Publication
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.