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.
