Pointwise well-posedness results for degenerate It\^{o}-SDEs with locally bounded drifts
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...
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Main Authors | , |
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Format | Journal Article |
Language | English |
Published |
20.05.2024
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Subjects | |
Online Access | Get full text |
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Summary: | 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. |
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DOI: | 10.48550/arxiv.2405.12048 |