Nash inequality for diffusion processes associated with Dirichlet distributions

Abstract

For any N ⩾ 2 and α = (α₁,…, α_N+1) ∈ (0, ∞)^N+1, let µ_a⁽ⁿ⁾ be the Dirichlet distribution with parameter α on the set Δ(N):= [x ∈ [0,1]^N: Σ₁⩽i⩽_Nx_i ⩽ ]. The multivariate Dirichlet diffusion is associated with the Dirichlet form

$${\mathcal E}_\alpha ^{(N)}(f,f): = \sum\limits_{n = 1}^N {\int_{{\Delta ^{(N)}}} {\left( {1 - \sum\limits_{1 \leqslant i \leqslant N} {{x_i}} } \right){x_n}{{({\partial _n}f)}^2}(x)\mu _\alpha ^{(N)}} (dx)}$$

with Domain ${\mathcal D}({\mathcal E}_\alpha ^{(N)})$ being the closure of C¹(Δ^(N)). We prove the Nash inequality

$$\mu _\alpha ^{(N)}({f^2})C{\mathcal E}_\alpha ^{(N)}{(f,f)^{p/(p + 1)}}\mu _\alpha ^{(N)}{(|f|)^{2/(p + 1)}},\;\;\;f \in {\mathcal D}({\mathcal E}_\alpha ^{(N)}),\mu _\alpha ^{(N)}(f) = 0,$$

for some constant C > 0 and p = (α_N+1–1)⁺ + Σ_i=1^N 1 V (2α_i), where the constant p is sharp when max₁⩽i⩽_Nα_i ⩽ 1/2 and α_N+1 ⩾ 1. This Nash inequality also holds for the corresponding Fleming-Viot process.