The method of stochastic characteristics for linear second-order hypoelliptic equations

We study hypoelliptic stochastic differential equations (SDEs) and their connection to degenerate-elliptic boundary value problems on bounded or unbounded domains. In particular, we provide probabilistic conditions that guarantee that the formal stochastic representation of a so-lution is smooth on the interior of the domain and continuously approaches the prescribed boundary data at a given boundary point. The main general results are proved using fine properties of the process stopped at the bound-ary of the domain combined with hypoellipticity of the operators associated to the SDE. The main general results are then applied to deduce proper-ties of the associated Green's functions and to obtain a generalization of Bony's Harnack inequality. We moreover revisit the transience and recur-rence dichotomy for hypoelliptic diffusions and its relationship to invariant measures.