Singular integral operators and stochastic processes (A02)

我们发展了与欧几里德空间或图的子集上的马尔可夫过程有关的非局部和离散微分算子的数学理论。我们关注的结果对于理解独立于状态空间的规则假设或介质的非均匀性的潜在动力学至关重要。在这个项目中，我们研究了以下两个相互关联的部分，它们超越了当前的研究并提供了几个新的数学挑战：(I)具有对数可微阶的非局部算子和（II）图上的非局部算子和马尔可夫过程。

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.