Random fields as generalized white noise functionals
Consider a generalized white noise functional, i.e., a vector field \(\{X(C),\;C \in \mathbb{C}\}\) with \(\mathbb{C} = \{C \mid C =\) simple, closed, convex manifold diffeomorphic to the sphere \(S^{d-1}\}\). It is assumed that \(X(C) \in S^* \supset L^ 2 \supset S\), where \(L^ 2 =\) Hilbert space of ordinary white noise functionals while \(S,S^*\) denote a test functional space and its dual, respectively. Let \(\mu_ C\) be a white noise measure on \(E(C)^* \supset L^ 2 (C,d \sigma) \supset E(C) =\) nuclear space induced by a Gaussian measure \(\mu\) on \(R^ d\) with the characteristic functional \(\exp (-{1 \over 2} \| \xi \|^ 2)\). The author's main objective is to describe a stochastic variational equation for generalized white noise functionals of the form \[ X(C) = \Phi \Bigl( \int_ CF \bigl( C,x(u),u \bigr) du \Bigr),\quad x \in E^* (\mu). \] For a class of quantum-mechanical, Schrödinger type, operators \(A\) (= polynomials of degree \(\leq 2\) in \(\partial_ t\), \(\partial^*_ t\), \(t \in R^ d)\) \(X(C)\) was found to be a solution to the variational equation \({\delta X (C) \over \delta C} (\rho) = AX(C)\), \(\rho \in C\), which is seen as an extension of an analogous classical result from the theory of diffusion processes \(\{X(t),\;t \geq 0\}\).
convex manifold diffeomorphic to the sphere
nuclear space induced by a Gaussian measure
scientific article; zbMATH DE number 627451