James Coe | Searcharxiv

arXiv2025

Quasi-Gaussianity of the 2D stochastic Navier-Stokes equations

We study the qualitative properties of solutions to the 2D stochastic Navier-Stokes equations with forcing that is white in time and coloured in space. Our main result shows that the unique invariant measure of this system is equivalent to that of the corresponding Ornstein-Uhlenbeck process. Our method relies on a generalization of the "time-shifted Girsanov method" of [MS05, MRS22] to compare the laws of time marginals for dissipative SPDEs. This generalisation allows to not only compare solutions to a nonlinear equation to those of the corresponding linear equation, but also to directly compare two nonlinear equations. We use this to establish equivalence of the Navier-Stokes system to a "twisted" nonlinear system that leaves the Gaussian measure invariant. We further apply this method to establish similar equivalence statements for a family of hypoviscous Navier-Stokes equations.

math.PR↗

arXiv2024

Sharp quasi-invariance threshold for the cubic Szeg\H{o} equation

We consider the 1-dimensional cubic Szeg\H{o} equation with data distributed according to the Gaussian measure with inverse covariance operator $(1-\partial_x^2)^\frac s2$, where $s>\frac12$. We show that, for $s>1$, this measure is quasi-invariant under the flow of the equation, while for $s<1$, $s\neq \frac34$, the transported measure and the initial Gaussian measure are mutually singular for almost every time. This is the first observation of a transition from quasi-invariance to singularity in the context of the transport of Gaussian measures under the flow of Hamiltonian PDEs.

math.AP↗