We consider the coadjoint action of a Loop group of a compact group on the dual of the corresponding centrally extended Loop algebra and prove that a Brownian motion in a Cartan subalgebra conditioned to remain in an affine Weyl chamber—which can be seen as a space time conditioned Brownian motion—is distributed as the radial part process of a Brownian sheet on the underlying Lie algebra.
"Kirillov–Frenkel character formula for loop groups, radial part and Brownian sheet." Ann. Probab. 47 (2) 1036 - 1055, March 2019. https://doi.org/10.1214/18-AOP1278