Spectral gap for projective processes of linear SPDEs

TL;DR

This paper investigates the spectral gap and invariant measures of the angular component of solutions to linear hyperviscous SPDEs on a torus, introducing a novel Lyapunov functional based on the energy median to analyze dynamics.

Contribution

It introduces a new Lyapunov functional for the angular component of linear SPDEs, providing conditions for existence, uniqueness, and spectral gaps of invariant measures in the projective space.

Findings

Established existence and uniqueness of invariant measures under certain conditions.

Proved spectral gaps for the angular component when hyperviscosity exceeds a threshold.

Derived lower bounds on top Lyapunov exponents using the energy median technique.

Abstract

This work studies the angular component $ \pi_{t} = u_{t} / \| u_{t} \| $ associated to the solution $ u $ of a vector-valued linear hyperviscous SPDE on a $d$-dimensional torus $$\mathrm{d} u^{\alpha} =- \nu^{\alpha} (- \Delta)^{\mathbf{a} } u^{\alpha} \mathrm{d} t + (u \cdot \mathrm{d} W)^{\alpha} \;,\quad \alpha \in \{ 1, \dots, m \} $$ for $ u \colon \mathbb{T}^{d} \to \mathbb{R}^{m} $, $ \mathbf{a} \geqslant 1 $ and a sufficiently smooth and non-degenerate noise $ W $. We provide conditions for existence, as well as uniqueness and spectral gaps (if $ \mathbf{a} > d/2$) of invariant measures for $ \pi $ in the projective space. Our proof relies on the introduction of a novel Lyapunov functional for $\pi_{t}$, based on the study of dynamics of the ``energy median'': the energy level $M$ at which projections of $u$ onto frequencies with energies less or more than $M$ have about equal $L^2$ norm. This technique is applied to obtain -- in an infinite-dimensional setting without order preservation -- lower bounds on top Lyapunov exponents of the equation, and their uniqueness via Furstenberg-Khasminskii formulas.