Logarithmic Sobolev inequalities on loop groups

Abstract

Let G be a compact Lie group. Denote by m the Brownian bridge measure on the loop group Y ≡ { g ϵ C([0, 1]; G): g(0) = g(1) = e }. The finite energy subgroup of Y determines in a natural way a gradient operation for functions on Y. The following logarithmic Sobolev inequality is proven, ∝ f 2, log ¦f¦dm ⩽ ∝ {¦ gradf(y)¦ 2 + V(y) f (y) 2} dm + ∥f∥ 2 log∥f∥ wherein ∥f∥ denotes the L 2( m) norm and V is a potential which is quadratic in the associated Lie algebra valued Brownian motion. The inequality is derived by a method of inheritance from the known inequality for the G valued Brownian motion.