Summary
One-dimensional stochastic Ising systems with a local mean field interaction (Kac potential) are investigated. It is shown that near the critical temperature of the equilibrium (Gibbs) distribution the time dependent process admits a scaling limit given by a nonlinear stochastic PDE. The initial conditions of this approximation theorem are then verified for equilibrium states when the temperature goes to its critical value in a suitable way. Earlier results of Bertini-Presutti-Rüdiger-Saada are improved, the proof is based on an energy inequality obtained by coupling the Glauber dynamics to its voter type, linear approximation.
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Fritz, J., Rüdiger, B. Time dependent critical fluctuations of a one dimensional local mean field model. Probab. Th. Rel. Fields 103, 381–407 (1995). https://doi.org/10.1007/BF01195480
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DOI: https://doi.org/10.1007/BF01195480