McKean-Vlasov flights without moments: well-posedness, propagation of chaos and a critical threshold for law-dependent jumps
Abstract
We study McKean-Vlasov equations driven by an isotropic 2s-stable Lévy noise, s∈(0,1), whose solutions may have no finite mean, with coefficients that are Lipschitz with respect to a bounded Wasserstein-type distance. When the jump noise is additive, we prove strong well-posedness for every initial law and every s, without any moment assumption, and propagation of chaos at the rate N−1/2 for kernel interactions. When the size of the jumps depends on the law, we prove the same results for s>1/2, and we show by an example that uniqueness fails for every s<1/2. The value s=1/2 thus separates well-posedness from non-uniqueness; the critical case itself remains open. These results answer, for stable noise, the question of uniqueness without moments raised by Jourdain, Méléard and Woyczyński and by Cavallazzi.
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BibTeX
@misc{jeddaoui2026mckeanvlasov,
title = {{McKean-Vlasov flights without moments: well-posedness, propagation of chaos and a critical threshold for law-dependent jumps}},
author = {Houssine El Jeddaoui},
howpublished = {HAL},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02859},
eprint = {2611.02859},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05847009v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - McKean-Vlasov flights without moments: well-posedness, propagation of chaos and a critical threshold for law-dependent jumps
AU - Houssine El Jeddaoui
PY - 2026
DA - 2026/11/01
PB - HAL
DO - 10.48550/arxiv.2611.02859
UR - https://hal.science/hal-05847009v1
AB - We study McKean-Vlasov equations driven by an isotropic 2s-stable Lévy noise, s∈(0,1), whose solutions may have no finite mean, with coefficients that are Lipschitz with respect to a bounded Wasserstein-type distance. When the jump noise is additive, we prove strong well-posedness for every initial law and every s, without any moment assumption, and propagation of chaos at the rate N^−1/2 for kernel interactions. When the size of the jumps depends on the law, we prove the same results for s>1/2, and we show by an example that uniqueness fails for every s<1/2. The value s=1/2 thus separates well-posedness from non-uniqueness; the critical case itself remains open. These results answer, for stable noise, the question of uniqueness without moments raised by Jourdain, Méléard and Woyczyński and by Cavallazzi.
ER -