The Lie Bracket of Adapted Vector Fields on Wiener Spaces

The Lie Bracket of Adapted Vector Fields on Wiener Spaces 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

The Lie Bracket of Adapted Vector Fields on Wiener Spaces

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Publisher
Springer-Verlag
Copyright
Copyright © Inc. by 1999 Springer-Verlag New York
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s002459900103
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Apr 1, 2024

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