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. 


[![enter image description here][1]][1]

[![enter image description here][2]][2]



  [1]: https://i.sstatic.net/L8b90.png
  [2]: https://i.sstatic.net/vi8GC.png