Transportation cost inequalities on path and loop groups

Abstract

Let G be a connected Lie group with the Lie algebra G . The action of Cameron–Martin space H( G) on the path space P e ( G) introduced by L. Gross (Illinois J. Math. 36 (1992) 447) is free. Using this fact, we define the H-distance on P e ( G), which enables us to establish a transportation cost inequality on P e ( G). This method will be generalized to the path space over the loop group L e(G) , so that we obtain a transportation cost inequality for heat measures on L e(G) .