Degenerate McKean--Vlasov SDE with Yamada--Osgood Coefficients and Arbitrary Modulus of Continuity in Wasserstein Distance
Abstract
We establish the well-posedness of strong solutions for a class of degenerate mean-field stochastic differential equations with coefficients satisfying non-Lipschitz conditions. The diffusion coefficient is not required to be non-degenerate. Both the diffusion and the drift coefficients may depend on the law through the Kantorovich-Rubinstein metric with an arbitrary modulus of continuity. The drift and the diffusion satisfy a dimension-dependent Osgood-type condition, that becomes particularly sharp in dimension one. The proof combines the classical Yamada--Watanabe argument with Schauder's fixed-point theorem and does not rely on Gaussian estimates, allowing us to cover degenerate diffusions where such estimates are not available. This fills a gap in the existing literature.
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BibTeX
@misc{buckdahn2026degenerate,
title = {{Degenerate McKean--Vlasov SDE with Yamada--Osgood Coefficients and Arbitrary Modulus of Continuity in Wasserstein Distance}},
author = {Rainer Buckdahn and Paolo Di Tella and Juan Li},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05091},
eprint = {2611.05091},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05091},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Degenerate McKean--Vlasov SDE with Yamada--Osgood Coefficients and Arbitrary Modulus of Continuity in Wasserstein Distance
AU - Rainer Buckdahn
AU - Paolo Di Tella
AU - Juan Li
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05091
UR - https://arxiv.org/abs/2611.05091
AB - We establish the well-posedness of strong solutions for a class of degenerate mean-field stochastic differential equations with coefficients satisfying non-Lipschitz conditions. The diffusion coefficient is not required to be non-degenerate. Both the diffusion and the drift coefficients may depend on the law through the Kantorovich-Rubinstein metric with an arbitrary modulus of continuity. The drift and the diffusion satisfy a dimension-dependent Osgood-type condition, that becomes particularly sharp in dimension one. The proof combines the classical Yamada--Watanabe argument with Schauder's fixed-point theorem and does not rely on Gaussian estimates, allowing us to cover degenerate diffusions where such estimates are not available. This fills a gap in the existing literature.
ER -