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

For any $N\ge 2$ and $\mathring{a}:=({\mathring{a}}_1,\cdots ,{\mathring{a}}_{N+1})\in (0,\mathrm{\infty }{)}^{N+1}$, let ${\mu }_{\mathring{a}}^{(N)}$ be the corresponding Dirichlet distribution on $\DD:= \big{ x=(x_i){1\le i\le N}\in [0,1]^N:\ \sum{1\le i\le N} x_i\le 1\big}.$WeprovethePoincar\acute{e}inequality$$\mu^{(N)}{\aa}(f^2)\le \ff 1 {\aa{N+1}} \int_{\DD}\Big{\Big(1-\sum_{1\le i\le N} x_i\Big) \sum_{n=1}^N x_n(\pp_n f)^2\Big}\mu^{(N)}\aa(\d x)+\mu^{(N)}{\aa}(f)^2,\ f\in C^1(\DD)$$andshowthattheconstant$\ff 1 {\aa_{N+1}}$ is sharp. Consequently, the associated diffusion process on $\text{DD}$ 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 Poincar'e inequality is extended to the infinite-dimensional setting, and the spectral gap of the corresponding discrete model is derived.