Pointwise well-posedness results for degenerate Itô-SDEs with locally bounded drifts

Building on recent results, where Itô-SDEs with possibly degenerate and discontinuous dispersion coefficient and highly singular drift were analyzed with respect to a given (sub-)invariant measure, and led to an SDE for a.e. starting point, we develop here additional elliptic regularity results for PDEs and consider Itô-SDEs with further regularity assumptions on the coefficients to provide a pointwise analysis for every starting point in Euclidean space, [Formula: see text]. Our main result is (weak) well-posedness, i.e. weak existence and uniqueness in law, which we obtain under our main assumption for any locally bounded drift and arbitrary starting point among all solutions that satisfy the condition to spend zero time at the points of degeneracy of the dispersion coefficient. The condition is satisfied by the weak solutions that we construct and moreover we obtain weak existence under broader assumptions than those needed for well-posedness. In particular, for weak existence the drift does not need to be locally bounded. Finally, since there are weak solutions that spend a strictly positive amount of time at the points of degeneracy of the dispersion coefficient, the condition to spend zero time at the points of degeneracy appears to be necessary for uniqueness in law.