Density-Dependent McKean--Vlasov Diffusions: Subgaussian Occupancy Bounds and Polynomial Propagation of Chaos
Abstract
We study the local density-dependent diffusion dYt=−Ξ(pt(Yt))∇Φ(Yt) dt+√2 dWt and a clipped, randomly shifted histogram particle approximation on ℝd. The central difficulty is that the empirical density is evaluated at the particles' locations and re-enters their drift, while the confining force ∇Φ may be unbounded. We provide a path-space entropy proof under two verifiable analytic conditions: a uniform pointwise Gaussian envelope for the true density pt, and a Gaussian--polynomial bound for its spatial gradient ∇ pt. The potential is allowed to have a gradient of at most linear growth. The probabilistic input is a weighted exponential occupancy estimate under the independent product law. It is proved by Poissonizing the system at total intensity N−1, performing a one-cell leave-one-out estimate bounded via Poisson information, using Gaussian cell summability, and de-Poissonizing. For every fixed time horizon T, we obtain Ent(PtN,k|pt⊗ k)≤ CT k(h2(1+|log h|)+(h−d+log N)/N). Consequently, selecting the optimally balanced bandwidth hasymp (Nlog N)−1/(d+2) yields a total variation error of ‖ PtN,k−pt⊗ k‖TV≤ CT√k N−1/(d+2)(log N)d/[2(d+2)] for fixed k. This includes the usual Ornstein--Uhlenbeck density and the density-dependent OU model whenever the PDE estimates hold on the considered interval. Furthermore, the histogram estimator offers a scalable approach for particle approximations. Using occupied-cell hashing, one algorithm step evaluates in expected O(dLN) operations under standard constant-time hashing assumptions. For a fixed dimension and number of shifts, this requires expected O(N) time, avoiding the O(N2) evaluation cost typical of standard kernel density estimators.
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BibTeX
@misc{belomestny2026densitydependent,
title = {{Density-Dependent McKean--Vlasov Diffusions: Subgaussian Occupancy Bounds and Polynomial Propagation of Chaos}},
author = {Denis Belomestny and Ekaterina Morozova},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.19583},
eprint = {2607.19583},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.19583},
}
RIS
TY - UNPB
TI - Density-Dependent McKean--Vlasov Diffusions: Subgaussian Occupancy Bounds and Polynomial Propagation of Chaos
AU - Denis Belomestny
AU - Ekaterina Morozova
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2607.19583
UR - https://arxiv.org/abs/2607.19583
AB - We study the local density-dependent diffusion dY_t=−Ξ(p_t(Y_t))∇Φ(Y_t) dt+√2 dW_t and a clipped, randomly shifted histogram particle approximation on ℝ^d. The central difficulty is that the empirical density is evaluated at the particles' locations and re-enters their drift, while the confining force ∇Φ may be unbounded. We provide a path-space entropy proof under two verifiable analytic conditions: a uniform pointwise Gaussian envelope for the true density p_t, and a Gaussian--polynomial bound for its spatial gradient ∇ p_t. The potential is allowed to have a gradient of at most linear growth. The probabilistic input is a weighted exponential occupancy estimate under the independent product law. It is proved by Poissonizing the system at total intensity N−1, performing a one-cell leave-one-out estimate bounded via Poisson information, using Gaussian cell summability, and de-Poissonizing. For every fixed time horizon T, we obtain Ent(P_t^N,k|p_t^⊗ k)≤ C_T k(h^2(1+|log h|)+(h^−d+log N)/N). Consequently, selecting the optimally balanced bandwidth hasymp (Nlog N)^−1/(d+2) yields a total variation error of ‖ P_t^N,k−p_t^⊗ k‖_TV≤ C_T√k N^−1/(d+2)(log N)^d/[2(d+2)] for fixed k. This includes the usual Ornstein--Uhlenbeck density and the density-dependent OU model whenever the PDE estimates hold on the considered interval. Furthermore, the histogram estimator offers a scalable approach for particle approximations. Using occupied-cell hashing, one algorithm step evaluates in expected O(dLN) operations under standard constant-time hashing assumptions. For a fixed dimension and number of shifts, this requires expected O(N) time, avoiding the O(N^2) evaluation cost typical of standard kernel density estimators.
ER -