Prof. Yana Kinderknecht (Univ. Kassel): Subordination principle, stochastic solutions and Feynman-Kac formulae for generalized time fractional evolution equations

EINLADUNG ZUM INSTITUTSKOLLOQUIUM
MATHEMATIK UND NATURWISSENSCHAFTEN
am Montag, den 05.06.2023, um 17:15, im Hörsaal 1409

Livestream:  https://uni-kassel.cloud.panopto.eu/Panopto/Pages/Sessions/List.aspx?folderID=d9242d60-5cb4-4afc-ac3d-adb700e7b246
 

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

Prof. Dr. Yana Kinderknecht (Univ. Kassel)

Abstract:

We consider generalized time-fractional evolution equations of the form

u(t) = u₀ + ∫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 (on a Banach space). In particular, L may be the generator L₀ of a Markov process χ on some state space Q, or  L := L₀ + 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; some of these equations are used in models of anomalous diffusion.

We show that the 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.

The talk is based on the joint work with Ch. Bender and M. Bormann.