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We study the Glauber dynamics of a two dimensional Blume-Capel model (ordilute Ising model) with Kac potential parametrized by $(\beta,\theta)$ - the"inverse temperature" and the "chemical potential". We prove that the locallyaveraged spin field rescales to the solution of the dynamical $\Phi^4$ equationnear a curve in the $(\beta,\theta)$ plane and to the solution of the dynamical$\Phi^6$ equation near one point on this curve. Our proof relies on a discreteimplementation of Da Prato-Debussche method as in a result by Mourrat-Weber butan additional coupling argument is needed to show convergence of the linearizeddynamics.

Glauber dynamics of 2D Kac-Blume-Capel model and their stochastic PDE limits