Moderate deviations for randomly perturbed dynamical systems

A Moderate Deviation Principle is established for random processes arising as small random perturbations of one-dimensional dynamical systems of the form Xn=f(Xn−1). Unlike in the Large Deviations Theory the resulting rate function is independent of the underlying noise distribution, and is always quadratic. This allows one to obtain explicit formulae for the asymptotics of probabilities of the process staying in a small tube around the deterministic system. Using these, explicit formulae for the asymptotics of exit times are obtained. Results are specified for the case when the dynamical system is periodic, and imply stability of such systems. Finally, results are applied to the model of density-dependent branching processes.