A Microlocal Approach to Renormalization in Stochastic PDEs

TL;DR

This paper introduces a microlocal analysis-based framework for renormalization in non-linear stochastic PDEs, avoiding traditional regularization and subtraction of infinities, inspired by quantum field theory methods.

Contribution

It develops a novel algebraic and microlocal approach to renormalization in stochastic PDEs, enabling analysis without regularization schemes.

Findings

Successfully applied to stochastic ^d model

Provides a regularization-free renormalization method

Bridges quantum field theory techniques with stochastic PDE analysis

Abstract

We present a novel framework for the study of a large class of non-linear stochastic PDEs, which is inspired by the algebraic approach to quantum field theory. The main merit is that, by realizing random fields within a suitable algebra of functional-valued distributions, we are able to use techniques proper of microlocal analysis which allow us to discuss renormalization and its associated freedomw without resorting to any regularization scheme and to the subtraction of infinities. As an example of the effectiveness of the approach we apply it to the perturbative analysis of the stochastic $\Phi^3_d$ model.