Nonlinear transformations of the canonical Gauss measure on Hilbert space and absolute continuity
type: Item
instance of: scholarly article
title: Nonlinear transformations of the canonical Gauss measure on Hilbert space and absolute continuity
zbMATH Open document ID: 0803.60050
DOI: 10.1007/BF00994912
author: Rajeeva L. Karandikar
author: Gopinath Kallianpur
published in: Acta Applicandae Mathematicae
publication date: 1995-01-03T00:00:00Z
review text: Let \((H, B, \mu)\) be an abstract Wiener space. The present paper investigates the absolute continuity of the cylindrical measure induced by the nonlinear transformation \(T(h)= h+ F(h)\), \(h\in H\), of the canonical Gauss measure on \(H\). The result is stated in terms of absolute continuity of \(\mu\circ \widetilde {T}^{-1}\) w.r.t. \(\mu\), where \(\widetilde {T}\omega= \omega+ \widetilde{F} (\omega)\), \(\omega\in B\), denotes the lifting of \(T\) onto \(B\). If, under a weak Gateaux- differentiability assumption on \(F\), for some increasing sequence of finite-dimensional orthogonal projectors \(P_ n\) acting on \(H\), \(P_ n\uparrow I\), the transformations \(T_ n(h)= h+ P_ n F(P_ n k)\), \(h\in H\), are bijective, all the corresponding Carleman-Fredholm determinants are everywhere different from zero and the densities of \(\widetilde{T}_ n\) w.r.t. \(\mu\) form a uniformly integrable family, then \(\mu_ 0 \widetilde{T}^{-1} \ll\mu\) and the corresponding density is given by Kusuoka's formula. In the sequel the authors study the stochastic integral occuring in this formula in some detail and provide sufficient conditions for the uniform integrability. The requirement of the existence of a sequence \(T_ n\) of 1-1 transformations approximating \(T\) corresponds to the assumption of \textit{D. Nualart} and \textit{M. Zakai} [Probab. Theory Relat. Fields 73, 255-280 (1986; Zbl 0601.60053)] that \(\widetilde {T}\) can be approximated by a sequence of smooth finite-dimensional 1-1 transformations \(\widetilde {T}_ n\). But the problem which continues to remain open is the presentation of general assumptions which guarantee the existence of such an approximating sequence.
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60B11
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 28C20
Mathematics Subject Classification ID: 46G12
zbMATH DE Number: 627452
zbMATH Keywords: nonlinear transformation of Wiener measure
zbMATH Keywords: liftings
zbMATH Keywords: Gokhberg-Krejn factorization
zbMATH Keywords: Ramer-Itô integral
zbMATH Keywords: abstract Wiener space
zbMATH Keywords: absolute continuity
zbMATH Keywords: cylindrical measure
zbMATH Keywords: canonical Gauss measure
zbMATH Keywords: weak Gateaux- differentiability
zbMATH Keywords: Carleman-Fredholm determinants
zbMATH Keywords: Kusuoka's formula
zbMATH Keywords: uniform integrability
reviewed by: Rainer Buckdahn
MaRDI profile type: Publication
cites work: Anticipative Girsanov transformations
cites work: The Transformation of Wiener Integrals by Nonlinear Transformations
cites work: Q3957682
cites work: Equivalence and perpendicularity of Gaussian processes
cites work: Q5571194
cites work: Q5590313
cites work: Integration and Nonlinear Transformations In Hilbert Space
cites work: Homogeneous Chaos, p-Forms, Scaling and the Feynman Integral
cites work: An RKHS approach to detection and estimation problems-- III: Generalized innovations representations and a likelihood-ratio formula
cites work: Non-anticipative representations of equivalent Gaussian processes
cites work: Q3908216
cites work: Q3999268
cites work: Q3038308
cites work: Q3973616
cites work: On nonlinear transformations of Gaussian measures
cites work: On the structure of independence on Wiener space
full work available at URL: https://doi.org/10.1007/bf00994912
OpenAlex ID: W2056501228
description: scientific article; zbMATH DE number 627452