Large Deviations in Fast–Slow Systems

The incidence of rare events in fast–slow systems is investigated via analysis of the large deviation principle (LDP) that characterizes the likelihood and pathway of large fluctuations of the slow variables away from their mean behavior—such fluctuations are rare on short time-scales but become ubiquitous eventually. Classical results prove that this LDP involves an Hamilton–Jacobi equation whose Hamiltonian is related to the leading eigenvalue of the generator of the fast process, and is typically non-quadratic in the momenta—in other words, the LDP for the slow variables in fast–slow systems is different in general from that of any stochastic differential equation (SDE) one would write for the slow variables alone. It is shown here that the eigenvalue problem for the Hamiltonian can be reduced to a simpler algebraic equation for this Hamiltonian for a specific class of systems in which the fast variables satisfy a linear equation whose coefficients depend nonlinearly on the slow variables, and the fast variables enter quadratically the equation for the slow variables. These results are illustrated via examples, inspired by kinetic theories of turbulent flows and plasma, in which the quasipotential characterizing the long time behavior of the system is calculated and shown again to be different from that of an SDE.