Chernoff approximation for semigroups generated by killed Feller processes and Feynman formulae for time-fractional Fokker-Planck-Kolmogorov equations

TL;DR

This paper develops Chernoff approximations and Feynman formulae for semigroups generated by killed Feller processes, enabling direct calculation and simulation of solutions to time-fractional evolution equations, including distributed order cases.

Contribution

It introduces a novel Chernoff approximation approach that converts into Feynman formulae for killed Feller processes and extends these methods to approximate solutions of time-fractional evolution equations.

Findings

Feynman formulae derived for killed Feller process semigroups.

New approximation method for time-fractional evolution equations.

Explicit Feynman formulae for distributed order time-fractional diffusion equations.

Abstract

Semigroups, generated by Feller processes killed upon leaving a given domain, are considered. These semigroups correspond to Cauchy-Dirichlet type initial-exterior value problems in this domain for a class of evolution equations with (possibly non-local) operators. The considered semigroups are approximated by means of the Chernoff theorem. For a class of killed Feller processes, the constructed Chernoff approximation converts into a Feynman formula, i.e. into a limit of $n$-fold iterated integrals of certain functions as $n\to\infty$. Representations of solutions of evolution equations by Feynman formulae can be used for direct calculations and simulation of underlying stochasstic processes. Further, a method to approximate solutions of time-fractional (including distributed order time-fractional) evolution equations is suggested. This method is based on connections between time-fractional and time-non-fractional evolution equations as well as on Chernoff approximations for the latter ones. Moreover, this method leads to Feynman formulae for solutions of time-fractional evolution equations. To illustrate the method, a class of distributed order time-fractional diffusion equations is considered; Feynman formulae for solutions of the corresponding Cauchy and Cauchy-Dirichlet problems are obtained.