发表机构
Duisburg-Essen University（杜伊斯堡-埃森大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究密度依赖的McKean-Vlasov扩散及直方图粒子近似，在两个解析条件下用路径空间熵证明，得出相关占有界和误差估计，还介绍了直方图估计器的可扩展性及低评估成本。
AI 中文摘要
我们研究局部密度依赖扩散\(dY_t = -\Xi(p_t(Y_t))\nabla\Phi(Y_t)dt+\sqrt{2}dW_t\)以及在\(\mathbb{R}^d\)上的一种截断、随机移位的直方图粒子近似。核心困难在于经验密度在粒子位置处评估并重新进入其漂移，同时限制力\(\nabla\Phi\)可能无界。我们在两个可验证的解析条件下提供了路径空间熵证明：真实密度\(p_t\)的一致逐点高斯包络，以及其空间梯度\(\nabla p_t\)的高斯-多项式界。势函数的梯度允许最多线性增长。概率输入是独立乘积律下的加权指数占有估计。通过在总强度\(N - 1\)下对系统进行泊松化，执行通过泊松信息界定的单单元留一估计，使用高斯单元可和性以及去泊松化来证明。对于每个固定的时间范围\(T\)，我们得到\(\text{Ent}(P_t^{N,k}|p_t^{\otimes k})\leq C_Tk(h^2(1 + |\log h|)+(h^{-d}+\log N)/N)\)。因此，选择最优平衡带宽\(h\asymp (N\log N)^{-1/(d + 2)}\)对于固定的\(k\)会产生总变差误差\(\Vert P_t^{N,k}-p_t^{\otimes k}\Vert_{\text{TV}}\leq C_T\sqrt{k}N^{-1/(d + 2)}(\log N)^{d/[2(d + 2)]}\)。这包括通常的奥恩斯坦 - 乌伦贝克密度以及只要在考虑区间上偏微分方程估计成立的密度依赖奥恩斯坦 - 乌伦贝克模型。此外，直方图估计器为粒子近似提供了一种可扩展的方法。使用占用单元哈希，在标准常数时间哈希假设下，一个算法步骤在期望\(O(dLN)\)操作中进行评估。对于固定维度和移位数量，这需要期望\(O(N)\)时间，避免了标准核密度估计器典型的\(O(N^2)\)评估成本。
英文摘要
We study the local density-dependent diffusion $dY_t=-Ξ(p_t(Y_t))\nablaΦ(Y_t)\,dt+\sqrt2\,dW_t$ 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 $\nablaΦ$ 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.
Comments14 pages
