Appl Math Optim 39:179–210 (1999)
1999 Springer-Verlag New York Inc.
The Lie Bracket of Adapted Vector Fields on
B. K. Driver
Department of Mathematics, 0112, University of California, San Diego,
La Jolla, CA 92093-0112, USA
Communicated by M. R¨ockner
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
(σ ) = P
(σ ) where P
(σ ) denotes
stochastic parallel translation up to time s along a Wiener path σ ∈ W (M) and
is an adapted T
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
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
is again a vector
ﬁeld on W (M) of the same form.
Key Words. Wiener measure, Itˆo development map, Lie bracket, Integration by
AMS Classiﬁcation. Primary 60H07, 60D05, Secondary 58D15.
,·,·,∇, o) be given, where M is a compact connected manifold (without
boundary) of dimension d, ·,· is a Riemannian metric on M, ∇ is a ·,·-compatible
covariant derivative, and o is a ﬁxed base point in M. Let T = T
and R = R
the torsion and curvature of ∇, respectively.
This research was partially supported by NSF Grant No. DMS 92-23177.