Spectral gap for projective processes of linear SPDEs

Abstract

This work studies the angular component ${\pi }_t=u_t/\Vert u_t\Vert$ associated to the solution $u$ of a vector-valued linear hyperviscous SPDE on a $d$-dimensional torus $$\mathrm{d}{u}^{\alpha }=-{\nu }^{\alpha }(-\Delta {)}^{\mathbf{a}}{u}^{\alpha }\mathrm{d}t+(u\cdot \mathrm{d}W{)}^{\alpha },\alpha \in \{1,\dots ,m\}$$ for $u:{\mathbb{T}}^d\to {\mathbb{R}}^m$, $\mathbf{a}⩾1$ 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.

73 pages, 2 figures, revised high-frequency stochastic instability section