Anticipating flows on the Wiener space generated by vector fields of low regularity (Q2563445)

An abstract Wiener space is a triple \((\mu,H,B)\) consisting of a Gaussian measure \(\mu\) on a separable Banach space \(B\) with covariance given by the inner product on the Hilbert space \(H\). The \(H\)-derivative of a function \(f\) on \(B\) is the Gateaux derivative of \(f\) in the \(H\)-directions. The author first defines Sobolev spaces for Banach-space-valued Wiener functionals using the operator norm for the \(H\)-derivatives instead of the Hilbert-Schmidt norm. The \(H\)-derivative is constructed differently from that used in the Malliavin calculus. He constructs finite-dimensional approximations of a Wiener functional in one of these \(H\)-Sobolev spaces. Next the author considers the question of extending a one-parameter group \(Q_t\) of unitary operators on \(H\) to a one-parameter group \(\widetilde{Q}_t\) of measure-preserving transformations on \(B\), and obtains a sufficient condition for this. The main results of the paper are generalizations of the results of \textit{A. B. Cruzeiro} [J. Funct. Anal. 54, 206-227 (1983; Zbl 0524.47028); ibid. 58, 335-347 (1984; Zbl 0551.47019)] concerning the existence and uniqueness of flows associated to vector fields on Sobolev spaces in the sense of the Malliavin calculus that are strongly admissible in the sense of the Cameron-Martin theorem for abstract Wiener space. The author's generalizations allow the vector field to be the sum of a possibly unbounded skew-adjoint operator and a nonlinear part whose regularity is measured by the new \(H\)-Sobolev norms. He also has weaker exponential conditions and is able to prove the semigroup property for the flows as well as being able to omit the regularity conditions from the uniqueness result. The paper contains a substantial amount of background material.