We study the mean first-passage time (MFPT) for asymmetric continuous time random walks in continuous space characterised by finite mean waiting times and jump amplitudes with both finite average and finite variance. We derive an inhomogeneous Wiener-Hopf integral equation that allows the exact estimation of the MFPT, which depends on the whole distribution of the jump amplitudes, but on the average of the waiting times only. Thus, our findings hold for general non-Markovian processes, since Markovianity emerges solely with an exponential distribution of the waiting times. Through the paradigmatic case study of a
general class of asymmetric distributions of the jump-amplitudes that is exponential towards the boundary and arbitrary in the opposite direction, we show, that only the average of the jump amplitudes in the opposite direction of the boundary contributes to the MFPT. Moreover, we determine a length-scale, which depends only of the distribution of jumps in the direction of the boundary, such that for initial positions close to the boundary the MFPT depends on the specific whole distribution of jump amplitudes, in opposition to the appearing universality for initial positions far away from the boundary.PRE2018-08442

Dahlenburg, M.

Pagnini, G.

English

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Wiener–Hopf Integral Equations in Mean First-passage Time Problemsfor Continuous-time Random WalksMarcus Dahlenburga,b,* and Gianni Pagninia,caBCAM–Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, BasqueCountry – SpainbInstitute for Physics & Astronomy, University of Potsdam, 14476 Potsdam, GermanycIkerbasque–Basque Foundation for Science, Plaza Euskadi 5, 48009 Bilbao, Basque Country – Spain*mdahlenburg@bcamath.orgAbstractWe study the mean first-passage time (MFPT) for asymmetric continuous time random walksin continuous space characterised by finite mean waiting times and jump amplitudes withboth finite average and finite variance. We derive an inhomogeneous Wiener-Hopf integralequation that allows the exact estimation of the MFPT, which depends on the whole dis-tribution of the jump amplitudes, but on the average of the waiting times only. Thus, ourfindings hold for general non-Markovian processes, since Markovianity emerges solely withan exponential distribution of the waiting times. Through the paradigmatic case study of ageneral class of asymmetric distributions of the jump-amplitudes that is exponential towardsthe boundary and arbitrary in the opposite direction, we show, that only the average of thejump amplitudes in the opposite direction of the boundary contributes to the MFPT. More-over, we determine a length-scale, which depends only of the distribution of jumps in thedirection of the boundary, such that for initial positions close to the boundary the MFPT de-pends on the specific whole distribution of jump amplitudes, in opposition to the appearinguniversality for initial positions far away from the boundary.Keywords: mean first-passage time, continuous-time random walk, Wiener–Hopfintegral equation1. Finite-mean first-passage timeWe consider a one-dimensional random walk in continuous space x ∈ R and continuous time t ≥ 0.In particular, we study a continuous time random walk (CTRW) characterised by a distribution q(ξ ) ofjump amplitudes and by a waiting times distribution between subsequent jumps ψ(τ), which normaliseaccording to∫∞−∞ q(ξ )dξ = 1 and∫∞0 ψ(τ)dτ = 1, respectively. If the distribution of the waiting-timesis exponential then the process is Markovian [1]. We assume that drawings of the jump-sizes and ofthe waiting-times are statistically independent, and both are independent and identically distributed (iid)random variables.Let Xt be the walker’s position at time t, and let the notation Xt−τ |y denoting the actual position ofthe random walker at time t under the condition that the walker was previously in y ∈ R at τ ∈ [0, t],which is the duration of the first random waiting time. Using the idea of the first renewal picture [2, 3],the walker may rest until time t in the starting point x0 ∈ R with probability Ψ(t) = 1−∫ t0 ψ(τ)dτ or, itmay be at time t in Xt−τ starting from the new initial-like position x0+ξ , ξ ∈R, with the complementaryprobability∫ t0 ψ(τ)dτ , in which ξ declares a random jump amplitude that may occurs at the new initialdatum τ ∈ [0, t]. Thus, in formulae, for the walkers position Xt |x0 we may writeXt |x0 =x0; withprobability :Ψ(t) = 1−∫ t0ψ(τ)dτ ,Xt−τ |(x0 +ξ ); withprobability :∫ t0ψ(τ)dτ .(1)M. Dahlenburg et al.The conditional probability density function (PDF) p(x, t;x0) of the statistically homogeneous process(1) may be described by the implicit integral equationp(x, t;x0) = Ψ(t)δ (x− x0)+∫ t0ψ(τ)∫∞−∞q(ξ − x0)p(x, t − τ;ξ )dξ dτ , (2)that in Fourier-Laplace space domain becomes the well-known Montroll-Weiss equation [4]̂̃p(κ,s;x0) = Ψ̃(s)exp(ikx0)1− ψ̃(s) q̂(κ) . (3)We assume a CTRW with finite mean waiting times, i.e., 0< ⟨τ⟩=∫∞0 τ ψ(τ)dτ <∞, and finite aver-age and finite variance of the jump sizes −∞< ⟨ξ ⟩=∫∞−∞ ξ q(ξ )dξ <∞ ,0< ⟨ξ 2⟩=∫∞−∞ ξ2 q(ξ )dξ <∞ .The Montroll-Weiss equation (3) may be re-arranged [5] such that in the diffusive limit t, |x| → ∞ of theCTRW, for which applies ψ̃(s)≃ 1− s⟨τ⟩+O(s2), respectively q̂(κ)≃ 1+ iκ⟨ξ ⟩−κ2⟨ξ 2⟩/2+O(k3),we may obtainŝ̃p(κ,s;x0)− exp(ikx0) = [iκv−Dκ2] ̂̃p(κ,s;x0), (4)with the parameters v = ⟨ξ ⟩/⟨τ⟩ and D = ⟨ξ 2⟩/2⟨τ⟩. Formula (4) proves the Central Limit Theoreminsofar as its backward transformation into physically space becomes the advection-diffusion equation∂ p(x, t;x0)∂ t=−v ∂ p(x, t;x0)∂x+D∂ 2 p(x, t;x0)∂x2, v ∈ R , D ≥ 0 . (5)Plugging into an absorbing boundary at x = 0 by assuming that x0 > 0 we may obtain the MFPT of anadvective-diffusive system (5) from the literature [6], by remembering that sgn{v}= sgn{⟨ξ ⟩}T (x0) =−x0v> 0 , with ⟨ξ ⟩< 0 . (6)2. An inhomogeneous Wiener–Hopf integral equation for the MFPTThe finite MFPT (6) derived in Section 1 holds in the limits t, |x| → ∞, or analogously s,κ → 0, inwhich the behaviour is governed by an advective-diffusive dynamic formula (5).To avoid the asymptotic limit we have a look on the survival probability that may be estimated byΛ(x0, t) =∫∞0pabs(x, t;x0)dx , t ≥ 0 , x0 > 0 , (7)where pabs(x, t;x0) is the conditional PDF with an absorbing boundary at x = 0 that can be calculated bypabs(x, t;x0) = Ψ(t)δ (x− x0)+∫ t0ψ(τ)∫∞0q(ξ − x0)pabs(x, t − τ;ξ )dξ dτ . (8)In equation (1), the variable ξ represents the size of the first jump, while in (2) and (8) ξ denotes theposition immediately after the first jump events. The absorbing boundary was implemented in equation(8) through the spatial integration ξ that does not cover the full space (in contrast to formula (2)) butonly the positive semi-infinite space above the absorbing boundary. Through the manipulation of thelower bound of the spatial integration the trajectories that pass the boundary at the first jump event wereabsorbed. Due to the implicitly of the integral equation, the random path is propagated in such a way thatan occurrence of the particle in the negative area after any jump event is accompanied by its absorption.From the formulas (7) and (8) we obtainΛ(x0, t) = Ψ(t)+∫ t0ψ(τ)∫∞0q(ξ − x0)Λ(ξ , t − τ)dξ dτ , (9)and later, through the limit s → 0 of the Laplace transform of (9) the MFPT T (x0) becomesT (x0) = ⟨τ⟩+∫∞0q(ξ − x0)T (ξ )dξ , x0 > 0 , (10)that is an inhomogeneous Wiener–Hopf integral equation whose kernel is a PDF [7]. Formula (10) statesthat the exact MFPT depends only on the average of the waiting times, but on the whole distribution ofthe jump amplitudes, although in the asymptotic limit, formula (6), the MFPT depends on the mean ofboth, waiting-time and jump sizes, only.Wiener–Hopf integral equations in mean first-passage time problems for continuous-time random walks3. Solution method applied to an asymmetric double-exponential jump distributionIn order to calculate the solution of (10), in the spirit of the Wiener–Hopf technique [8], we firstgeneralise the MFPT to T : R→ R and introduce ρ(ξ ) = q(−ξ ), ξ ∈ R, such that we obtain the inho-mogeneous Wiener–Hopf equation in its actual formT (x0) = f (x0)+∫∞0ρ(x0 −ξ )T (ξ )dξ , x0 ∈ R . (11)For our concern, we may setT (x0) ={T+(x0) ; if x0 > 0 ,T−(x0) ; if x0 ≤ 0 ,, f (x0) ={⟨τ⟩> 0; if x0 > 0 ,0; if x0 ≤ 0 .(12)The spatial distinction provided by (12) allows us to re-write equation (11)T+(x0) = ⟨τ⟩+∫∞0ρ(x0 −ξ )T+(ξ )dξ , x0 > 0 ,T−(x0) =∫∞0ρ(x0 −ξ )T+(ξ )dξ , x0 ≤ 0 .(13)In Fourier-space that lead us to the following pairsT̂±(κ) =±∫ ±∞0exp(+iκx0)T±(x0)dx0 , T±(x0) =12π∫L±exp(−iκx0)T̂±(κ)dκ ,where L± are proper integration paths in the complex plain.The general solution of formula (13) is explicitly given in Fourier space and readsT̂+(κ) =−T̂−(κ)[1− ρ̂(κ)]− i ⟨τ⟩κ [1− ρ̂(κ)]. (14)By investigating the paradigmatic case of an asymmetric double-exponential jumps’ distributionρ(ξ ) ={(1−b)exp(−ξ/ℓ)/ℓ ; if ξ ≥ 0 ,abexp(aξ/ℓ)/ℓ ; if ξ < 0 ,with 0 ≤ b ≤ 1, ℓ≥ 0, a ≥ 0 and the average of the jump amplitudes⟨ξ ⟩=∫∞−∞ξ q(ξ )dξ =−∫∞−∞ξ ρ(ξ )dξ =(1+aa)ℓb− ℓ , (15)the solution set may be obtained through formula (14)T+(x0) = T−(0){ℓa⟨ξ ⟩− (ℓ−a⟨ξ ⟩)exp(−a⟨ξ ⟩x0/ℓ2)a⟨ξ ⟩}++ ⟨τ⟩{− x0⟨ξ ⟩+⟨ξ ⟩ℓ(1−a)+ ℓ2a⟨ξ ⟩2− (⟨ξ ⟩+ ℓ)(ℓ−a⟨ξ ⟩)exp(−a⟨ξ ⟩x0/ℓ2)a⟨ξ ⟩2}. (16)The non-unique solution, formula (16), is only in agreement with the asymptotic limit for x0 → ∞,equation (6), if T−(0) =−⟨τ⟩(ℓ+ ⟨ξ ⟩)/⟨ξ ⟩ such that the unique MFPT is finally given byT+(x0) =−⟨τ⟩⟨ξ ⟩(ℓ+ x0) , for x0 > 0 , ⟨ξ ⟩< 0 , ⟨τ⟩> 0 . (17)The result (17) is indeed in agreement with the literature [9] and we may see that the asymptotic limitcan be used as approximation if x0 ≫ ℓ : T+(x0)∼−x0/v by remembering that v = ⟨ξ ⟩/⟨τ⟩. Moreover,through formula (15) the condition that ⟨ξ ⟩< 0 can be used to define the range for the parameter setting0 ≤ b < a/(1+a) that guarantees a finite MFPT.M. Dahlenburg et al.4. Indirect estimation of the jump-sizes distributionThe identification of the MFPT of a CTRW in continuous space with the inhomogeneous Wiener-Hopf integral equation [10] offers the possibility to determine the jump amplitudes PDF for a givenMFPT by distinguishing between the jump directions governed by the probability b ∈ [0,1]ρ(ξ ) ={(1−b)ρ>(ξ ) ; if ξ ≥ 0 ,bρ<(ξ ) ; if ξ < 0 ,(18)for which it yields∫ 0−∞ ρ<(ξ )dξ = 1 and∫∞0 ρ>(ξ )dξ = 1 without limitation. Moreover, we set that∫ 0−∞ξ ρ<(ξ )dξ =−ℓ/a , and∫∞0ξ ρ>(ξ )dξ = ℓ , ℓ > 0 , a > 0, (19)to obtain the same average as provided by equation (15).Plugging in (17), (18) and (19) into equation (13) leads us to the relation−⟨τ⟩⟨ξ ⟩(ℓ+ x0) = ⟨τ⟩−b⟨τ⟩⟨ξ ⟩(ℓ+ x0 +ℓa)− (1−b)⟨τ⟩⟨ξ ⟩∫ x00(ℓ+ξ )ρ>(x0 −ξ )dξ , x0 > 0.In Laplace-space we may explicitly obtain ρ̃>(s) that may be transformed back into physically spaceρ̃>(s) =1ℓs+1⇔ ρ>(ξ ) =exp(−ξ/ℓ)ℓ, ξ ≥ 0 , ℓ > 0. (20)We derived an exponential distribution of the jump amplitudes towards the absorbing boundary, but thereis no restriction for the jumps away from the target besides the constrain (19). Thus, the PDF of jumpsin the opposite direction of the boundary may be chosen arbitrarily and we obtain the same result (17).5. ConclusionWe studied the problem of a finite MFPT of a CTRW in continuous space characterised by finitemean waiting times and by jump amplitudes with both finite average and finite variance. In particular, weobtain a inhomogeneous Wiener–Hopf integral equation that allows for an exact calculation of the MFPTby avoiding asymptotic limits. This formula results to depend on the whole PDF of the jump amplitudesand on the mean value of the waiting times only such that our findings holds for general non-Markovianprocesses. The derived Wiener–Hopf integral equation (10) has been used for the paradigmatic case studyof an asymmetric double-exponential distribution of the jump amplitudes, whose MFPT turns out to bevalid for a more general class of asymmetric distributions of the jump-sizes, namely exponential towardsthe boundary and arbitrary in the opposite direction. Especially, we show that when the jumps towardsthe boundary are exponentially distributed, the MFPT is indeed independent of the jump amplitudesPDF in the opposite direction. Furthermore, a length-scale emerges that depends only on the distributionof the jump amplitudes in the direction of the boundary, which defines the transition to the asymptoticbehaviour (6). Thus, in opposition to the universal MFPT for starting points that are far-away from theboundary, for starting points that are close to the boundary this universality is lost.References[1] F. Mainardi, M. Raberto, R. Gorenflo, and E. Scalas, Physica A 287, 468–481, (2000).[2] A. Pal, A. Kundu, and M. R. Evans, J. Phys. A 49, 225001, (2016).[3] M. Dahlenburg, A. V. Chechkin, R. Schumer, and R. Metzler, Phys. Rev. E 103, 052123 (2021).[4] E. W. Montroll and G. H. Weiss, J. Math. Phys. 6, 167–181, (1965).[5] V. Méndez, A. Masó–Puigdellosas, T. Sandev, Phys. Rev. E 103, 022103, (2021).[6] A. V. Chechkin, R. Metzler, V. Y. Gonchar, J. Klafter, and L. V. Tanatarov, J. Phys. A 36, L537–L544, (2003).[7] F. Spitzer, Duke Math. J. 24, 327–343, (1957).[8] F. G. Leppington, Modern Methods in Analytical Acoustics, Springer–Verlag GmbH (1992)[9] S. G. Kou and H. Wang, Adv. Appl. Prob. 35, 504 (2003).[10] M. Dahlenburg and G. Pagnini, J. Phys. A: Math. Theor. 55, 505003 (2022).

Wiener-Hopf Integral Equations in Mean First-passage Time Problems for Continuous-time Random Walks

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Wiener-Hopf Integral Equations in Mean First-passage Time Problems for Continuous-time Random Walks

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