Density-Dependent McKean--Vlasov Diffusions: Subgaussian Occupancy Bounds and Polynomial Propagation of Chaos

Abstract

We study the local density-dependent diffusion $d{Y}_t=-\hat{I}ž(p_t(Y_t))\nabla \hat{I}¦(Y_t)dt+\sqrt{2}d{W}_t$ and a clipped, randomly shifted histogram particle approximation on ${\mathbb{R}}^d$. The central difficulty is that the empirical density is evaluated at the particles' locations and re-enters their drift, while the confining force $\nabla \hat{I}¦$ 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 $\nabla p_t$. 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 $\mathrm{Ent}(P_t^{N,k}|p_t^{\otimes k})\le C_{T}k(h^2(1+|\log h|)+(h^{-d}+\log N)/N)$. Consequently, selecting the optimally balanced bandwidth $h\asymp (N\log N{)}^{-1/(d+2)}$ yields a total variation error of $\Vert P_t^{N,k}-p_t^{\otimes k}{\Vert }_{\mathrm{TV}}\le C_T\sqrt{k}{N}^{-1/(d+2)}(\log N{)}^{d/[2(d+2)]}$ 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.

11 pages