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July 2020 Elliptic stochastic quantization
Sergio Albeverio, Francesco C. De Vecchi, Massimiliano Gubinelli
Ann. Probab. 48(4): 1693-1741 (July 2020). DOI: 10.1214/19-AOP1404


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.


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Sergio Albeverio. Francesco C. De Vecchi. Massimiliano Gubinelli. "Elliptic stochastic quantization." Ann. Probab. 48 (4) 1693 - 1741, July 2020.


Received: 1 December 2018; Revised: 1 August 2019; Published: July 2020
First available in Project Euclid: 20 July 2020

zbMATH: 07224958
MathSciNet: MR4124523
Digital Object Identifier: 10.1214/19-AOP1404

Primary: 60H15
Secondary: 81Q60 , 82B44

Keywords: dimensional reduction , elliptic stochastic partial differential equations , Euclidean quantum field theory , Stochastic quantisation , supersymmetry , Wiener space

Rights: Copyright © 2020 Institute of Mathematical Statistics


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Vol.48 • No. 4 • July 2020
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