The paper analyzes a model in surface growth where the uniqueness of weak solutions seems to be out of reach. We prove existence of a weak martingale solution satisfying energy inequalities and having the Markov property. Furthermore, under nondegeneracy conditions on the noise, we establish that any such solution is strong Feller and has a unique invariant measure.
"Markovianity and ergodicity for a surface growth PDE." Ann. Probab. 37 (1) 275 - 313, January 2009. https://doi.org/10.1214/08-AOP403