Dynamical large deviations of linear diffusions

Linear diffusions are used to model a large number of stochastic processes in physics, including small mechanical and electrical systems perturbed by thermal noise, as well as Brownian particles controlled by electrical and optical forces. Here we use techniques from large deviation theory to study the statistics of time-integrated functionals of linear diffusions, considering three classes of functionals or observables relevant for nonequilibrium systems which involve linear or quadratic integrals of the state in time. For these, we derive exact results for the scaled cumulant generating function and the rate function, characterizing the fluctuations of observables in the long-time limit, and study in an exact way the set of paths or effective process that underlies these fluctuations. The results give a complete description of how fluctuations arise in linear diffusions in terms of effective forces that remain linear in the state or, alternatively, in terms of fluctuating densities and currents that solve Riccati-type equations. We illustrate these results using two common nonequilibrium models, namely, transverse diffusions in two dimensions involving a nonconservative rotating force, and two interacting particles in contact with heat baths at different temperatures.