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 $\kappa > 0$: $$\partial_{t} u(t,x) = \kappa \Delta u(t,x) + \xi(t, x) u(t,x), \quad u(0, x) = u_{0}(x), \qquad t > 0, x \in \mathbb{T}.$$ 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 Bou\'e--Dupuis formulas, as well as on Doob's H-transform and on tools from singular stochastic PDEs.