Exact moments in a continuous time random walk with complete memory of its history

TL;DR

This paper introduces a continuous-time random walk model with complete memory, deriving exact moments and revealing new superdiffusive transitions that are not captured by traditional Fokker-Planck approaches.

Contribution

It provides exact expressions for moments of a memory-dependent continuous-time random walk and uncovers novel superdiffusive transitions due to memory effects.

Findings

Exact moments for the displacement distribution are derived.

New superdiffusive transitions are identified in the parameter space.

Memory effects lead to behaviors not captured by Fokker-Planck approximations.

Abstract

We present a continuous time generalization of a random walk with complete memory of its history [Phys. Rev. E 70, 045101(R) (2004)] and derive exact expressions for the first four moments of the distribution of displacement when the number of steps is Poisson distributed. We analyze the asymptotic behavior of the normalized third and fourth cumulants and identify new transitions in a parameter regime where the random walk exhibits superdiffusion. These transitions, which are also present in the discrete time case, arise from the memory of the process and are not reproduced by Fokker-Planck approximations to the evolution equation of this random walk.