2026/07/21 by Denis Belomestny, Ekaterina Morozova
#math.PR
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 h\asymp (Nlog N)-1/(d+2) yields a total variation error of \Vert PtN,k-pt⊗ k\VertTV≤ 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.