Fokker-Planck Equations on Homogeneous Lie Groups and Probabilistic Counterparts

We address the well-posedness of subelliptic Fokker–Planck equations arising from stochastic control problems, as well as the properties of the associated diffusion processes. Here, the main difficulty arises from the possible 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 the existence and uniqueness of the energy solution and its representation as the transition density of the underlying subelliptic diffusion process. Moreover, we show its H\"older continuity in time w.r.t. the Fortet–Mourier distance, where the H\"older seminorm depends on the degree of homogeneity of the vector fields. Finally, we provide a probabilistic proof of the Feyman–Kac formula as a consequence of the uniform boundedness in finite time intervals of all moments.