Transition density estimates for Lévy-type processes

这个研究项目的目的是获得热核估计所谓的Lévy型过程;这些是Feller（跳跃）过程，由具有负定（因此，粗糙和非经典）符号的伪微分算子生成。Lévy型过程在局部上看起来像Lévy过程，但它们的性质可能会根据空间中的位置而改变。典型的例子是由变阶分数拉普拉斯算子生成的类稳定过程。如果不存在主导扩散部分，则对这类过程的转移函数知之甚少，并且对随机性质（局部时间、样本路径渐近性、短时行为等）的研究也不多。还处于起步阶段我们的目标是联合收割机分析和随机方法来获得短期热核估计Levy型过程。我们的观点是将转移函数理解为相应的Kolmogorov向后方程的基本解，并且我们希望使用偏微分方程理论的方法来获得保证我们可以用收敛级数表示基本解的一般条件（参数构造）。这里的主要问题是，大多数随机有趣的符号不属于任何经典符号类，我们需要为这种设置开发新的技术。作为一个应用，我们想得到一个Lévy型过程（Chung型重对数律）的样本路径的短时渐近性，其标度函数可以根据过程的生成元的符号显式导出。

The aim of this research project is to obtain heat kernel estimates for so-called Lévy-type processes; these are Feller (jump) processes which are generated by pseudo differential operators with negative definite (hence, rough and non-classical) symbols. Lévy-type processes look locally like Lévy processes, but their character may change depending on the position in space. Typical examples are stable-like processes which are generated by variable-order fractional Laplacians. If there is no dominating diffusion part, not much is known on the transition function of such processes, and the study of stochastic properties (local times, sample path asymptotics, short-time behaviour etc.) is in its infancy. Our aim is to combine analytic and stochastic methods to derive short-time heat kernel estimates for Levy-type processes. Our point of view is to understand the transition function as the fundamental solution of the corresponding Kolmogorov backwards equation, and we want to use methods from the theory of partial differential equations to obtain general conditions which guarantee that we can represent the fundamental solution in terms of convergent series (parametrix construction). The main problem here is that most stochastically interesting symbols are not in any of the classical symbol classes, and that we need to develop new techniques for this setting. As an application we want to derive short-time asymptotics for the sample paths of a Lévy-type process (Chung-type law of the iterated logarithm) with a scaling function which can be explicitly derived in terms of the symbol of the generator of the process.