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

TL;DR

This paper establishes a subordination principle and derives Feynman-Kac formulae for generalized time-fractional evolution equations involving various operators, including Markov generators and Schrödinger operators, using stochastic processes like subordinate Markov processes and generalized Brownian motions.

Contribution

It introduces a broad subordination principle and Feynman-Kac representations for diverse fractional evolution equations with general kernels and operators, extending existing frameworks.

Findings

Subordination principle applies to generalized fractional evolution equations.

Feynman-Kac formulae are derived using subordinate Markov and Gaussian processes.

Includes formulas involving generalized grey Brownian motion and self-similar processes.

Abstract

We consider generalized time-fractional evolution equations of the form $$u(t)=u_0+\int_0^tk(t,s)Lu(s)ds$$ with 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 $\xi$ on some state space $Q$, or $L:=L_0+b\nabla+V$ for a suitable potential $V$ and drift $b$, or $L$ generating subordinate semigroups or Schr\"{o}dinger type groups. This class of evolution equations includes in particular time- and space- fractional heat and Schr\"odinger 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.