Elliptic stochastic quantization

We prove an explicit formula for the law in zero of the solution of a class of elliptic SPDE in 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 Rd+1 dimensional parabolic SPDE and its 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 non-supersymmetric, elementary proof seems only to be available in the Gaussian case.