Discretisations of rough stochastic PDEs
The aim of this paper is to develop a general framework for spatial discretization of the class of parabolic stochastic partial differential equations (PDEs) of the form \[ \partial_tu=Au+F(u,\xi), \] where \(A\) is an elliptic differential operator, \(\xi\) is a rough noise, and \(F\) is a nonlinear function in \(u\) which is affine in \(\xi\). In the discretization, the spatial variable takes values in the dyadic grid with mesh tending to zero. The solutions of the initial PDE are provided in the framework of the theory of regularity structures, and they are functions in time. A particular example, prototypical for the class of the equations under consideration, is the dynamics model which is called \(\Phi^4\) model in dimension 3, or \(\Phi^4_3\), that is the same. This model is considered on the torus, and operator \(A\) contains Laplace operator. It is proved that the dynamical \(\Phi^4_3\) model, discretized to the dyadic grid, converges after renormalization to its continuous counterpart. The convergence is in probability and in the norm of the functional spaces introduced in the main theorem. 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.
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