Nash inequality for diffusion processes associated with Dirichlet distributions

For any N ⩾ 2 and α = (α1,…, αN+1) ∈ (0, ∞)N+1, let µa(n) be the Dirichlet distribution with parameter α on the set Δ(N):= [x ∈ [0,1]N: Σ1⩽i⩽Nxi ⩽ ]. The multivariate Dirichlet diffusion is associated with the Dirichlet form Eα(N)(f,f):=∑n=1N∫Δ(N)(1−∑1⩽i⩽Nxi)xn(∂nf)2(x)μα(N)(dx)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal E}_\alpha ^{(N)}(f,f): = \sum\limits_{n = 1}^N {\int_{{\Delta ^{(N)}}} {\left( {1 - \sum\limits_{1 \leqslant i \leqslant N} {{x_i}} } \right){x_n}{{({\partial _n}f)}^2}(x)\mu _\alpha ^{(N)}} (dx)}$$\end{document} with Domain D(Eα(N))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal D}({\mathcal E}_\alpha ^{(N)})$$\end{document} being the closure of C1(Δ(N)). We prove the Nash inequality μα(N)(f2)CEα(N)(f,f)p/(p+1)μα(N)(|f|)2/(p+1),f∈D(Eα(N)),μα(N)(f)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _\alpha ^{(N)}({f^2})C{\mathcal E}_\alpha ^{(N)}{(f,f)^{p/(p + 1)}}\mu _\alpha ^{(N)}{(|f|)^{2/(p + 1)}},\;\;\;f \in {\mathcal D}({\mathcal E}_\alpha ^{(N)}),\mu _\alpha ^{(N)}(f) = 0,$$\end{document} for some constant C > 0 and p = (αN+1–1)+ + Σi=1N 1 V (2αi), where the constant p is sharp when max1⩽i⩽Nαi ⩽ 1/2 and αN+1 ⩾ 1. This Nash inequality also holds for the corresponding Fleming-Viot process.