Indeed as mentioned in the comments, there is a natural analogue whose Dirichlet form is in terms of a Gaussian kernel eg. see section 3 in "Ornstein-Uhlenbeck Type Processes on Wasserstein Space"

See also the reference G. Da Prato, J. Zabaczyk, Ergodicity for Infinite-Dimensional Systems at 5.2.9 and 6.2.1.

Indeed as mentioned in the comments, there is a natural analogue whose Dirichlet form is in terms of a Gaussian kernel eg. see sections 3,4 in "Ornstein-Uhlenbeck Type Processes on Wasserstein Space"

First they construct the generator and Dirichlet form $dG_{Q}$ (for covariance kernel $Q$) on the tangent space. And then we can use the exponential map sending $Exp: T_{x}\to M$ to precompose to create a Dirichlet form measure $dN_{Q}=dG_{Q}\circ Exp^{-1}$ on the manifold.