The elliptic stochastic quantization of some two dimensional Euclidean QFTs

We study a class of elliptic SPDEs with additive Gaussian noise on R2 × 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 (Ann. Probab. 48 (2020) 1693–1741), the identity between the law of the solution to the SPDE evaluated at the origin with a Gibbs type measure on an abstract Wiener space over M. The results are then applied to the elliptic stochastic quantization equation for the scalar field with polynomial interaction over T2, and with exponential interaction over R2 (known also as Høegh-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 R2+2 is derived as well as the dimensional reduction for the values α of the “charge parameter” (formula persented) for which the model has an Euclidean invariant, reflection positive probability measure (hence also permitting to get the corresponding relativistic invariant model on the two dimensional Minkowski space).