We study an infinite white $\alpha$-stable systems with unbounded interactions, and prove the existence of a solution by Galerkin approximation and an exponential mixing property by an $\alpha$-stable version of gradient bounds.
"Existence and Exponential Mixing of Infinite White $\alpha$-Stable Systems with Unbounded Interactions." Electron. J. Probab. 15 1994 - 2018, 2010. https://doi.org/10.1214/EJP.v15-831