Nash inequality for Diffusion Processes Associated with Dirichlet Distributions

Abstract

For any $N\ge 2$ and $\alpha =({\alpha }_1,\cdots ,{\alpha }_{N+1})\in (0,\infty {)}^{N+1}$, let ${\mu }_{\alpha }^{(N)}$ be the Dirichlet distribution with parameter $\alpha$ on the set ${\Delta }^{(N)}:=\{x\in [0,1{]}^N:{\sum }_{1\le i\le N}{x}_i\le 1\}.$ The multivariate Dirichlet diffusion is associated with the Dirichlet form $${\scr E}_α^{(N)}(f,f):= \sum_{n=1}^N \int_{ Δ^{(N)}} \bigg(1-\sum_{1\le i\le N}x_i\bigg) x_n(\partial_n f)^2(x)\,μ^{(N)}_α(d x)$$ with Domain ${\scr D}({\scr E}_α^{(N)})$ being the closure of $C^1({\Delta }^{(N)})$. We prove the Nash inequality $$μ_α^{(N)}(f^2)\le C {\scr E}_α^{(N)}(f,f)^{\frac p{p+1} }μ_α^{(N)} (|f|)^{\frac 2 {p+1}},\ \ f\in {\scr D}({\scr E}_α^{(N)}), μ_α^{(N)}(f)=0$$ for some constant $C>0$ and $p=({\alpha }_{N+1}-1{)}^{+}+{\sum }_{i=1}^{N}1\vee (2{\alpha }_i),$ where the constant $p$ is sharp when ${\max }_{1\le i\le N}{\alpha }_i\le 1/2$ and ${\alpha }_{N+1}\ge 1$. This Nash inequality also holds for the corresponding Fleming-Viot process.

arXiv comment20 pages

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