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

Asymptotic derivation of Langevin-like equation with non-Gaussian noise and its analytical solution