Lyapunov exponents in a slow environment

Motivated by the evolution of a population in a slowly varying random environment, we consider the 1D Anderson model on finite volume, with viscosity κ>0: (0.1)∂tu(t,x)=κΔu(t,x)+ξ(t,x)u(t,x),u(0,x)=u0(x),t>0,x∈T.The noise ξ is chosen constant on time intervals of length τ>0 and sampled independently after a time τ. We prove that the Lyapunov exponent λ(τ) is positive and near τ=0 follows a power law that depends on the regularity on the driving noise. As τ→∞ the Lyapunov exponent converges to the average top eigenvalue of the associated time-independent Anderson model. The proofs make use of a solid control of the projective component of the solution and build on the Furstenberg–Khasminskii and Boué–Dupuis formulas, as well as on Doob’s H-transform and on tools from singular stochastic PDEs.