SPECTRAL GAP FOR PROJECTIVE PROCESSES OF LINEAR SPDES

The goal of this work is to initiate the study of lower bounds for Lyapunov exponents of stochastic partial differential equations(SPDEs).To this end, we consider as a toy model the angular component πt=ut/|| ut|| associated to the solution u of a vector-valued linear hyperviscous SPDE on ad-dimensional torus (formula Presented) for u∶Td→Rm,a⩾1 and a sufficiently smooth and non-degeneratenoise W. We provide conditions for existence, as well as uniqueness and spectral gaps (if a>d/2) of in variant 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.