Minimax rate of estimation for invariant densities associated to continuous stochastic differential equations over anisotropic Hölder classes

We study the problem of the nonparametric estimation for the density π$$ \pi $$ of the stationary distribution of a d$$ d $$‐dimensional stochastic differential equation (Xt)t∈[0,T]$$ {\left({X}_t\right)}_{t\in \left[0,T\right]} $$. From the continuous observation of the sampling path on [0,T]$$ \left[0,T\right] $$, we study the estimation of π(x)$$ \pi (x) $$ as T$$ T $$ goes to infinity. For d≥2$$ d\ge 2 $$, we characterize the minimax rate for the L2$$ {\mathbf{L}}^2 $$‐risk in pointwise estimation over a class of anisotropic Hölder functions π$$ \pi $$ with regularity β=(β1,…,βd)$$ \beta =\left({\beta}_1,\dots, {\beta}_d\right) $$. For d≥3$$ d\ge 3 $$, our finding is that, having ordered the smoothness such that β1≤⋯≤βd$$ {\beta}_1\le \cdots \le {\beta}_d $$, the minimax rate depends on whether β2