Ultra slow sub-logarithmic diffusion of a sluggish random walker subject to resetting with memory

Abstract

We solve a model of sluggish stochastic motion in which a Brownian particle diffuses with a diffusion coefficient that decays algebraically with the distance to the origin, as $|x{|}^{-\alpha }$. Additionally, the particle resets with a constant rate $r$ to positions previously visited in the past, so that frequently visited regions are more likely to be revisited. An exact expression is obtained at all times for the position distribution in arbitrary spatial dimensions. At late times, the typical displacement of the walker from the origin grows extremely slowly, as $[\ln (rt){]}^{1/(\alpha +2)}$, and the position distribution tends to a scaling law. For any $\alpha >0$, the scaling function has a bimodal shape with a minimum at $x=0$ and non-Gaussian tails. Although the mean square displacement is hard to compute, some generalized moments of this process can be calculated exactly at all times in one dimension, and are shown to be closely related to the moments of the well-studied model with a constant diffusion coefficient.

22 pages, 4 figures

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