Abstract: 

Many natural phenomena, such as heat flow or fluid motion, are effectively described by PDEs. To obtain more realistic models, one may account for random effects, like external shocks or unpredictable microscopic fluctuations, which naturally leads to incorporating stochasticity into the picture and to studying the resulting SPDEs. This immediately raises a basic question: what is the right notion of “noise” in an infinite-dimensional setting, and how can it be defined in a mathematically rigorous way? The aim of this talk is to introduce cylindrical Lévy processes and explain how they provide a natural framework for modeling jump-type noise in infinite dimensions. We will give an informal discussion of stochastic evolution equations driven by such noise and why the central analytical challenge is to make sense of stochastic integrals with respect to cylindrical Lévy processes. After outlining the main ideas and results of the corresponding integration theory, we return to SPDEs to illustrate how these tools lead to the well-posedness of a wide class of stochastic evolution equations driven by cylindrical Lévy noise.