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The Lie Bracket of Adapted Vector Fields on Wiener Spaces
(UCSD Preprint, September 7, 1995.)
Let $W(M)$ be the based (at $o\in M)$ path space of a compact
Riemannian manifold $M$ equipped with Wiener measure $\nu .$ This paper
is devoted to considering vector fields on $W(M)$ of the form $X_s^h(\sigma )=P_s(\sigma
)h_s(\sigma )$ where $P_s(\sigma )$ denotes
stochastic parallel translation
up to time $s$ along a Wiener path $\sigma\in W(M)$ and $\{h_s\}_{
s\in [0,1]}$ is an
adapted $T_oM$-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.
The manuscript is available as a
LaTeX file (93K)
and as a DVI file (132K).
August 8, 1996
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