A Novel Perturbative Approach to Stochastic Partial Differential Equations

The topic of this thesis is the application of techniques proper of algebraic quantum field theory (AQFT) to the analysis of stochastic partial differential equations (SPDEs), in particular to non- linear ones. Despite being apparently so far apart, these two frameworks have a lot in common and, probably, the most unexpected shared feature is the need of invoking renormalization. Chapter 1 is devoted to recollecting some basic material about stochastic partial differential equations, starting from some motivating examples, presenting a brief survey of the theory of regularity structures and highlighting some notable technical results. In this chapter also some further results of the author are discussed, in particular concerning a microlocal version of the Young's product theorem and the formulation on smooth manifolds of the reconstruction theorem in the framework of coherent germs of distributions. The remaining Chapters are devoted to the main contribution of this Ph.D. thesis, namely the microlocal approach to SPDEs. This provides a novel framework for the perturbative analysis of a vast class of non-linear SPDEs. In particular, adapting techniques proper of AQFT, such as microlocal analysis and the theory of the scaling degree, it allows to deal with renormalization avoiding any regularization procedures and subtraction of in infinities. On the contrary, it allows the explicit construction of finite renormalization constants and the classification of the ambiguities arising as a consequence of the renormalization procedure. The last chapter is devoted to the application of the general machinery discussed in the previous two chapters to a specific example, namely the stochastic quantization equation. In this chapter we make some explicit computations at first order in perturbation theory both for the expectation value of the solution and for the two-point correlation function.