We prove weak Poincaré inequalities on domains which are inverse images of open sets in Wiener spaces under continuous functions of Brownian rough paths. The result is applicable to Dirichlet forms on loop groups and connected open subsets of path spaces over compact Riemannian manifolds.
"Weak Poincaré inequalities on domains defined by Brownian rough paths." Ann. Probab. 32 (4) 3116 - 3137, October 2004. https://doi.org/10.1214/009117904000000478