Wong-Zakai approximations of SDEs and SPDEs with jump noise

由布朗运动或l<s:1>过程驱动的随机常微分方程或偏微分方程对于模拟各种现实世界现象是必不可少的。然而，他们的解决方案已经通过构造只是方便的数学理想化的实际过程。大约50年前，Wong和Zakai建议将随机微分方程视为由噪声过程的顺路径正则（如光滑）近似驱动的普通随机方程的极限。在这种方法中，布朗运动可以被看作是短距离混沌运动（扩散）的理想化，而跳跃则是非常快速连续的长距离异常转变的理想化。对于随机微分方程，已知极限过程在布朗情况下求解Stratonovich方程，在一般情况下求解正则（Marcus）方程。在物理、水文学和工程学的例子的激励下，我们将支持由lsamvy噪声驱动的随机常微分方程和偏微分方程的Wong—Zakai型近似。重点将放在非标准Skorokhod拓扑中不连续极限的正则逼近的收敛性和极限随机方程中适当校正项的识别上。项目的一个重点将放在整个空间的平流-扩散方程上，其中lsamvy噪声作用于输运项，这可以与湍流扩散率有关。对于有界域上的平流扩散方程，边界上的lsamvy噪声将模拟污染物瞬间释放到地下水中。最后，我们探讨了利用确定性求解器在Wong-Zakai近似下求解spde的数值方法。在这个项目中获得的结果，除了它们的数学价值外，应该有助于更深入地了解物理和应用科学中由lsamvy驱动的动力学和数值。

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.