Stochastic solutions for space-time fractional evolution equations on bounded domain

TL;DR

This paper develops stochastic solutions for space-time fractional evolution equations on bounded domains, proving existence, uniqueness, and providing a stochastic representation, including new results on inhomogeneous Caputo equations.

Contribution

It introduces a novel stochastic representation for space-time fractional equations with nonlocal initial conditions and proves key theoretical properties.

Findings

Established existence and uniqueness of solutions.

Derived a new stochastic representation for the equations.

Proved a new result on inhomogeneous Caputo evolution equations.

Abstract

Space-time fractional evolution equations are a powerful tool to model diffusion displaying space-time heterogeneity. We prove existence, uniqueness and stochastic representation of classical solutions for an extension of Caputo evolution equations featuring nonlocal initial conditions. We discuss the interpretation of the new stochastic representation. As part of the proof a new result about inhomogeneous Caputo evolution equations is proven.