Vanishing of one-dimensional L2-cohomologies of loop groups

Abstract

Let G be a simply connected compact Lie group. Let L e (G) be the based loop group with the base point e which is the identity element. Let ν e be the pinned Brownian motion measure on L e (G) and let α ∈ L 2 ( 1 T * L e (G), ν e ) ∩ D ∞,p ( 1 T * L e (G), ν e ) (1 < p < 2) be a closed 1-form on L e (G). Using results in rough path analysis, we prove that there exists a measurable function f on L e (G) such that df = α. Moreover we prove that dim ker = 0 for the Hodge-Kodaira type operator acting on 1-forms on L e (G).

Asymptotics of spectral gaps on loop spaces over a class of Riemannian manifolds

“…Here 2/3 < θ < 1 and m is a sufficiently large positive number. It is proved in [8] that Ω c is a slim set in the sense of Malliavin with respect to the Brownian motion measure µ. However, it is easy to check that the same result holds for the Brownian motion measure µ λ with variance 1/λ for any λ > 0.…”

“…Let Ω be all elements b belonging to the Wiener space W n such that b(N ) 1 s,t and b(N ) 2 s,t converge in the Besov type norm • 4m,2θ and • 2m,θ respectively ( [8]). Here 2/3 < θ < 1 and m is a sufficiently large positive number.…”

“…However, the probability distribution of b does not charge the slim sets in the sense of Malliavin. Hence, we need to consider Brownian rough paths for all Brownian paths except a slim set ( [8]). After preparation of necessary estimates from rough paths (see Lemma 5.3), we prove Theorem 3.2.…”

Large deviations for small noise hypoelliptic diffusion bridges on sub-Riemannian manifolds

“…Take κ > 0 so small that n j=1 [a j − κ, a j + κ] ⊂ U holds. For this κ, we can find a sequence {λ k } ∞ k=1 of smooth and non-decreasing function on R such that (1)…”

De Rham–Hodge decomposition and vanishing of harmonic forms by derivation operators on the Poisson space

“…The Weitzenböck formula can be used to prove the vanishing of harmonic differential forms, cf. [1] in the case of loop groups.…”