Space-time fractional equations and the related stable processes at random time

TL;DR

This paper studies fractional space-time equations and their solutions via stable processes at random times, providing probabilistic representations and generalizations including the fractional telegraph equation and n-dimensional Gauss-Laplace law.

Contribution

It introduces a unified stochastic framework for fractional equations using stable processes and inverse stable subordinators, extending classical models and solving generalized telegraph and diffusion equations.

Findings

Solutions are represented by compositions of stable processes and inverse subordinators.

The framework includes the fractional telegraph equation as a special case.

As parameters vary, solutions connect to the n-dimensional Gauss-Laplace law.

Abstract

In this paper we consider the general fractional equation \sum_{j=1}^m \lambda_j \frac{\partial^{\nu_j}}{\partial t^{\nu_j}} w(x_1,..., x_n ; t) = -c^2 (-\Delta)^\beta w(x_1,..., x_n ; t), for \nu_j \in (0,1], \beta \in (0,1] with initial condition w(x_1,..., x_n ; 0)= \prod_{j=1}^n \delta (x_j). The solution of the Cauchy problem above coincides with the distribution of the n-dimensional process \bm{S}_n^{2\beta} \mathcal{L} c^2 {L}^{\nu_1,..., \nu_m} (t) \r, t>0, where \bm{S}_n^{2\beta} is an isotropic stable process independent from {L}^{\nu_1,..., \nu_m}(t) which is the inverse of {H}^{\nu_1,..., \nu_m} (t) = \sum_{j=1}^m \lambda_j^{1/\nu_j} H^{\nu_j} (t), t>0, with H^{\nu_j}(t) independent, positively-skewed stable r.v.'s of order \nu_j. The problem considered includes the fractional telegraph equation as a special case as well as the governing equation of stable processes. The composition \bm{S}_n^{2\beta} (c^2 {L}^{\nu_1,..., \nu_m} (t)), t>0, supplies a probabilistic representation for the solutions of the fractional equations above and coincides for \beta = 1 with the n-dimensional Brownian motion at the time {L}^{\nu_1,..., \nu_m} (t), t>0. The iterated process {L}^{\nu_1,..., \nu_m}_r (t), t>0, inverse to {H}^{\nu_1,..., \nu_m}_r (t) =\sum_{j=1}^m \lambda_j^{1/\nu_j} _1H^{\nu_j} (_{2}H^{\nu_j} (_3H^{\nu_j} (... _{r}H^{\nu_j} (t)...))), t>0, permits us to construct the process \bm{S}_n^{2\beta} (c^2 {L}^{\nu_1,..., \nu_m}_r (t)), t>0, the distribution of which solves a space-fractional generalized telegraph equation. For r \to \infty and \beta = 1 we obtain a distribution which represents the n-dimensional generalisation of the Gauss-Laplace law and solves the equation \sum_{j=1}^m \lambda_j w(x_1,..., x_n) = c^2 \sum_{j=1}^n \frac{\partial^2}{\partial x_j^2} w(x_1,..., x_n).