Fokker-Planck equations on homogeneous Lie groups and probabilistic counterparts

The well-posedness of Fokker-Planck equations defined on homogeneous Lie groups and the properties of diffusion processes associated with such equations are addressed. The main difficulty arises from the polynomial growth of the coefficients, which is related to the growth of the family of vector fields generating the first layer of the associated Lie algebra. We prove that the Fokker-Planck equation has a unique energy solution, which we can represent as the transition density of the underlying subelliptic diffusion process. Moreover, we show its Holder continuity in time, where the Holder seminorm depends on the degree of homogeneity of the vector fields. Finally, we provide a probabilistic proof of the Feyman-Kac formula, also as a consequence of the uniform boundedness in finite time intervals of all moments.