\(H\)-\(C^1\) maps and elliptic SPDEs with polynomial and exponential perturbations of Nelson's Euclidean free field
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 \(\psi\).
elliptic stochastic partial differential equations
nonlinear transformations on Wiener space
scientific article; zbMATH DE number 1888373
scientific article