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

References in corpus (12)

• Distributed Order Calculus and Equations of Ultraslow Diffusion
• Distributed-order fractional Cauchy problems on bounded domains
• Fractional diffusion equations and processes with randomly varying time
• Space-time fractional diffusion on bounded domains
• Brownian subordinators and fractional Cauchy problems
• Non-Markovian diffusion equations and processes: analysis and simulations
• Fokker-Planck-Kolmogorov equations associated with SDEs driven by time-changed fractional Brownian motion
• SDEs driven by a time-changed Lévy process and their associated time-fractional order pseudo-differential equations
• Quasi-Feynman formulas -- a method of obtaining the evolution operator for the Schroedinger equation
• Reflected Spectrally Negative Stable Processes and their Governing Equations
• The Euler scheme for Feller processes
• Smooth Contractive Embeddings and Application to Feynman Formula for Parabolic Equations on Smooth Bounded Domains

Cited by in corpus (1)

• Time-changed fractional Ornstein-Uhlenbeck process