Zeta function regularized Laplacian on the smooth Wasserstein space above the unit circle

Via elements of second order differential geometry on smooth Wasserstein spaces of probability measures we give an explicit formula for a Laplacian in the case that the Wasserstein space is based on the unit circle. The Laplacian on this infinite dimensional manifold is calculated as trace of the Hessian in the sense of Zeta function regularization. Its square field operator is the square norm of the Wasserstein gradient.