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

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 \(\...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2022-06
Main Authors: Bender, Christian, Bormann, Marie, Butko, Yana A
Format: Article
Language:English
Subjects:
Brownian motion
Chaos theory
Euclidean space
Evolution
Gaussian process
Markov processes
Mathematical analysis
Operators (mathematics)
Self-similarity
Semigroups
Stochastic processes
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2202.01655