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For the 2D matrix Langevin dynamics that corresponds to the continuous-timelimit of the product of some $2 \times 2$ random matrices, the finite-timeLyapunov exponent can be written as an additive functional of the associatedRiccati process submitted to some Langevin dynamics on the infinite periodicring. Its large deviations properties can be thus analyzed from two points ofview that are equivalent in the end by consistency but give differentperspectives. In the first approach, one starts from the large deviations atlevel 2.5 for the joint probability of the empirical density and of theempirical current of the Riccati process and one performs the appropriateEuler-Lagrange optimization in order to compute the cumulant generatingfunction of the Lyapunov exponent. In the second approach, this cumulantgenerating function is obtained from the spectral analysis of the appropriatetilted Fokker-Planck operator. The associated conditioned process obtained viathe generalization of Doob's h-transform allows to clarify the equivalence withthe first approach. Finally, applications to one-dimensional AndersonLocalization models are described in order to obtain explicitly the firstcumulants of the finite-size Lyapunov exponent.

Large deviations of the Lyapunov exponent in 2D matrix Langevin dynamics with applications to one-dimensional Anderson Localization models