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 π t = u t / ‖ u t ‖ π ₜ = uₜ / \| uₜ \| associated to the solution u u of a vector-valued linear hyperviscous SPDE on a d d -dimensional torus d u α = − ν α ( − Δ ) a u α d t + ( u ⋅ d W ) α , α ∈ { 1 , … , m } {equation*} \,duα =- ν α (- Δ )^{{ a} } uα \,dt + (u · \,dW)α \;, α ∈ \{ 1, , m \} {equation*} for u : T