  The influence of multiplicative stochastic perturbations on the class of
asymptotically Hamiltonian systems on the plane is investigated. It is assumed
that disturbances do not preserve the equilibrium of the corresponding limiting
system and their intensity decays in time with power-law asymptotics. The paper
discusses the long-term asymptotic behaviour of solutions and its dependence on
the structure and parameters of perturbations. In particular, it is shown that
perturbed trajectories can tend to the equilibrium of the limiting system or
new stochastically stable states can arise. The proposed analysis is based on a
combination of the averaging method and the construction of stochastic Lyapunov
functions. The results obtained are applied to the problem of capture into
parametric autoresonance in the presence of noise.
