Program

Lecture 01: Operator semigroups, evolution equations and Markov processes We introduce the objects mentioned in the title, outline the interply between them and present some cornerstones of the Theory of Operator Semigroups. 

Lecture 02: Feller semigroups / Feller processes 

We discuss an important subclass of Markov processes: Feller processes and in particu lar Lévy processes. We establish properties of the corresponding semigroups and integro differential evolution equations (e.g., equations with fractional Laplacians and relativistic Hamiltonians), find connections between convolution semigroups of measures, infinite divisible distributions, Lévy processes and continuous negative definite functions. 

Lecture 03: Construction and approximation of operator semigroups Differrent approaches to construct/approximate an operator semigroup will be discussed. We start with some standard procedures, discuss the perturbation techniques, Bernstein functions and subordination of operator semigroups / stochastic processes in the sence of Bochner. 

Lecture 04: Chernoff approximation of evolution semigroups 

We present the Chernoff theorem and its corollaries, construct some Chernoff approxima tions for Feller semigroups (in particular, for Feller diffusions) and obtain the Feynman– Kac formula for a particle in external potentials. We present Chernoff approximations for operator semigroups with additively andmultiplicatively perturbed generators.We discuss the relations of Chernoff approximations of operator semigroups with numerical schemes for PDEs and SDEs. 

Lecture 05: Stochastic processes and evolution equations in the models of anoma lous diffusion 

We present a general model of continuous time random walks (CTRWs), leading to dif ferent types of diffusion (standard diffusion, subdiffusion, superdiffusion, fractional dif fusion) and obtain governing equations for probability density functions of the processes being the scaling limits of CTRWs. In the regime of standard diffusion, one obtains the stan dard diffusion equation. In some particular cases of other regimes of CTRWs, the govern ing equations are actually time- or/and space-fractional diffusion equations. We discuss a large class of generalized time-fractional evolution equations, the subordination structure of their solutions (and hence their relation to semigroups and Markov processes), and dif ferent classes of underlying stochastic processes. In particular, we present Feymnan-Kac formulae for solutions of such equations on the base of Markov processes time-changed by inverse subordinators, and on the base of randomly scaled Gaussian processes (such as Generalized Grey Brownian Motion (GGBM)).