The invariant measures with finite first moment of the smoothing process are characterized in terms of the harmonic functions. It is also shown that under some restriction on the initial configuration, the difference between the mass at a given site and the mean of the masses of its neighbours converges in probability to zero.
Enrique D. Andjel. "Invariant Measures and Long Time Behaviour of the Smoothing Process." Ann. Probab. 13 (1) 62 - 71, February, 1985. https://doi.org/10.1214/aop/1176993066