Resonantly driven Brownian motion: Basic concepts and exact results

We consider the Brownian motion of a particle in a force field exposed to external periodic modulations. Discussed are ergodic properties, the long-time behavior of correlation functions, and spectral densities for general (nonlinear) force fields. For a parabolic potential we present analytical results for the quasispectrum (quasieigenvalues and quasieigenfunctions) at all values of the damping constant as well as spectral densities and cycle-averaged probability distributions. The theory is also applied to a Brownian oscillator with memory damping, yielding additional resonances between the external field and internal degrees of freedom.