On the uniqueness of Markovian self-adjoint extension of diffusion operators on infinite dimensional spaces
Let (H,B,\(\mu)\) be an abstract Wiener space, \(\{e_ i\}^{\infty}_{i=1}\subset B^*\) be a complete orthonormal basis of H and \(FC_ 0^{\infty}=\{f:\) \(f(x)=\tilde f(<e_{i_ 1},x>,...,<e_{i_ n},x>)\), \(x\in B\), \(\tilde f\in C_ 0^{\infty}(R^ n)\), \(n\geq 1\}\). Sufficient conditions are found for a positive function \(\rho\) (x), \(x\in B\), such that the Friedrichs extension of the symmetric operator \[ Su=2^{-1}{\mathcal L}u+\rho^{- 1}<D\rho,Du>_ H,\quad u\in FC_ 0^{\infty}, \] is the unique self- adjoint extension which generates a strongly continuous Markov semi-group on \(L_ 2(\rho^ 2\mu)\), where \({\mathcal L}\) is the Ornstein-Uhlenbeck generator.
scientific article; zbMATH DE number 3988412