  We asymptotically derive a non-linear Langevin-like equation with
non-Gaussian white noise for a wide class of stochastic systems associated with
multiple stochastic environments, by developing the expansion method in our
previous paper [K. Kanazawa et al., arXiv: 1407.5267 (2014)]. We further obtain
a full-order asymptotic formula of the steady distribution function in terms of
a large friction coefficient for a non-Gaussian Langevin equation with an
arbitrary non-linear frictional force. The first-order truncation of our
formula leads to the independent-kick model and the higher-order correction
terms directly correspond to the multiple-kicks effect during relaxation. We
introduce a diagrammatic representation to illustrate the physical meaning of
the high-order correction terms. As a demonstration, we apply our formula to a
granular motor under Coulombic friction and get good agreement with our
numerical simulations.
