Tamed Euler Approximation of Invariant Measures for Weakly Interacting Particle Systems and McKean--Vlasov Equations
Abstract
We analyze explicit tamed Euler approximations for weakly interacting particle systems and McKean--Vlasov diffusions with additive noise and polynomially weighted locally Hölder, possibly superlinear drift.Under dissipativity at infinity and suitable stability assumptions, we establish time-uniform W1 bounds by reflection coupling, separating relaxation, time-discretization, and finite-particle errors.For small step sizes, we prove existence and uniqueness of numerical invariant laws and bound their error relative to the continuous-time particle invariant law uniformly in N in a normalized joint W1 distance. We further bound the error of the one-particle numerical invariant marginal relative to the McKean--Vlasov invariant law by O(θmin{α,β/2}+N−1/2), where θ is the step size, N the particle number, α∈(0,1/2] the taming exponent, and β∈(0,1] the Hölder exponent. Analytical examples illustrate the framework for smooth and non-Lipschitz drifts.
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BibTeX
@misc{cheng2026tamed,
title = {{Tamed Euler Approximation of Invariant Measures for Weakly Interacting Particle Systems and McKean--Vlasov Equations}},
author = {Lingyan Cheng and Ruinan Li and Shiyu Liu and Wei Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29314},
eprint = {2610.29314},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29314},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Tamed Euler Approximation of Invariant Measures for Weakly Interacting Particle Systems and McKean--Vlasov Equations
AU - Lingyan Cheng
AU - Ruinan Li
AU - Shiyu Liu
AU - Wei Liu
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29314
UR - https://arxiv.org/abs/2610.29314
AB - We analyze explicit tamed Euler approximations for weakly interacting particle systems and McKean--Vlasov diffusions with additive noise and polynomially weighted locally Hölder, possibly superlinear drift.Under dissipativity at infinity and suitable stability assumptions, we establish time-uniform W_1 bounds by reflection coupling, separating relaxation, time-discretization, and finite-particle errors.For small step sizes, we prove existence and uniqueness of numerical invariant laws and bound their error relative to the continuous-time particle invariant law uniformly in N in a normalized joint W_1 distance. We further bound the error of the one-particle numerical invariant marginal relative to the McKean--Vlasov invariant law by O(θ^min{α,β/2}+N^−1/2), where θ is the step size, N the particle number, α∈(0,1/2] the taming exponent, and β∈(0,1] the Hölder exponent. Analytical examples illustrate the framework for smooth and non-Lipschitz drifts.
ER -