Discretisations of rough stochastic PDEs

TL;DR

This paper introduces a framework for discretising parabolic stochastic PDEs within the regularity structures theory, demonstrating convergence of the dynamical _3 model on dyadic grids and confirming invariance of the _3 measure.

Contribution

It provides a novel general framework for spatial discretisations of rough stochastic PDEs and proves convergence of the _3 model on dyadic grids after renormalisation.

Findings

The _3 model on dyadic grids converges to the continuous model after renormalisation.

The _3 measure with small coupling constant is invariant under the discretised dynamics.

Solutions to the discretised _3 model have almost surely infinite lifetime for almost every initial condition.

Abstract

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 $\Phi^4_3$ model on the dyadic grid converges after renormalisation to its continuous counterpart. This result in particular implies that, as expected, the $\Phi^4_3$ 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.