Time-changed processes governed by space-time fractional telegraph equations

TL;DR

This paper constructs and analyzes time-changed stochastic processes governed by space-time fractional telegraph equations, revealing their connections to stable processes, inverse subordinators, and classical processes like Brownian motion and telegraph processes.

Contribution

It introduces a unifying framework linking space-time fractional telegraph equations with compositions of stable processes and inverse subordinators, extending classical stochastic process models.

Findings

Established distributional equivalences between processes and Brownian motion at random times.

Connected fractional telegraph equations to compositions of stable processes and inverse subordinators.

Analyzed special cases including telegraph process at Brownian time and iterated Brownian motion.

Abstract

In this work we construct compositions of processes of the form \bm{S}_n^{2\beta}(c^2 \mathpzc{L}^\nu (t) \r, t>0, \nu \in (0, 1/2], \beta \in (0,1], n \in \mathbb{N}, whose distribution is related to space-time fractional n-dimensional telegraph equations. We present within a unifying framework the pde connections of n-dimensional isotropic stable processes \bm{S}_n^{2\beta} whose random time is represented by the inverse \mathpzc{L}^\nu (t), t>0, of the superposition of independent positively-skewed stable processes, \mathpzc{H}^\nu (t) = H_1^{2\nu} (t) + (2\lambda \r^{\frac{1}{\nu}} H_2^\nu (t), t>0, (H_1^{2\nu}, H_2^\nu, independent stable subordinators). As special cases for n=1, \nu = 1/2 and \beta = 1 we examine the telegraph process T at Brownian time B (Orsingher and Beghin) and establish the equality in distribution B (c^2 \mathpzc{L}^{1/2} (t)) \stackrel{\textrm{law}}{=} T (|B(t)|), t>0. Furthermore the iterated Brownian motion (Allouba and Zheng) and the two-dimensional motion at finite velocity with a random time are investigated. For all these processes we present their counterparts as Brownian motion at delayed stable-distributed time.