Pointwise well-posedness results for degenerate It\^{o}-SDEs with locally bounded drifts

TL;DR

This paper establishes pointwise well-posedness for degenerate Itô-SDEs with locally bounded drifts, extending previous results by analyzing solutions from every starting point and considering degeneracy in the dispersion coefficient.

Contribution

It introduces new elliptic regularity results and proves weak well-posedness for degenerate SDEs with minimal assumptions on the drift and initial conditions.

Findings

Weak existence and uniqueness in law for degenerate SDEs.

Solutions can hit degeneracy points which form a measure zero set.

Broader conditions allow for arbitrary starting points without local boundedness of drift.

Abstract

Building on results developed in https://doi.org/10.48550/arXiv.2404.14902, where It\^{o}-SDEs with possibly degenerate and discontinuous dispersion coefficient and measurable drift were analyzed with respect to a given (sub-)invariant measure, we develop here additional elliptic regularity results for PDEs and consider the same equations with some further regularity assumptions on the coefficients to provide a pointwise analysis for every starting point in Euclidean space, $d\ge 2$. 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 spend zero time at the points of degeneracy of the dispersion coefficient. The points of degeneracy form a $d$-dimensional Lebesgue measure zero set, but may be hit by the weak solutions. Weak existence for arbitrary starting point is obtained under broader assumptions. In particular, in that case the drift does not need to be locally bounded.