Long–time behaviour of Hamiltonian PDEs driven by an additive one dimensional standard Brownian motion and their numerical discretisations

Abstract

This thesis investigates the long–time behaviour of Hamiltonian partial differential equations driven by an additive one dimensional standard Brownain motion. The study aims to understand how random fluctuations affect fundamental properties, mostly in classical systems, analytically and numerically. Deterministic Hamiltonian ODEs and PDEs are first examined, after which stochastic perturbations are introduced and analyzed using stochastic differential equations, Itô calculus and numerical methods. A numerical integrator called Strang–AVF is derived, which preserves the long–time behaviour of the expected value of the Hamiltonian in stochastic systems.