Let G be a compact connected
Lie group with identity element e, and let PeG denote the space of continuous maps
y : [0,1] → G such that y(0) = e. When equipped with the natural group
structure and sup metric, PeG becomes an interesting example of an infinite
dimensional nonlinear topological group. The purpose of this paper is to
consider certain aspects of analysis on PeG. Stimulated by a theorem of M.
Malliavin and P. Malliavin, we prove the existence of a natural Brownian
motion on PeG which depends only on a choice of bi-invariant metric for G.
Our main results, however, concern the heat semigroup associated to the
Brownian motion on PeG. We identify the action of the generator of this
semigroup when applied to certain highly regular functions, with a result similar
to that obtained earlier by L. Gross in the (linear) abstract Wiener space
context.
