Lectures on singular stochastic PDEs (Q2789578)

This book consists of 6 chapters. Chapter 1 is an introduction. It contains an overview of results concerning basic problems on singular stochastic differential parabolic equations. Chapter 2 is devoted to the notion of energy solution. The authors introduce distributions defined on the \(d\)-dimensional torus and consider the stochastic Burgers equation on the torus. The Ornstein-Uhlenbeck process is studied and the Itō formula for functions of the Ornstein-Uhlenbeck process is derived. The notion of an energy solution to the stochastic Burgers equation is presented and the existence of solutions is proved. In Chapter 3, a collection of some classical results from harmonic analysis is presented. The definition of Besov spaces is introduced and Poisson summation, the Bernstein inequality and Besov embeddings are presented. Chapter 4 is devoted to diffusion in a random environment. The authors consider a homogenization problem for the linear heat equation with random potential. Then, the two-dimensional generalized parabolic Anderson model is considered. In Chapter 5, the authors explain how to use paraproducts and the paracontrolled ansatz in order to keep the non-linear effect of the singular data under control. In Chapter 6, the stochastic Burgers equation is considered and it is shown how to apply paracontrolled distributions in order to obtain the existence and uniqueness of solutions in the non-stationary case.