  Open system dynamics in a classical setting is microscopically governed by
the structure of the thermal environment which influences the dynamics of the
probe particle (free or in an external potential). Nonlinear baths have
recently been shown to impart interesting nonequilibrium correlations in the
dissipative dynamics affecting Generalised Langevin Equations and the
Fluctuation Dissipation Relations. In the following work, we investigate some
aspects of nonlinear baths with rigour relying on perturbative expansions to
deal with nonlinear equations. Firstly, the question of noise spectrum emerging
from such nonlinearities are addressed and the Markovian limit is explored;
super-Ohmic corrections to the linear Ohmic spectrum is deduced. Velocity
correlations of a probe system under such approximations are studied in detail.
In a second part to the paper, a quenched initial thermal bath is modelled via
nonlinearities and Louivillean evolution is applied to evaluate subsequent
correlations out of equilibrium. In all these problems, weak system-bath
coupling is assumed and quartic baths are used for specific calculations.
