Transition density estimates for Lévy-type processes

• Applicant: Professor Dr. René Leander Schilling
• Subject Area: Mathematics
• Term: from 2013 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 239237733

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.

• DFG Programme: Research Grants