Random time-change with inverses of multivariate subordinators: governing equations and fractional dynamics

TL;DR

This paper extends the theory of Markov processes composed with inverse subordinators to multivariate cases, deriving governing equations and demonstrating applications in modeling anisotropic anomalous diffusion.

Contribution

It introduces a generalized framework for multivariate inverse subordinators and derives their governing equations, expanding the understanding of fractional dynamics in complex systems.

Findings

Derived integro-differential equations for multivariate inverse subordinators.

Connected the theoretical framework to models of anisotropic anomalous diffusion.

Provided a weak limit model of continuous-time random walks in anisotropic media.

Abstract

It is well-known that compositions of Markov processes with inverse subordinators are governed by integro-differential equations of generalized fractional type. This kind of processes are of wide interest in statistical physics as they are connected to anomalous diffusions. In this paper we consider a generalization; more precisely we mean componentwise compositions of $\mathbb{R}^d$-valued Markov processes with the components of an independent multivariate inverse subordinator. As a possible application, we present a model of anomalous diffusion in anisotropic medium, which is obtained as a weak limit of suitable continuous-time random walks.