Article

Probability Theory and Related Fields

, Volume 103, Issue 3, pp 409-429

First online:

Random rotations of the Wiener path

  • A. S. ÜstünelAffiliated withDépartement Réseaux, ENSTDepartment of Electrical Engineering, Technion-Israel Institute of Technology
  • , M. ZakaiAffiliated withDépartement Réseaux, ENSTDepartment of Electrical Engineering, Technion-Israel Institute of Technology

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Summary

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 quasi-nilpotent operator onH for everyhH. 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, hH, thenR is an isometry and ∇Rh is quasi-nilpotent for allHH. 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