Biased continuous-time random walks with Mittag-Leffler jumps

TL;DR

This paper introduces a novel class of biased continuous-time random walks with Mittag-Leffler jumps, providing explicit formulas, diffusion limits, and connections to fractional calculus, with potential applications in stochastic processes on digraphs.

Contribution

It constructs new Laplacian matrix functions for random walks, defines the space-time Mittag-Leffler process, and derives explicit solutions and diffusion limits, advancing the theory of fractional stochastic processes.

Findings

Explicit formulas for state probabilities of the process

Diffusion limit reveals connections to Prabhakar fractional calculus

Potential applications in space-time generalizations of Poisson processes

Abstract

We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps subordinated to a (continuous-time) fractional Poisson process. We call this process `{\it space-time Mittag-Leffler process}'. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a `well-scaled' diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the `state density kernel' solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time fractional Mittag-Leffler process. The approach of construction of good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs.