On heterogeneous diffusion processes and the formation of spatial-temporal nonlocality

N S Arkashov; V A Seleznev

doi:10.1063/5.0159907

On heterogeneous diffusion processes and the formation of spatial-temporal nonlocality

Chaos. 2023 Jul 1;33(7):073145. doi: 10.1063/5.0159907.

Authors

N S Arkashov¹, V A Seleznev²

Affiliations

¹ Sobolev Institute of Mathematics, 4, Acad. Koptyug Ave., 630090 Novosibirsk, Russia.
² Department of Engineering Mathematics, Novosibirsk State Technical University, 20, Karl Marx Ave., 630073 Novosibirsk, Russia.

PMID: 37466421
DOI: 10.1063/5.0159907

Abstract

Heterogeneous diffusion processes defined as a solution to the overdamped Langevin equation with multiplicative noise, the amplitude of which has a power-law space-dependent form, are studied. Particular emphasis is on discrete analogs of these processes, for which, in particular, an asymptotic estimate of their variance behavior in time is obtained. In addition, a class of processes formed by deformation of the discrete analog of the fractional Brownian motion using the Cantor ladder and its inverse transformation is considered. It is found that such a class turns out to be close in structure to discrete analogs of heterogeneous processes. This class of processes allows us to illustrate geometrically the emergence of sub- and superdiffusion transport regimes. On the basis of discrete analogs of heterogeneous processes and memory flow phenomenology, we construct a class of random processes that allows us to model nonlocality in time and space taking into account spatial heterogeneity.