Spectral gap for projective processes of linear SPDEs

This work studies the angular component πt=ut/‖ut‖ associated to the solution u of a vector-valued linear hyperviscous SPDE on a d-dimensional torus duα=−να(−Δ)auαdt+(u⋅dW)α,α∈{1,…,m} for u:𝕋d→ℝm, a⩾1 and a sufficiently smooth and non-degenerate noise W. We provide conditions for existence, as well as uniqueness and spectral gaps (if a>d/2) of invariant measures for π in the projective space. Our proof relies on the introduction of a novel Lyapunov functional for πt, 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 L2 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.