1. The Lie Bracket of Adapted Vector Fields on Wiener Spaces
- Author
-
Bruce K. Driver
- Subjects
Path (topology) ,Pure mathematics ,Control and Optimization ,Applied Mathematics ,Mathematical analysis ,Commutator (electric) ,Riemannian manifold ,Differential operator ,Measure (mathematics) ,law.invention ,Semimartingale ,law ,Lie algebra ,Vector field ,Mathematics - Abstract
Let W.M/ be the based (at o2 M/ path space of a compact Riemannian manifold M equipped with Wiener measure": This paper is devoted to considering vector fields on W.M/ of the form X h.ae/ D Ps.ae/hs.ae/ where Ps.ae/ denotes stochastic parallel translation up to time s along a Wiener path ae 2 W.M/ and fhsgs2(0;1) is an adapted To 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
- Published
- 1999
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