We consider one dimensional, attractive stochastic Ising models with finite range interactions in which the particles move according to the stirring process. We prove that the process is exponentially ergodic provided the migration rate is sufficiently small.
Claudia Neuhauser. "One Dimensional Stochastic Ising Models with Small Migration." Ann. Probab. 18 (4) 1539 - 1546, October, 1990. https://doi.org/10.1214/aop/1176990630