pushed down by μℓt\mu\ell_t. Except at μ=1\mu=1, where Lévy's identity — the theorem that ∣Bt∣−ℓt|B_t|-\ell_t is itself a Brownian motion — makes it a Brownian motion, PRBM is not a Markov process, and it arises in Le Gall–Yor winding asymptotics (the large-time behavior of the winding of planar Brownian motion around points), in Carmona–Petit–Yor Ray–Knight-type theorems (results describing the local times of a process stopped at a hitting time, viewed as a Markov process in the level variable), and as the scaling limit of self-interacting random walks (walks whose transition mechanism depends on their own past). For PRBM, Shi–Werner proved the occupation-time dichotomy with exponent 1/(2μ)1/(2\mu) for the positive-side occupation time Atμ=∫0t1{Xs>0} dsA_t^\mu=\int_0^t\mathbf{1}_{\{X_s>0\}}\,ds and exponent 1/21/2 for the negative-side time A~tμ=t−Atμ\tilde A_t^\mu=t-A_t^\mu, together with a small-time version obtained by time inversion, and a last-passage-time test with exponent η=min⁡(1/2,1/(2μ))\eta=\min(1/2,1/(2\mu)) for the last zero of XX before tt. Their method has two halves: an 'easy half' via the first Borel–Cantelli lemma on exponentially spaced grids, fed by fixed-time tail estimates; and a 'hard half' that, for the stable case, uses the self-similarity of ZZ to show that the log-rescaled process Yu=e−u/αZeuY_u=e^{-u/\alpha}Z_{e^u} is a stationary, irreducible, Harris-recurrent Markov process (Harris recurrence meaning that every set of positive invariant-measure mass is hit infinitely often almost surely; an Ornstein–Uhlenbeck-type embedding), to which the Azéma–Duflo–Revuz theorem — a limit theorem for additive functionals over return times to a set of a recurrent Markov process — is applied before a second Borel–Cantelli argument. The required hitting-time inputs for PRBM are the Carmona–Petit–Yor identities, notably P(T−a<Tb)=(b/(a+b))1/μ\mathbb{P}(T_{-a}<T_b)=(b/(a+b))^{1/\mu} for hitting −a-a before bb and Corollary 3.4.1 of their 1994 paper, which gives the full law of the hitting time Tμ,a=inf⁡{u:∣Bu∣−μℓu=a}T^{\mu,a}=\inf\{u:|B_u|-\mu\ell_u=a\} of a positive level through an explicit Laplace-transform identity and a series density; a Tauberian theorem — a result converting the behavior of a Laplace transform near 0 into the tail behavior of the distribution — is then the natural route from such an identity to the exact asymptotic of P(T1>t)\mathbb{P}(T_1>t). Bertoin (1995), in parallel work on integral tests for diffusions, observed that his method cannot handle the occupation time of a general Lévy process precisely because jumps across 0 destroy the excursion-theoretic arguments (based on Itô's decomposition of the path into excursions away from 0), and credited Shi–Werner with the stable case; the step beyond strict stability and self-similarity has remained the acknowledged gap.

Problem Statement

The problem is to settle the unfinished directions of Shi–Werner (1995) on almost-sure occupation-time asymptotics, in any mathematically rigorous form that covers at least one substantive core.

Core 1 (primary). Establish — or rigorously refute — an almost-sure integral-test dichotomy for the occupation time of the positive half-line, Ht=∫0t1{Xs>0} dsH_t=\int_0^t \mathbf{1}_{\{X_s>0\}}\,ds, for process classes beyond the strictly stable one. Concretely: for a Lévy process XX (or a random walk) outside the strictly stable class, identify hypotheses under which there is a critical function of the process's fluctuation characteristics (in the strictly stable case the criterion is ∫∞f(x)ρ dx/x\int^\infty f(x)^{\rho}\,dx/x with ρ=P(Z1>0)\rho=\mathbb{P}(Z_1>0); Brownian motion is the case ρ=1/2\rho=1/2, recovering Hobson's test) such that lim inf⁡t→∞Ht/(tf(t))\liminf_{t\to\infty} H_t/(t f(t)) equals 0 when the integral diverges and +∞+\infty when it converges, and determine the correct replacement for the exponent ρ\rho in general (for instance whether it is governed by the large-time behavior of t↦P(Xt>0)t\mapsto\mathbb{P}(X_t>0), by persistence exponents — the exponential decay rates of the probability that the process stays on one side of 0 over long intervals — or by no single parameter at all). A valid answer may: prove the test for a meaningfully broader class than strictly stable processes (for example oscillating Lévy processes — those with lim sup⁡Xt=+∞\limsup X_t=+\infty and lim inf⁡Xt=−∞\liminf X_t=-\infty almost surely — with jumps of both signs, spectrally one-sided Lévy processes — those whose jumps all have the same sign — or random walks in the domain of attraction of a stable law, meaning that their rescaled partial sums converge in distribution to a stable law), provided it recovers the Shi–Werner stable test and the Hobson/Brownian test as special cases; or disprove the existence of any universal test of this shape by an explicit construction. Identifying the right hypotheses is part of the problem, not a defect of an answer; arbitrary restrictions that trivialize the question (for example, processes that never jump across 0) do not count.

Core 2 (primary). Determine the exact asymptotic of P(T1>t)\mathbb{P}(T_1>t) as t→∞t\to\infty, with explicit constant, where T1=inf⁡{s:Xs=1}T_1=\inf\{s:X_s=1\} is the hitting time of level 1 by the perturbed reflecting Brownian motion Xt=∣Bt∣−μℓtX_t=|B_t|-\mu\ell_t, μ>0\mu>0. The source used only the polynomial lower bound P(T1>t)≥c3 t−1/(2μ)\mathbb{P}(T_1>t)\ge c_3\,t^{-1/(2\mu)}. The answer must reproduce the known μ=1\mu=1 case, where Lévy's identity (under which ∣Bt∣−ℓt|B_t|-\ell_t is itself a Brownian motion) reduces T1T_1 to a Brownian hitting time with tail constant 2/π\sqrt{2/\pi}.

Core 3 (completeness item, declared routine by the source). Write out and verify the small-time (t→0+t\to 0+) analogue of the strictly stable occupation-time integral test, which the source left to the reader.

A resolution of any of Core 1 (for a non-trivial class), Core 2, or a rigorous negative result of the kind described in Core 1 constitutes an answer; the small-time analogue alone does not.

References

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