Singular integral operators and stochastic processes (A02)

我们发展了一个关于欧氏空间或图的子集上的马尔可夫过程的非局部和离散微分算子的数学理论。我们把重点放在结果上，这些结果对于理解潜在的动力学是至关重要的，而不是关于状态空间的正则性假设或介质的不均匀。在这个项目中，我们研究了以下两个相互关联的部分，这两个部分超越了现有的研究，并提出了几个新的数学挑战：(I)具有可微对数阶的非局部算子和(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.