Long-time behavior of McKean--Vlasov stochastic systems with singular coefficients
Abstract
We develop a quantitative framework for the long-time behavior of McKean--Vlasov stochastic differential equations with singular coefficients and possible phase transitions. The framework separates the existence of invariant measures from their uniqueness. For existence, we introduce a generalized Lyapunov condition with a one-level trapping mechanism, which requires dissipativity at only one admissible moment level rather than global contraction. This permits distribution dependence with supercritical growth and thus is more compatible with phase-transition models. For uniqueness and convergence, we establish an anchored uniqueness and quantitative ergodicity principle. An explicit L1-smallness condition, expressed through a convolution kernel built from derivative estimates of the anchored frozen semigroup, yields uniqueness and transfers exponential or polynomial mixing rates of the frozen dynamics to the nonlinear McKean--Vlasov system. Moreover, the principle is sensitive to the choice of topology: different distances lead to different perturbation kernels and hence different stability thresholds. A local version gives local uniqueness and quantitative attraction near a prescribed equilibrium. We apply the framework to two representative models. For non-symmetric granular media dynamics, we derive two explicit uniqueness thresholds that reveal the topology-sensitive nature of the theory. For the dynamical Curie--Weiss model, the Wasserstein-1 criterion recovers the sharp bifurcation threshold up to the critical equality. We identify all invariant measures, establish basin-dependent exponential convergence in the phase-transition regime, and show that the loss of anchored smallness at criticality leads to polynomial slowing down.
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BibTeX
@misc{hua2026longtime,
title = {{Long-time behavior of McKean--Vlasov stochastic systems with singular coefficients}},
author = {Shangyun Hua and Tao Wang and Longjie Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27267},
eprint = {2609.27267},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27267},
}
RIS
TY - UNPB
TI - Long-time behavior of McKean--Vlasov stochastic systems with singular coefficients
AU - Shangyun Hua
AU - Tao Wang
AU - Longjie Xie
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27267
UR - https://arxiv.org/abs/2609.27267
AB - We develop a quantitative framework for the long-time behavior of McKean--Vlasov stochastic differential equations with singular coefficients and possible phase transitions. The framework separates the existence of invariant measures from their uniqueness. For existence, we introduce a generalized Lyapunov condition with a one-level trapping mechanism, which requires dissipativity at only one admissible moment level rather than global contraction. This permits distribution dependence with supercritical growth and thus is more compatible with phase-transition models. For uniqueness and convergence, we establish an anchored uniqueness and quantitative ergodicity principle. An explicit L^1-smallness condition, expressed through a convolution kernel built from derivative estimates of the anchored frozen semigroup, yields uniqueness and transfers exponential or polynomial mixing rates of the frozen dynamics to the nonlinear McKean--Vlasov system. Moreover, the principle is sensitive to the choice of topology: different distances lead to different perturbation kernels and hence different stability thresholds. A local version gives local uniqueness and quantitative attraction near a prescribed equilibrium. We apply the framework to two representative models. For non-symmetric granular media dynamics, we derive two explicit uniqueness thresholds that reveal the topology-sensitive nature of the theory. For the dynamical Curie--Weiss model, the Wasserstein-1 criterion recovers the sharp bifurcation threshold up to the critical equality. We identify all invariant measures, establish basin-dependent exponential convergence in the phase-transition regime, and show that the loss of anchored smallness at criticality leads to polynomial slowing down.
ER -