On fully mixed and multidimensional extensions of the Caputo and Riemann-Liouville derivatives, related Markov processes and fractional differential equations

TL;DR

This paper introduces multidimensional and mixed extensions of Caputo and Riemann-Liouville derivatives, linking them to Markov processes and fractional differential equations, providing new well-posedness results and a unified probabilistic framework.

Contribution

It develops fully mixed and multidimensional generalizations of fractional derivatives and connects them to Markov processes, advancing the theoretical understanding of fractional PDEs.

Findings

Established well-posedness of the extended derivatives

Connected derivatives to Markov processes with jump interruptions

Unified existing fractional derivative theories

Abstract

From the point of view of stochastic analysis the Caputo and Riemann-Liouville derivatives of order $\al \in (0,2)$ can be viewed as (regularized) generators of stable L\'evy motions interrupted on crossing a boundary. This interpretation naturally suggests fully mixed, two-sided or even multidimensional generalizations of these derivatives, as well as a probabilistic approach to the analysis of the related equations. These extensions are introduced and some well-posedness results are obtained that generalize, simplify and unify lots of known facts. This probabilistic analysis leads one to study a class of Markov processes that can be constructed from any given Markov process in $\R^d$ by blocking (or interrupting) the jumps that attempt to cross certain closed set of 'check-points'.