Discretisations of rough stochastic PDEs
Provides a rigorous discretization framework for rough stochastic PDEs, with implications for the Φ^4_3 model in mathematical physics.
The paper develops a general framework for spatial discretizations of parabolic stochastic PDEs and proves that the dynamical Φ^4_3 model on the dyadic grid converges after renormalization to its continuous counterpart, implying invariance of the Φ^4_3 measure and almost surely infinite solution lifetime.
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 $Φ^4_3$ model on the dyadic grid converges after renormalisation to its continuous counterpart. This result in particular implies that, as expected, the $Φ^4_3$ 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.