Singular integral operators and stochastic processes (A02)

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We develop a mathematical theory for nonlocal and discrete differential operators, which are related to Markov processes on subsets of the Euclidean space or graphs. We focus on results which are crucial for an understanding of the underlying dynamics independent of regularity assumptions on the state space or inhomogeneities of the medium. In this project, we study the following two interrelated parts, which go beyond current research and provide several new mathematical challenges: (I) Nonlocal operators with logarithmic order of differentiability and (II) Nonlocal operators and Markov processes on graphs.