Wong Zakai type approximations of SDEs and SPEDEs with jump noise

Stochastic ordinary or partial differential equations driven by a Brownian motion or Lévy processes are indispensable for the modelling of various real world phenomena. However their solutions are already by construction just convenient mathematical idealizations of real processes. About 50 years ago, Wong and Zakai suggested to treat stochastic differential equations as limits of the ordinary random equations driven by path-wise regular (e.g. smooth) approximations of the noise process. In this approach, Brownian motions can be seen as idealization of short range chaotic motions (diffusion), whereas jumps appear as idealizations of very fast continuous long range anomalous transitions. It is known in case of stochastic differential equations, that the limiting process solves the Stratonovich equation in the Brownian case, and the canonical (Marcus) equation in the general case with jump noise. Motivated by examples form physics, hydrology and engineering, we are going to underpin the Wong--Zakai type approximations for stochastic ordinary and partial differential equations driven by Lévy noise. The main emphasis will be made on the convergence of the regular approximations to a discontinuous limit in the non-standard Skorokhod topology and to the identification of proper correction terms in the limiting stochastic equation. One focus of the project will be set on the advection-diffusion equations in the whole space with Lévy noise acting on the transport term, which can be related to the turbulent diffusivity. For advection-diffusion equations on bounded domains, Lévy noise on the boundary will mimic an instantaneous release of a contaminant into a ground water. Finally, we explore the numerical methods of solving SPDEs with the help of deterministic solvers applied to the Wong-Zakai approximations. The results obtained in the project, besides their mathematical value, should contribute to a deeper understanding of the Lévy driven dynamics and numerics in physics and applied sciences.

Eines der aktuellsten Forschungsgebiete in der stochastischen Analysis sind stochastische partielle Differentialgleichungen. Dies betrifft sowohl die Theorie als auch die Anwendungen. Hier betrachtet man systeme die normalerweise mittels partiellen Differentialgleichungen beschrieben werden wo man neben den deterministischen Einflüssen zusätzlich einen stochastischen Störfaktoren (Rauschen) betrachtet. Beispielsweise konnen durch stochastische partielle Differentialgleichungen Prozesse in der Populationsgenetik, die Ausbreitung von Epidemien, die Ausbreitung von Schadstoffen in der Atmosphaere, zufaeallige Schwingungen und vieles anderes modelliert werden. Um aber diese zu modellieren, muss man den zufaelligen Stoerterm auch approximieren. In diesen Projekt haben wir eine bestimmte Art den Stoerterm zu approximieren untersucht und untersucht mit welcher Gleichung der Prozess beschrieben werden kann, falls man die Approximation verfeinert und die Approximation konvergiert.