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arXiv · 2202.01655

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

Christian Bender·Marie Bormann·Yana A. Butko
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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 $ξ$ 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.

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Christian Bender, Marie Bormann, Yana A. Butko. 2022-06-23. Subordination principle and Feynman-Kac formulae for generalized time-fractional evolution equations. https://doi.org/10.1007/s13540-022-00082-8

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