Verification of Lyapunov functions for the analysis of stochastic Liénard equations

Versions of stochastic Liénard equations perturbed by both additive and multiplicative white noise are considered. We discuss existence, uniqueness, continuity, boundedness and moment stability of solutions with the help of several Lyapunov-type functions. The Lyapunov functions are explicitly found to control uniform moment boundedness and stability. A new matching condition on the interaction of resistance and restoring force plays an essential role to guarantee stability (uniform boundedness) of pp-th moments by validation of those Lyapunov functions.