# Elliptic stochastic quantization

## TL;DR

This paper establishes an explicit formula for the law of solutions to a class of elliptic SPDEs in two dimensions, revealing a dimensional reduction phenomenon linking elliptic SPDEs to Gibbs measures, and clarifies the supersymmetric proof of this reduction.

## Contribution

It provides a rigorous proof of dimensional reduction for elliptic SPDEs using supersymmetric quantum field theory, fixing gaps in previous proofs and connecting elliptic SPDEs with supersymmetric models.

## Key findings

- Derived explicit law formula for elliptic SPDE solutions in 2D.
- Confirmed dimensional reduction links elliptic SPDEs to Gibbs measures.
- Fixed gaps in previous supersymmetric proofs of dimensional reduction.

## Abstract

We prove an explicit formula for the law in zero of the solution of a class of elliptic SPDE in $\mathbb{R}^2$. This formula is the simplest instance of dimensional reduction, discovered in the physics literature by Parisi and Sourlas (1979), which links the law of an elliptic SPDE in $d + 2$ dimension with a Gibbs measure in $d$ dimensions. This phenomenon is similar to the relation between an $\mathbb{R}^{d + 1}$ dimensional parabolic SPDE and its $\mathbb{R}^d$ dimensional invariant measure. As such, dimensional reduction of elliptic SPDEs can be considered a sort of elliptic stochastic quantisation procedure in the sense of Nelson (1966) and Parisi and Wu (1981). Our proof uses in a fundamental way the representation of the law of the SPDE as a supersymmetric quantum field theory. Dimensional reduction for the supersymmetric theory was already established by Klein et al. (1984). We fix a subtle gap in their proof and also complete the dimensional reduction picture by providing the link between the elliptic SPDE and the supersymmetric model. Even in our $d = 0$ context the arguments are non-trivial and a non-supersymmetric, elementary proof seems only to be available in the Gaussian case.