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

We study the local density-dependent diffusion $dY_t=-\Xi(p_t(Y_t))\nabla\Phi(Y_t)\,dt+\sqrt2\,dW_t$ 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 $\nabla\Phi$ 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 $\operatorname{Ent}(P_t^{N,k}|p_t^{\otimes k})\leq C_T k(h^2(1+|\log h|)+(h^{-d}+\log N)/N)$.

Consequently, selecting the optimally balanced bandwidth $h\asymp (N\log N)^{-1/(d+2)}$ yields a total variation error of $\Vert P_t^{N,k}-p_t^{\otimes k}\Vert_{\operatorname{TV}}\leq C_T\sqrt{k}\,N^{-1/(d+2)}(\log N)^{d/[2(d+2)]}$ 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.