We have developed the theory of stochasitc rotational relation to include symmetric top molecules. A general theorem has been derived which can be used to determine the form of the nonlinear coupling between the rotor and its surrounding bath. This theorem has been used to derive the form of the coupling for the flat rotors with three-fold symmetry, such as symmetric triflourobenzene, and for tetrahedrons, such as methane. In the former molecule, two independent stochastic coefficients arise, corresponding the randomization of angular momentum and perpendicular, respectively, to the symmetry axis. In the tetrahedron, only on independent stochastic variable arises. Both results are in agreement with expectations. The equations contain several features derived earlier for on dimensional rotor, including multiplicative (rather than additive) noise, a diffusion """"""""constant"""""""" that depends on the rotational motion, and an abnormal diffusion term. In the limit of white noise, these equations reduce to the standard linearly coupled equations. With colored noise, however, the linearly and nonlinearly coupled equations are distinct and the unique features of the nonlinearly coupled equations drive the dynamics in a fundamentally different way Development is continuing to extend these results to other systems and applications.
