From the heat measure to the pinned Wiener measure on loop groups

Abstract

For each L p -Wasserstein distance ( p > 1 ) with the cost function induced by the L 2 -distance on loop groups, we show that there exists a unique optimal transport map solving the Monge–Kantorovich problem when the initial probability measure is absolutely continuous with respect to the heat kernel measure. In particular, this provides us a family of measurable maps on loop groups which push the heat kernel measure forward to the pinned Wiener measure.