Approximations of dispersive PDEs in the presence of low-regularity randomness

Abstract

We introduce a new class of numerical schemes which allow for low regularity approximations to the expectation $ \mathbb{E}(|u_{k}(τ, v^η)|^2)$, where $u_k$ denotes the $k$-th Fourier coefficient of the solution $u$ of the dispersive equation and $ v^η(x) $ the associated random initial data. This quantity plays an important role in physics, in particular in the study of wave turbulence where one needs to adopt a statistical approach in order to obtain deep insight into the generic long-time behaviour of solutions to dispersive equations. Our new class of schemes is based on Wick's theorem and Feynman diagrams together with a resonance based discretisation (see arXiv:2005.01649) set in a more general context: we introduce a novel combinatorial structure called paired decorated forests which are two decorated trees whose decorations on the leaves come in pair. The character of the scheme draws its inspiration from the treatment of singular stochastic partial differential equations via Regularity Structures. In contrast to classical approaches, we do not discretize the PDE itself, but rather its expectation. This allows us to heavily exploit the optimal resonance structure and underlying gain in regularity on the finite dimensional (discrete) level.