Spectral gap for projective processes of linear SPDEs

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.