Piecewise linear approximation for the dynamical $\Phi^4_3$ model

TL;DR

This paper develops a piecewise linear approximation scheme for the dynamical $\

Contribution

It introduces a novel piecewise linear approximation method for the $\

Findings

Convergence of the approximations to the true solution.

Implementation of renormalisation via time-dependent functions.

Extension of regularity structures to piecewise linear noise.

Abstract

We construct a piecewise linear approximation for the dynamical $\Phi_3^4$ model on $\mathbb{T}^3$ by the theory of regularity structures in [Hai14]. For the dynamical $\Phi^4_3$ model it is proved in [Hai14] that a renormalisation has to be performed in order to define the nonlinear term. Compared to the results in [Hai14] we consider piecewise linear approximations to space-time white noise and prove that the solutions to the approximating equations converge to the solution to the dynamical $\Phi^4_3$ model. The renormalisation in this case corresponds to adding the solution multiplied by a function depending on $t$ in the approximating equation.