The logarithmic Sobolev inequality for the Wasserstein diffusion

We prove that the Dirichlet form associated with the Wasserstein diffusion on the set of all probability measures on the unit interval, introduced in von Renesse and Sturm (Entropic measure and Wasserstein diffusion. Ann Probab, 2008) satisfies a logarithmic Sobolev inequality. This implies hypercontractivity of the associated transition semigroup. We also study functional inequalities for related diffusion processes.