# The elliptic stochastic quantization of some two dimensional Euclidean QFTs

## TL;DR

This paper investigates elliptic stochastic partial differential equations on two-dimensional spaces, establishing a connection between their solutions and Gibbs measures, with applications to quantum field models like Liouville theory.

## Contribution

It introduces a dimensional reduction principle for elliptic SPDEs with Gaussian noise, linking solutions to Gibbs measures in a broad setting, and applies this to quantum field models with polynomial and exponential interactions.

## Key findings

- Established existence and uniqueness of solutions for exponential interactions on .2.
- Proved dimensional reduction for the elliptic SPDEs under general conditions.
- Derived invariant measures for quantum field models with specific charge parameters.

## Abstract

We study a class of elliptic SPDEs with additive Gaussian noise on $\mathbb{R}^2 \times M$, with $M$ a $d$-dimensional manifold equipped with a positive Radon measure, and a real-valued non linearity given by the derivative of a smooth potential $V$, convex at infinity and growing at most exponentially. For quite general coefficients and a suitable regularity of the noise we obtain, via the dimensional reduction principle discussed in our previous paper on the topic, the identity between the law of the solution to the SPDE evaluated at the origin with a Gibbs type measure on the abstract Wiener space $L^2 (M)$. The results are then applied to the elliptic stochastic quantization equation for the scalar field with polynomial interaction over $\mathbb{T}^2$, and with exponential interaction over $\mathbb{R}^2$ (known also as H{\o}eg-Krohn or Liouville model in the literature). In particular for the exponential interaction case, the existence and uniqueness properties of solutions to the elliptic equation over $\mathbb{R}^{2 + 2}$ is derived as well as the dimensional reduction for the values of the ``charge parameter'' $\sigma = \frac{\alpha}{2\sqrt{\pi}} < \sqrt{4 \left( 8 - 4 \sqrt{3} \right) \pi} \simeq \sqrt{4.23\pi}$, for which the model has an Euclidean invariant probability measure (hence also permitting to get the corresponding relativistic invariant model on the two dimensional Minkowski space).