H\"older continuity of Lyapunov exponents for non-invertible and non-compact random cocycles

Abstract

We study the regularity of Lyapunov exponents for random linear cocycles taking values in $\Mat_m(\R)$ and driven by i.i.d. processes. Under three natural conditions - finite exponential moments, a spectral gap between the top two Lyapunov exponents, and quasi-irreducibility of the associated semigroup - we prove that the top Lyapunov exponent is H\"older continuous with respect to the Wasserstein distance. In the final section, we apply the main results to Schr\"odinger cocycles with unbounded potentials.