Poincar\'e Inequality for Dirichlet Distributions and Infinite-Dimensional Generalizations

For any $N\ge 2$ and $\aa:=(\aa_1,\cdots, \aa_{N+1})\in (0,\infty)^{N+1}$, let $\mu^{(N)}_{\aa}$ 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 Poincar\'e inequality $$\mu^{(N)}_{\aa}(f^2)\le \ff 1 {\aa_{N+1}} \int_{\DD}\Big\{\Big(1-\sum_{1\le i\le N} x_i\Big) \sum_{n=1}^N x_n(\pp_n f)^2\Big\}\mu^{(N)}_\aa(\d x)+\mu^{(N)}_{\aa}(f)^2,\ f\in C^1(\DD)$$ and show that the constant $\ff 1 {\aa_{N+1}}$ is sharp. Consequently, the associated diffusion process on $\DD$ converges to $\mu^{(N)}_{\aa}$ in $L^2(\mu^{(N)}_{\aa})$ at the exponentially rate $\aa_{N+1}$. The whole spectrum of the generator is also characterized. Moreover, the sharp Poincar\'e inequality is extended to the infinite-dimensional setting, and the spectral gap of the corresponding discrete model is derived.