Cylindrical Lévy processes are a natural extension of cylindrical Brownian motion which has been the standard model of random perturbations of partial differential equations and other models in infinite dimensions for the last 50 years. Here, the attribute cylindrical refers to the fact that cylindrical Brownian motions are not classical stochastic processes attaining values in the underlying space but are generalised objects. The reasons for the choice of cylindrical but not classical Brownian motion can be found in the facts that there does not exist a classical Brownian motion with independent components in an infinite dimensional Hilbert space, and that cylindrical processes enable a very flexible modelling of random noise in time and space.

This talk is a very introductory presentation to cylindrical Lévy processes. We explain the difficulty to define random noises in infinite dimensions and explain the approach by cylindrical measures and cylindrical random variables, which are strongly related to other areas such as harmonic analysis and operator theory. We present some specific examples of cylindrical Lévy processes in detail and discuss their relations to other models of random perturbations in the literature. We explain how a theory of stochastic integration for cylindrical Lévy processes can be developed although standard approaches to stochastic integration cannot be applied, and how this theory can be used to derive a theory of stochastic partial differential equations driven by Cylindrical Lévy processes.