On Cameron-Martin Theorem and almost sure global existence

TL;DR

This paper explores the use of the Cameron-Martin theorem to establish almost sure global existence of solutions for certain nonlinear Hamiltonian PDEs with initial data combining smooth functions and rough random perturbations.

Contribution

It demonstrates how Cameron-Martin theorem can be applied to prove almost sure global existence for Hamiltonian PDEs with mixed deterministic and random initial data.

Findings

Almost sure global existence for specific Hamiltonian PDEs.

Application of Cameron-Martin theorem to PDEs with rough random perturbations.

Extension of known results with Gaussian measures.

Abstract

In this note, we discuss various aspects of invariant measures for nonlinear Hamiltonian PDEs. In particular, we show almost sure global existence for some Hamiltonian PDEs with initial data of the form: "smooth deterministic function + a rough random perturbation", as a corollary to Cameron-Martin Theorem and known almost sure global existence results with respect to Gaussian measures on spaces of functions.