Piecewise linear approximation for the dynamical $$\Phi_3^4$$Φ34 model

Abstract

We construct a piecewise linear approximation for the dynamical $$\Phi_3^4$$ model on $$\mathbb{T}^3$$. The approximation is based on the theory of regularity structures developed by Hairer (2014). They proved that renormalization in a dynamical $$\Phi_3^4$$ model is necessary for defining the nonlinear term. In contrast to Hairer (2014), we apply piecewise linear approximations to space-time white noise, and prove that the solutions of the approximating equations converge to the solution of the dynamical $$\Phi_3^4$$ model. In this case, the renormalization corresponds to multiplying the solution by a t-dependent function, and adding it to the approximating equation.