We discuss lower bounds on Lyapunov exponents for linear PDEs driven by random velocity fields, including the advection–diffusion equation and the linearized stochastic Navier–Stokes equations. In particular, we show that the exponential rate of decay of the L^2-norm for solutions to the advection–diffusion equation is optimal. A key ingredient in the proof is high-frequency stochastic instability, arising from the non-degeneracy of the driving noise. This talk is based on joint work with Martin Hairer, Tommaso Rosati, and Sam Punshon-Smith.