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. 

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

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