Local Poincare inequalities on loop spaces

Let M be a compact connected Riemannian manifold, and fix x, y is an element of M. For a sufficiently small constant R > 0, Poincare inequalities w.r.t. pinned Wiener measure with time parameter T > 0 are proven on the sets Omega(x,y)(R,N),N is an element of N, consisting of all continuous paths omega : [0,1] --> M such that w(0) = x, w(l) = y, and d(w(s), w(t)) < R if s, t is an element of [(i - 1)/N, i/N] for some integer i. Moreover, the asymptotic behaviour of the best constants in the Poincare inequalities as T goes to 0 is studied. It turns out that the asymptotic depends crucially on the Riemannian metric on M and, in particular, on the geodesics contained in Omega(x,y)(R,N). Key ingredients in the proofs are a bisection argument, estimates for finite-dimensional spectral gaps, and a crucial variance estimate by Malliavin and Stroock. (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.