Local Lyapunov Exponents : Sublimiting Growth Rates of Linear Random Differential Equations Linear differential systems with parameter excitation Locality and time scales of the underlying non degenerate stochastic system: Freidlin Wentzell theory Exit probabilities for degenerate systems Local Lyapunov exponents Establishing a new concept of local Lyapunov exponents the author brings together two separate theories namely Lyapunov exponents and the theory of large deviations. Specifically a linear differential system is considered which is controlled by a stochastic process that during a suitable noise intensity dependent time is trapped near one of its so called metastable states. The local Lyapunov exponent is then introduced as the exponential growth rate of the linear system on this time scale. Unlike classical Lyapunov exponents which involve a limit as time increases to infinity in a fixed system here the system itself changes as the noise intensity converges too