Applied Mathematics and Optimization

, Volume 39, Issue 2, pp 179–210 | Cite as

The Lie Bracket of Adapted Vector Fields on Wiener Spaces

  • B. K. Driver


Let W(M) be the based (at o∈ M) path space of a compact Riemannian manifold M equipped with Wiener measure ν . This paper is devoted to considering vector fields on W(M) of the form X s h ( σ )=P s ( σ )h s ( σ ) where P s ( σ ) denotes stochastic parallel translation up to time s along a Wiener path σ ∈ W(M) and {h s } s∈ [0,1] is an adapted T o M -valued process on W(M). It is shown that there is a large class of processes h (called adapted vector fields) for which we may view X h as first-order differential operators acting on functions on W(M) . Moreover, if h and k are two such processes, then the commutator of X h with X k is again a vector field on W(M) of the same form.

Key words. Wiener measure, Itô development map, Lie bracket, Integration by parts. AMS Classification. Primary 60H07, 60D05, Secondary 58D15. 


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Copyright information

© Springer-Verlag New York Inc. 1999

Authors and Affiliations

  • B. K. Driver
    • 1
  1. 1.Department of Mathematics, 0112, University of California, San Diego, La Jolla, CA 92093-0112, USA US

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