Chronological operator-valued Feynman-Kac formulae for generalized fractional evolutions

TL;DR

This paper develops a generalized operator-valued Feynman-Kac formula for solving complex fractional differential equations involving mixtures of Caputo derivatives, providing new integral representations even for standard cases.

Contribution

It introduces a novel non-commutative operator-valued Feynman-Kac approach for time-dependent fractional equations with arbitrary Caputo derivative mixtures, extending existing solution methods.

Findings

Derived explicit solutions for time-homogeneous cases using Mittag-Leffler functions.

Established a new integral representation for solutions in the time-dependent case.

Reinterpreted fractional equations as stationary problems with operator-valued Feynman-Kac formulas.

Abstract

We study the generalized fractional linear problem $D^{\nu}_{a+*} f(x) =A(x)f(x)+g(x)$, where $D^{\nu}$ is an arbitrary mixture of Caputo derivatives of order at most one and $A(x)$ a family of operators in a Banach space generating strongly continuous semigroups. For time homogeneous case, when $A(x)$ does not depend on time $x$, the solution is expressed by the generalized operator-valued Mittag-Leffler function. For the more involved time-dependent case we use the method of non-commutative operator-valued Feynman-Kac formula in combination with the probabilistic interpretation of Caputo derivatives suggested recently by the author to find the general integral representation of the solutions, which are new even for the case of the standard Caputo derivative $D^{\beta}_{a+*}$. In the point of view adopted here we analyse the fractional equations not as some 'exotic evolutions', but rather as 'standard' stationary problems leading to the stationary non-commutative operator-valued Feynman-Kac representation.