Lyapunov exponents in a slow environment

Abstract

Motivated by the evolution of a population in a slowly varying random environment, we consider the 1D Anderson model on finite volume, with viscosity $\kappa >0$: $${\partial }_{t}u(t,x)=\kappa \Delta u(t,x)+\xi (t,x)u(t,x),u(0,x)=u_0(x),t>0,x\in \mathbb{T}.$$ The noise $\xi$ is chosen constant on time intervals of length $\tau >0$ and sampled independently after a time $\tau$. We prove that the Lyapunov exponent $\lambda (\tau )$ is positive and near $\tau =0$ follows a power law that depends on the regularity on the driving noise. As $\tau \to \infty$ 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.

44 Pages