Asymptotic solutions of decoupled continuous-time random walks with superheavy-tailed waiting time and heavy-tailed jump length distributions

Abstract

We study the long-time behavior of decoupled continuous-time random walks characterized by superheavy-tailed distributions of waiting times and symmetric heavy-tailed distributions of jump lengths. Our main quantity of interest is the limiting probability density of the position of the walker multiplied by a scaling function of time. We show that the probability density of the scaled walker position converges in the long-time limit to a non-degenerate one only if the scaling function behaves in a certain way. This function as well as the limiting probability density are determined in explicit form. Also, we express the limiting probability density which has heavy tails in terms of the Fox H-function and find its behavior for small and large distances.

Limiting distributions of continuous-time random walks with superheavy-tailed waiting times

“…In contrast to Ref. [29], now we are concerned with the effects arising from the asymmetry of w(ξ). In addition, we are going to develop a numerical method for the simulation of these CTRWs and apply it to verify the theoretical predictions.…”

“…The second one, which describes all other cases, is constituted by a two-parametric (with parameters α and ϕ) probability density (3.20). If ϕ = 0, then the limiting probability density (3.20) is reduced to the symmetric one (2.11), whose properties is well established [29]. As for the non-symmetric case, it has never been studied.…”

“…Recently, we have partially solved this problem for a new class of CTRWs with superheavy-tailed distributions of waiting times [28,29]. These distributions are characterized by the following asymptotic behavior of the waiting-time density:…”

“…For the CTRWs characterized by arbitrary superheavy-tailed distributions of waiting time, the scaling function a(t) and the corresponding limiting probability density P(y) of Y (∞) have already been found for the jump densities having finite second moments [28] and symmetric heavy tails [29]. In the present work, we report a comprehensive theoretical and numerical studies of the long-time behavior of the reference CTRWs in the general case of asymmetric jump densities characterized by heavy tails.…”

“…[29] under the condition that heavy-tailed jump densities with α = 2 are symmetric. However, since the condition l 1 = 0 does not imply that w(−ξ) = w(ξ), the two-sided exponential density (3.42) corresponds to a more wide class jump densities characterized by the conditions α = 2 and l 1 = 0.…”