Invertibility of adapted perturbations of the identity on abstract Wiener space
This paper is an extension of the results published in [\textit{A. S. Üstünel}, J. Funct. Anal. 257, No. 11, 3655--3689 (2009; Zbl 1196.60104)] to the context of abstract Wiener spaces \((W, H, \mu)\) which satisfy the assumption that there exists a continuous and increasing sequence of projections on \(H\). This increasing resolution of the identity on \(H\) was used also in [\textit{A. S. Üstünel} and \textit{M. Zakai}, J. Funct. Anal. 143, No. 1, 10--32 (1997; Zbl 0873.60039)] to define the arrow of time, since the abstract Wiener space, unlike the classical Wiener space, does not possess a natural notion of time. The invertibility of an adapted perturbation of the identity is related to to the equality between the energy of the signal and the relative entropy of the measure it induces. This result is also extended to a probability measure which is absolutely continuous with respect to the Wiener measure and has finite entropy. The author also discusses the relation between the adapted perturbations of the identity problem, the Monge problem and the innovation conjecture.
scientific article; zbMATH DE number 6017629
scientific article