论文标题
相互作用粒子系统的非参数估计：McKean-Vlasov模型
Nonparametric estimation for interacting particle systems : McKean-Vlasov models
论文作者
论文摘要
我们考虑了一个由$ n $相互作用的粒子组成的系统，该系统受运输和扩散管辖的系统，该系统在平均场上限制了McKean-Vlasov方程的解决方案。从在固定时间范围内观察系统的轨迹，我们研究了相关的非线性fokker-Planck方程的解决方案的非参数估计，以及控制相互作用的漂移术语，在较大的人群中限制$ n \ rightarrow \ rightarrow \ infty $。根据Lepski的原则，我们构建数据驱动的内核估计器并确定甲骨文不平等。我们的结果基于麦基恩 - 维拉索夫模型中的新伯恩斯坦浓度不平等，以围绕其平均值（可能是独立利益）的经验度量。我们获得了构建在Fokker-Planck方程的解决方案图上的各向异性Hölder平滑度类别上的自适应估计器，并在微小含义上证明了它们的最佳性。在弗拉索夫模型的具体情况下，我们得出了相互作用潜力的估计量并确定其一致性。
We consider a system of $N$ interacting particles, governed by transport and diffusion, that converges in a mean-field limit to the solution of a McKean-Vlasov equation. From the observation of a trajectory of the system over a fixed time horizon, we investigate nonparametric estimation of the solution of the associated nonlinear Fokker-Planck equation, together with the drift term that controls the interactions, in a large population limit $N \rightarrow \infty$. We build data-driven kernel estimators and establish oracle inequalities, following Lepski's principle. Our results are based on a new Bernstein concentration inequality in McKean-Vlasov models for the empirical measure around its mean, possibly of independent interest. We obtain adaptive estimators over anisotropic Hölder smoothness classes built upon the solution map of the Fokker-Planck equation, and prove their optimality in a minimax sense. In the specific case of the Vlasov model, we derive an estimator of the interaction potential and establish its consistency.
