Poincaré Inequality for Dirichlet Distributions and Infinite-Dimensional Generalizations

International audienceFor any N ≥ 2 and α = (α 1 , · · · , α N +1) ∈ (0, ∞) N +1 , let µ(N) α be the corresponding Dirichlet distribution on ∆(N) :=x = (xi)1≤i≤N ∈ [0,1]N : |x|1 := P1≤i≤N xi ≤1. We prove the Poincar´e inequality µ(N) α (f2) ≤ 1 αN+1 Z∆(N)n1−|x|1 N X n=1 xn(∂nf)2oµ(N) α (dx) + µ(N) α (f)2, for f ∈ C1(∆(N)), and show that the constant 1 αN+1 is sharp. Consequently, the associated diffusion process on ∆(N) converges to µ(N) α in L2(µ(N) α ) at the exponentially rate αN+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