Scientific seminar on the differential and functional differential equation “Feynman-Kac formulae for time fractional evolution equations”
Speaker: Yana Kinderknecht (Butko), Saarland University, 66123 Saarbrucken, Germany.
Topic: Feynman-Kac formulae for time fractional evolution equations.
Abstract: The present talk serves as an introduction and an overview. We discuss the notion of anomalous diffusion, models of anomalous diffusion based on the use of Continuous Time Random Walks (CTRWs), evolution equations arising in such models. Typical representative of the considered class of equations is the analogue of the heat equation where the first time derivative is substituted by the fractional Caputo derivative of order $\beta\in(0,1)$. In the talk, we will discuss equations of a more general form: a class of integro-differential operators (generalizing the Caputo derivative) is being used instead of the first time derivative; a generator of a strongly continuous (e.g., Feller) semigroup is being used instead of the Laplacian. Solutions of such equations can be obtained as expectations of functionals of some stochastic processes (such representations of solutions of differential equations are usually called Feynmam-Kac formulae or stochastic representations). In this talk, we will discuss different classes of stochastic processes allowing to obtain stochastic representations for solutions of the considered evolution equations, advantages and disadvantages of the corresponding Feynman-Kac formulae.