Let K be a connected Lie group of compact type and let W(K) denote the
set of continuous paths in K, starting al the identity and with time-
interval [0, 1]. Then W(K) forms an infinite-dimensional group under t
he operation of pointwise multiplication. Let rho denote the Wiener me
asure on W(K). We construct an analog of the Segal-Bargmann transform
for W(K). Let K-C be the complexification of K, W(K-C) the set of cont
inuous paths in K-C starting at the identity, and mu the Wiener measur
e on W(K-C). Our transform is a unitary map of L-2(W(K), rho) onto the
''holomorphic'' subspace of L-2(W(K-C), mu). By analogy with the clas
sical transform, our transform is given by convolution with the Wiener
measure, followed by analytic continuation. We prove that the transfo
rm for W(K) is nicely related by means of the Ito map to the classical
Segal-Bargmann transform for the path-space in the Lie algebra of K.
(C) 1998 Academic Press.