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

Abstract

We consider generalized time-fractional evolution equations of the form $$u(t)=u_0+{\int }_0^{t}k(t,s)Lu(s)ds$$ 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 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.

References in corpus (1)

• Stochastic solutions of generalized time-fractional evolution equations

Cited by in corpus (2)

• Subdiffusion in the Presence of Reactive Boundaries: A Generalized Feynman-Kac Approach
• Fractional Ito Calculus for Randomly Scaled Fractional Brownian Motion and its Applications to Evolution Equations