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  Many random phenomena in real life and science do not evolve continuously in time, but rather have abrupt behaviour, manifested through a sudden change of states.Discontinuous stochastic processes model the dynamics of such random evolutions. The case when the evolution forgets its past leads to the Markov property of the stochastic process. Markov processes are, through their infinitesimal generators,related to functional analysis and theory of partial differential equations. The infinitesimal generator of a discontinuous Markov process is a non-local, integro-differential operator. The goal of this project is to study certain classes of discontinuous stochastic processes and non-local operators, their potential theory and stability theory, and connection to the theory of partial differential equations. Specifically, our objectives are to investigate the following problems: (1) Given a bounded open subset of the Euclidean space, we consider a discontinuous Markov process on that subset defined through its jumping kernel. The kernel takes into account the distances of points to the boundary of the subset, as well as the distance between the points. We address the question of constructing such processes and study their potential-theoretic, analytic and stabilityproperties, such as Green function and heat kernel estimates, boundary Harnack principle, ergodic properties and homogenization problem; (2) We want to extend some parts of the theory of semilinear equations for elliptic differential operators to non-local operators. We start with the fractional Laplacian and then move to more general non-local operators satisfying certain weak scaling properties. The final goal is to find a probabilistic representation of solutions of semilinear equations for non-local operators; (3) Applications of discontinuous stochastic processes to risk theory.

  The research methods we use come from probability theory, potential theory, partial differential equations and functional analysis. These methods include but are not limited to, Markov processesmethods, potential-theoretic methods (both probabilistic and analytic), methods from Dirichlet form theory, martingale methods, random measures methods, and PDE methods.

  The expected result of the proposed research is to further develop some aspects of the mathematical theory of discontinuous stochastic processes and non-local operators and to contribute to deeper understanding of the theory.