  Let $\mathcal P_2$ be the space of probability measures on $\R^d$ having
finite second moment, and consider the Riemannian structure on $\mathcal P_2$
induced by the intrinsic derivative on the $L^2$-tangent space. By using
stochastic analysis on the tangent space, we construct an Ornstein$-$Uhlenbeck
(OU) type Dirichlet form on $\mathcal P_2$ whose generator is formally given by
the intrinsic Laplacian with a drift. The log-Sobolev inequality holds and the
associated Markov semigroup is $L^2$-compact.
  Perturbations of the OU Dirichlet form are also studied.
