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

Fractional Calculus and Applied Analysis. 2018. Vol. 21. No. 5.

Butko Ya. A.

We consider operator semigroups generated by Feller processes killed upon leaving a given domain. 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
leads to a representation of the solution of the corresponding Cauchy–Dirichlet type problem by a Feynman formula, i.e. by a limit of n-fold iterated integrals of certain functions as n → ∞. Feynman formulae can be used
for direct calculations, modelling of underlying dynamics, simulation of underlying stochasstic processes. Further, a method to approximate solutions of time-fractional evolution equations is suggested. The method is based
on connections between time-fractional and time-non-fractional evolution equations as well as on Chernoff approximations for the latter ones. This method leads to Feynman formulae for solutions of time-fractional evolution
equations. A class of distributed order time-fractional equations is considered; Feynman formulae for solutions of the corresponding Cauchy and Cauchy–Dirichlet type problems are obtained.

Keywords: Feynman formula, Chernoff approximation, Approximation of operator semigroups, approximation of transition probabilities, killed Feller processes, initial-exterior value problems, distributed order time-fractional Fokker–Planck–Kolmogorov equations