The invariant density of diffusion processes which are small random perturbations of dynamical systems can be expanded in W.K.B. type, as the random effect disappears, in the set in which the Freidlin-Wentzell quasipotential $V(\cdot)$ is of $C^\infty$-class and each coefficient which appears in the expansion is of $C^\infty$-class.
Toshio Mikami. "Asymptotic Analysis of Invariant Density of Randomly Perturbed Dynamical Systems." Ann. Probab. 18 (2) 524 - 536, April, 1990. https://doi.org/10.1214/aop/1176990843