### 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

 08/15/2016 02:53 PM