Infinitesimal analysis of singular stochastic partial differential equations

Abstract

The last decade has seen a considerable rise in the study of singular stochastic partial differential equations (SPDEs) which turned into the birth of many celebrated techniques for the development of a solution theory for such kind of equations. The present thesis is devoted to the study of some problems involving singular SPDEs with approaches based on the study of the infinitesimal generator of the semigroup related to the solution to the equation under investigation. <br /> In the first part of the work, we study a probabilistic approach to singular SPDEs. More precisely, we deal with a martingale problem associated to the infinitesimal generator of the equation involved. Because of the irregular behaviour of the terms appearing in the equation, the first task is to give a meaning to the martingale problem itself, and only in a second moment one can proceed with studying existence and uniqueness for the martingale problem. In order to do so, we exploit stochastic calculus in infinite dimensions and the analysis of the infinitesimal generator corresponding to the solution of the equation, defining a suitable domain where we are able to solve the related Kolmogorov backward equation. As an application of the technique under consideration, we focus on (quasi-)stationary solutions to hyperviscous stochastic Navier–Stokes equation in two dimensions (both on the torus and on the plane). Such an approach was first developed for singular SPDEs by Gubinelli and Perkowski for the stochastic Burgers equation on the one-dimensional torus and on the real line. <br /> The second part of the thesis is concerned with Euclidean quantum field theory. We approach the problem of stochastic quantization by providing a differential characterization of quantum field theories through the study of a singular integration by parts formula. In particular, we focus on the case of exponential interactions (alias Høegh-Krohn model) on the whole plane and show existence and uniqueness of a measure solving the associated renormalized integration by parts, that is a suitable Euclidean Dyson–Schwinger equation. This is achieved requiring that the measure can be compared with a Gaussian free field (meaning that it has a finite Wasserstein-type distance from it) and studying the corresponding symmetric Fokker–Planck–Kolmogorov equation. More precisely, we get existence of solutions exploiting Lyapunov functions, and uniqueness by analyzing the resolvent equation associated to the infinitesimal generator. This allows us to characterize the invariant measure of the stochastic quantization equation as the only measure satisfying the integration by parts formula.