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Non-uniqueness for McKean--Vlasov SDEs with Hölder diffusion coefficients
Abstract
We show that uniqueness may fail for one-dimensional McKean--Vlasov stochastic differential equations with Hölder continuous diffusion coefficients. More precisely, for every α∈(1/2,1), we construct an equation with a diffusion coefficient that is α-Hölder continuous in the state variable, Lipschitz continuous in the measure variable, and of linear growth, for which uniqueness in law and pathwise uniqueness fail. The construction relies on a spreading estimate together with a Schauder fixed-point argument.
Related
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BibTeX
@misc{bergerhausen2026nonuniqueness,
title = {{Non-uniqueness for McKean--Vlasov SDEs with Hölder diffusion coefficients}},
author = {Martin Bergerhausen and Paul Nikolaev and David J. Prömel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15017},
eprint = {2610.15017},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15017},
}
RIS
TY - UNPB
TI - Non-uniqueness for McKean--Vlasov SDEs with Hölder diffusion coefficients
AU - Martin Bergerhausen
AU - Paul Nikolaev
AU - David J. Prömel
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.15017
UR - https://arxiv.org/abs/2610.15017
AB - We show that uniqueness may fail for one-dimensional McKean--Vlasov stochastic differential equations with Hölder continuous diffusion coefficients. More precisely, for every α∈(1/2,1), we construct an equation with a diffusion coefficient that is α-Hölder continuous in the state variable, Lipschitz continuous in the measure variable, and of linear growth, for which uniqueness in law and pathwise uniqueness fail. The construction relies on a spreading estimate together with a Schauder fixed-point argument.
ER -