Subordination principle and Feynman-Kac formulae for generalized time-fractional evolution equations

Abstract

We consider generalized time-fractional evolution equations of the formwith a fairly general memory kernel k and an operator L being the generator of a strongly continuous semigroup. In particular, L may be the generator L 0 of a Markov process ξ on some state space Q, or L ∶= L 0 + b∇ + V for a suitable potential V and drift b, or L generating subordinate semigroups or Schrödinger type groups. This class of evolution equations includes in particular time-and space-fractional heat and Schrödinger type equations. We show that a subordination principle holds for such evolution equations and obtain Feynman-Kac formulae for solutions of these equations with the use of different stochastic processes, such as subordinate Markov processes and randomly scaled Gaussian processes. In particular, we obtain some Feynman-Kac formulae with generalized grey Brownian motion and other related self-similar processes with stationary increments.

Subdiffusion in the Presence of Reactive Boundaries: A Generalized Feynman–Kac Approach

“…The main purpose of this paper is to provide such a generalization. We do this by considering an alternative generalized FKE [27,28] rather than the previously mentioned fractional FKE. The generalized FKE we use is in the form of an (imaginary time) time fractional Schrödinger equation [29][30][31][32] and governs subordinated forms of the functionals.…”