A remark on random perturbations of the nonlinear pendulum (Q1578588)

The authors consider the long-time behaviour of the nonlinear pendulum undergoing a small random perturbation. The fundamental equation considered is \[ \ddot X_t+\sin X_t=0, \] with attention also being paid to the more general process \[ \ddot{\widetilde X}^\varepsilon_t+ \sin\widetilde X^\varepsilon_t= \varepsilon b(\widetilde X^\varepsilon_t, \dot{\widetilde X}_t^\varepsilon)+ \sqrt\varepsilon \dot W_t, \] where \(b(x,y)\) is \(2\pi\)-periodic in \(x\). The term \(\varepsilon b(x,y)\) is of interest in a model of phase synchronization in radiophysics and in optimal stabilization. The perturbation is of the white-noise type.