Exact calculation of the mean first-passage time of continuous-time random walks by nonhomogeneous Wiener–Hopf integral equations

Abstract

We study the mean first-passage time (MFPT) for asymmetric continuous-time random walks in continuous-space characterised by waiting-times with finite mean and by jump-sizes with both finite mean and finite variance. In the asymptotic limit, this well-controlled process is governed by an advection-diffusion equation and the MFPT results to be finite when the advecting velocity is in the direction of the boundary. We derive a nonhomogeneous Wiener-Hopf integral equation that allows for the exact calculation of the MFPT by avoiding asymptotic limits and it emerges to depend on the whole distribution of the jump-sizes and on the mean-value only of the waiting-times, thus it holds for general non-Markovian random walks. Through the case study of a quite general family of asymmetric distributions of the jump-sizes that is exponential towards the boundary and arbitrary in the opposite direction, we show that the MFPT is indeed independent of the jump-sizes distribution in the opposite direction to the boundary. Moreover, we show also that there exists a length-scale, which depends only on the features of the distribution of jumps in the direction of the boundary, such that for starting points near the boundary the MFPT depends on the specific whole distribution of jump-sizes, in opposition to the universality emerging for starting points far-away from the boundary.

Sturm–Liouville systems for the survival probability in first-passage time problems

“…By taking into account (2.4), we have in the most general case that Λ~+false(z,sfalse) satisfies the equation normalddz{p(z) dnormaldzΛ~+(z,s)}+qfalse(zfalse) Λ~+false(z,sfalse)=−ψ~false(sfalse) Λ~+false(z,sfalse)+qfalse(zfalse) Ψ~false(sfalse).Hence, in the special case kfalse(xfalse)=normale−|x|/2, see, e.g. [16–19], it holds pfalse(zfalse)=1 and qfalse(zfalse)=−1 and, together with the Markovian assumption (4.6), equation (4.12) reduces to normald2Λ…”

“…In this section, first we directly solve (1.1) for the generating function by means of classical methods in complex analysis for the meaningful and manageable example with the exponential kernel kfalse(xfalse)=normale−|x|/2, see, e.g. [16–19], and later we solve the Sturm–Liouville system derived in §2 for the same special case.…”

“…If we consider the special case kfalse(xfalse)=normale−|x|/2, see, e.g. [16–19], then from (2.1) we have that kz′false(z,ξfalse)=−sgn⁡false(z−ξfalse) normale−|z−ξ|/2 and from (2.2) that pfalse(zfalse)=1, with z≠0. Therefore, by solving (2.3) with respect to qfalse(zfalse) in all points except z=ξ, it results qfalse(zfalse)=−1.…”

“…Moreover, we introduce also the probability that a given waiting interval between two consecutive jumps is greater than or equal to t, namely true0Ψfalse(tfalse)=1−∫0tψfalse(τfalse) dτ. In this continuous-time setting, we replace the notation of the survival probability in discrete epochs ϕnfalse(ξfalse) with the new notation Λfalse(ξ,tfalse) and now (1.1) reads [19, formula (16)] normalΛ~false(ξ,sfalse)=normalΨ~false(sfalse)+ψ~false(sfalse)∫0∞kfalse(z−ξfalse)normalΛ~false(z,sfalse) dz,where normalΛ~false(ξ,sfalse) stands for the Laplace transform of Λfalse(ξ,tfalse) with s∈C, and it holds snormalΨ~false(sfalse)=1−ψ~false(sfalse). After Laplace inversion and by setting …”

“…We consider now, as a meaningful and manageable example, the exponential kernel kfalse(xfalse)=normale−|x|/2, again, see, e.g. [16–19]. By applying the Fourier transform (3.3) to (4.2), we have Λ~^+false(ω,sfalse)=−ωfalse(1−iωfalse)Λ~−false(0,sfalse)+ifalse(1+ω2false)normalΨ~false(sfalse)ωfalse(snormalΨ~false(sfalse)+ω2false),where by exploiting the exponential kernel, we have k^false(ωfalse)=1/false(1+ω2false) and from the definition of Λ~−false(ξ,sfalse) it results Λ~^−false(ω,sfalse)=Λ~−false(0,sfalse)/…”