Wilson-It\^o diffusions

TL;DR

Wilson-Itô diffusions are a new class of stochastic fields on Euclidean space, described by stochastic differential equations, offering a non-perturbative quantization method that generalizes Wilson-Polchinski flow equations and is applicable to gauge theories.

Contribution

Introduction of Wilson-Itô diffusions as a novel non-perturbative quantization framework using stochastic dynamics and algebraic structures, independent of path-integral methods.

Findings

Defines Wilson-Itô diffusions via stochastic differential equations.

Shows these diffusions form a pre-factorization algebra.

Connects the framework to Wilson-Polchinski flow equations when a path-integral exists.

Abstract

We introduce Wilson-It\^o diffusions, a class of random fields on $\mathbb{R}^d$ that change continuously along a scale parameter via a Markovian dynamics with local coefficients. Described via forward-backward stochastic differential equations, their observables naturally form a pre-factorization algebra \`a la Costello-Gwilliam. We argue that this is a new non-perturbative quantization method applicable also to gauge theories and independent of a path-integral formulation. Whenever a path-integral is available, this approach reproduces the setting of Wilson-Polchinski flow equations.