Transition density estimates for Lévy-type processes

这项研究的目的是获得所谓的Lévie型过程的热核估计；这些过程是由具有负定(因此，粗糙和非经典)符号的伪微分算子生成的Feller(跳跃)过程。L-维型过程在局部上看起来像L-维过程，但它们的性质可能会随着空间位置的不同而改变。典型的例子是由变阶分数拉普拉斯过程产生的类稳定过程。如果没有主导扩散部分，人们对这类过程的转移函数以及随机性质(当地时间、样本路径渐近性、短时行为等)的研究知之甚少。正处于初级阶段。我们的目的是将解析方法和随机方法结合起来，得到Levy型过程的短时热核估计。我们的观点是将转移函数理解为相应的Kolmogorov倒向方程的基本解，并希望利用偏微分方程组理论的方法来获得保证我们可以用收敛级数(参数线构造)来表示基本解的一般条件。这里的主要问题是，大多数随机有趣的符号都不在任何经典符号类中，我们需要为这种设置开发新的技术。作为应用，我们想要得到Léviy型过程样本轨道的短时渐近性(重对数律)，该过程的尺度函数可以用该过程的生成元的符号显式地得到。

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.