Discretisations of rough stochastic PDEs

We develop a general framework for spatial discretisations of parabolic stochastic PDEs whose solutions are provided in the framework of the theory of regularity structures and which are functions in time. As an application, we show that the dynamical ${\mathrm{\Phi }}_3^4$ model on the dyadic grid converges after renormalisation to its continuous counterpart. This result in particular implies that, as expected, the ${\mathrm{\Phi }}_3^4$ measure with a sufficiently small coupling constant is invariant for this equation and that the lifetime of its solutions is almost surely infinite for almost every initial condition.