Poincaré Inequality for Dirichlet Distributions and Infinite-Dimensional Generalizations

Abstract

For any $N\ge 2$ and $\mathring{a}:=({\mathring{a}}_1,\cdots ,{\mathring{a}}_{N+1})\in (0,\infty {)}^{N+1}$, let ${\mu }_{\mathring{a}}^{(N)}$ be the corresponding Dirichlet distribution on $\DD:= \big\{ x=(x_i)_{1\le i\le N}\in [0,1]^N:\ \sum_{1\le i\le N} x_i\le 1\big\}.$ We prove the Poincaré inequality $$μ^{(N)}_å(f^2)\le \ff 1 {å_{N+1}} \int_{\DD}\Big\{\Big(1-\sum_{1\le i\le N} x_i\Big) \sum_{n=1}^N x_n(\pp_n f)^2\Big\}μ^{(N)}_å(\d x)+μ^{(N)}_å(f)^2,\ f\in C^1(\DD)$$ and show that the constant $\ff 1 {å_{N+1}}$ is sharp. Consequently, the associated diffusion process on $\DD$ converges to ${\mu }_{\mathring{a}}^{(N)}$ in $L^2({\mu }_{\mathring{a}}^{(N)})$ at the exponentially rate ${\mathring{a}}_{N+1}$. The whole spectrum of the generator is also characterized. Moreover, the sharp Poincaré inequality is extended to the infinite-dimensional setting, and the spectral gap of the corresponding discrete model is derived.

21 pages