Lattice approximation to the dynamical $\Phi_{3}^{4}$ model

Abstract

We study the lattice approximations to the dynamical $\Phi^{4}_{3}$ model by paracontrolled distributions proposed in [Forum Math. Pi 3 (2015) e6]. We prove that the solutions to the lattice systems converge to the solution to the dynamical $\Phi_{3}^{4}$ model in probability, locally uniformly in time. Since the dynamical $\Phi_{3}^{4}$ model is not well defined in the classical sense and renormalisation has to be performed in order to define the nonlinear term, a corresponding suitable drift term is added in the stochastic equations for the lattice systems.