Elliptic stochastic quantization

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 (Phys. Rev. Lett. 43 (1979) 744–745), 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 a $\mathbb{R}^{d+1}$ 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 (Phys. Rev. 150 (1966) 1079–1085) and Parisi and Wu (Sci. Sin. 24 (1981) 483–496). 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. (Comm. Math. Phys. 94 (1984) 459–482). 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 nontrivial and a nonsupersymmetric, elementary proof seems only to be available in the Gaussian case.