Abstract
Scaling limits of continuous‐time random walks are used in physics to model anomalous diffusion in which particles spread at a different rate than the classical Brownian motion. In this paper, we characterize the scaling limit of the average of multiple particles, independently moving as a continuous‐time random walk. The limit is taken by increasing the number of particles and scaling from microscopic to macroscopic view. We show that the limit is independent of the order of these limiting procedures and can also be taken simultaneously in both procedures. Whereas the scaling limit of a single‐particle movement has quite an obscure behavior, the multiple‐particle analogue has much nicer properties.
Highlights
Continuous-time random walks (CTRWs) were introduced in [24] to study random walks on a lattice
A CTRW is a random walk subordinated to a renewal process
The usual assumption is that the CTRW is uncoupled, meaning that the random walk is independent of the subordinating renewal process
Summary
Continuous-time random walks (CTRWs) were introduced in [24] to study random walks on a lattice. The usual assumption is that the CTRW is uncoupled, meaning that the random walk is independent of the subordinating renewal process. 214 Multiple-particle CTRW time is of the form A(E(t)), where A(t) is the scaling limit of the underlying random walk and E(t) is the hitting time process for a β-stable subordinator independent of A(t). We analyze the limiting behavior of the average over multiple infinite mean waiting time CTRWs in the context of operator self-similarity of stochastic processes. As shown in [21], the limiting process {M(t)}t≥0 of a single uncoupled CTRW has quite an obscure behavior, as it is not an operator-stable process nor it has independent increments (i.i.) It follows from [21, Theorem 4.6] that even for a Brownian motion {A(t)}t≥0, the distribution of M(t) is not even Gaussian. We conclude this paper by discussing an example of a so-called coupled CTRW, that is, a CTRW where the waiting times and the jumps are dependent
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