Complex systems play an important role in exact and life sciences, embracing a richness of systems such as liquid crystals, glasses, polymers, biopolymers, proteins, organisms and even ecosystems. Many of such systems can be modeled by evolution equations with non-local integro-differential operators (sometimes, the considered operators can be interpreted in terms of fractional derivatives). In many cases, such evolution equations are related to some stochastic processes (with or without memory). These relations can be established via Feynman–Kac formulae, i.e. representations of solutions of evolution equations in terms of expectations of functionals of stochastic processes. Feynman–Kac formulae allow to model the considered dynamics through simulation of underlying stochastic processes (via Monte-Carlo methods). Often, the relations between evolution equations and stochastic processes can be obtained also in terms of operator semigroups. This allows to use many tools of the Operator Semigroup Theory to construct or to approximate the solutions of the considered evolution equations. This all brings together Analysis, Stochastics and Mathematical Physics.

In this course, we first introduce the basic notions and outline some classical results of the Operator Semigroup Theory, consider interrelations between operator semigroups, evolution equations and stochastic processes. Further, we present our own results on this subject. In particular, we introduce the so-called Method of Chernoff Approximations for evolution semigroups. Chernoff approximations give rise to numerical schemes for solving the corresponding evolution equations and related stochastic differential equations; Chernoff approximations can often be interpreted as approximations of path integrals in Feynman-Kac formulas. We present some recent results on Chernoff approximations for subordinate semigroups, for Feller semigroups (Feller processes), for semigroups which are obtained from some original ones through different perturbation procedures. Further, we consider a general class of generalized time-fractional evolution equations containing a memory kernel k and an operator L being the generator of a strongly continuous semigroup. Such type of equations are used in particular in the models of anomalous diffusion. We present a subordination principle for such evolution equations, what allows to use all approximations and stochastic representations of the semigroup, generated by L, for solving this type of equations. Further, we establish 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.

Literature:

1. Christian Bender, Marie Bormann, and Yana A. Butko, Subordination principle and Feynman-Kac formulae for generalized time-fractional evolution equations, Fract. Calc. Appl. Anal. (2022). https://doi.org/10.1007/s13540-022-00082-8
    
2. Christian Bender and Yana A. Butko, Stochastic solutions of generalized time- fractional evolution equations, Fract. Calc. Appl. Anal. 25 N2, 488–519 (2022). https://doi.org/10.1007/s13540-022-00025-3
    
3. Ya.A. Butko. Chernoff approximation of subordinate semigroups. Stoch. Dyn., 18 N3 (2018), 1850021, 19 p. https://doi.org/10.1142/S0219493718500211
    
4. Ya. A. Kinderknecht (Butko). Habilitation script: Chernoff approximation of evolution semigroups generated by Markov processes. Feynman formulae and path integrals, 2018, xi+168 p. https://www.math.uni-sb.de/ag/fuchs/Menupkte/Publikationen/Public-Yana/Butko-Habilitation.pdf
    
5. A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983.
    
6. N. Jacob. Pseudo-differential operators and Markov processes. Vol.I—III. Imperial College Press, 2001.
    
7. K.J. Engel, R. Nagel. One-Parameter Semigroups for Linear Evolution Equations, Springer, 2000.
    
8. K.-I. Sato. Levy Processes and Infinitely Divisible Distributions. Cambridge University Press, 1999.