Probability Theory and Related Fields
, Volume 103, Issue 3, pp 409429
First online:
Random rotations of the Wiener path
 A. S. ÜstünelAffiliated withDépartement Réseaux, ENSTDepartment of Electrical Engineering, TechnionIsrael Institute of Technology
 , M. ZakaiAffiliated withDépartement Réseaux, ENSTDepartment of Electrical Engineering, TechnionIsrael Institute of Technology
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Let (W, H, μ) be an abstract Wiener space and letR(w) be a strongly measurable random variable with values in the set of isometries onH. Suppose that ∇Rh is smooth in the Sobolev sense and that it is a quasinilpotent operator onH for everyh∈H. It is shown that δ(R(w)h) is again a Gaussian (0, h _{ H } ^{2} )random variable. Consequently, if (e _{ i },i∈ℕ)⊂W ^{*} is a complete, orthonormal basis ofH, then\(\tilde w = \sum\nolimits_i {(\delta R(w)e_i )e_i } \) defines a measure preserving transformation, a “rotation”, onW. It is also shown that if for some strongly measurable, operator valued (onH) random variableR, δ(R(w+k)h) is (0, h _{ H } ^{2} )Gaussian for allk, h∈H, thenR is an isometry and ∇Rh is quasinilpotent for allH∈H. The relation between the stochastic calculi for these Wiener pathsw and\(\tilde w\), as well as the conditions of the inverbibility of the map\(w \to \tilde w\) are discussed and the problem of the absolute continuity of the image of the Wiener measure μ under Euclidean motion on the Wiener space (i.e.\(w \to \tilde w\) composed with a shift) is studied.
Mathematics Subject Classification
60G30 60H07 Title
 Random rotations of the Wiener path
 Journal

Probability Theory and Related Fields
Volume 103, Issue 3 , pp 409429
 Cover Date
 199509
 DOI
 10.1007/BF01195481
 Print ISSN
 01788051
 Online ISSN
 14322064
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 60G30
 60H07
 Industry Sectors
 Authors

 A. S. Üstünel ^{(1)} ^{(2)}
 M. Zakai ^{(1)} ^{(2)}
 Author Affiliations

 1. Département Réseaux, ENST, 46, rue Barrault, F75013, Paris, France
 2. Department of Electrical Engineering, TechnionIsrael Institute of Technology, 32000, Haifa, Israel