\(H\)-\(C^1\) maps and elliptic SPDEs with polynomial and exponential perturbations of Nelson's Euclidean free field
type: Item
instance of: scholarly article
title: \(H\)-\(C^1\) maps and elliptic SPDEs with polynomial and exponential perturbations of Nelson's Euclidean free field
zbMATH Open document ID: 1015.60052
DOI: 10.1016/S0022-1236(02)00008-3
author: Minoru W. Yoshida
author: S. Albeverio
published in: Journal of Functional Analysis
publication date: 2003-03-26T00:00:00Z
review text: Let \(\varphi\) be an \({\mathcal S}'({\mathbb R}^d)\)-valued centered Gaussian random variable with covariance \[ E\langle\varphi_1,\varphi\rangle\langle\varphi_2,\varphi\rangle= \int_{{\mathbb R}^d}((-\Delta+1)^{-1}\varphi_1(x))\varphi_2(x)dx,\quad \varphi_1,\varphi_2\in{\mathcal S}({\mathbb R}^d) \] (``Euclidean free field''). \(\varphi\) has a stochastic integral representation \(\varphi= (-\Delta+1)^{1/2}\dot W\), which can also be written as \((-\Delta +1)\varphi=(-\Delta +1)^{1/2}\dot W\), where \(\dot W\) is an \({\mathbb R}^d\)-white noise. It is shown that under a suitable (Girsanov) transformation of the distribution of \(\varphi\) on an appropriate subspace of \(\mathcal S'\) this equation transforms into a nonlinear one: \[ (-\Delta+1)\psi+V(\psi)=(-\Delta+1)^{1/2}\dot W, \] where \(V\) is either a Wick power or a Wick exponential of \(\psi\).
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60H40
Mathematics Subject Classification ID: 81T99
zbMATH DE Number: 1888373
zbMATH Keywords: elliptic stochastic partial differential equations
zbMATH Keywords: analysis on abstract Wiener space
zbMATH Keywords: nonlinear transformations on Wiener space
zbMATH Keywords: free Euclidean field
reviewed by: Tomasz Bojdecki
MaRDI profile type: Publication
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full work available at URL: s0022-1236(02)00008-3
OpenAlex ID: W1979870532
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