Probabilistic representation formula for the solution of fractional high order heat-type equations

TL;DR

This paper introduces a probabilistic method using complex plane random walks and subordinated processes to solve fractional high order heat-type equations, extending to both space and time fractional derivatives.

Contribution

It presents a novel probabilistic construction for solutions of fractional high order heat equations, including space and time fractional derivatives, using complex random walks and subordinated processes.

Findings

Probabilistic representation for space fractional derivatives of any order.

Extension to equations with Caputo time fractional derivatives.

Framework applicable to a broad class of fractional heat-type equations.

Abstract

We propose a probabilistic construction for the solution of a general class of fractional high order heat-type equations in the one-dimensional case, by using a sequence of random walks in the complex plane with a suitable scaling. A time change governed by a class of subordinated processes allows to handle the fractional part of the derivative in space. We first consider evolution equations with space fractional derivatives of any order, and later we show the extension to equations with time fractional derivative (in the sense of Caputo derivative) of order $\alpha \in (0,1)$.