Nash inequality for diffusion processes associated with Dirichlet distributions

Abstract

For any N\ge \mathrm{2} and \alpha ={(\alpha }_1,\mathrm{...},{\alpha }_{N+1})\in {(0,\infty )}^{N+1}, let {\mu }_{\alpha }^{(N)} be the Dirichlet distribution with parameter α on the set {\mathrm{\Delta }}^{(N)}:={\{x\in [0,1]}^N:{\displaystyle {\sum }_{\mathrm{1}\le i\le N}}{x}_i\le \mathrm{1}\}. The multivariate Dirichlet diffusion is associated with the Dirichlet form

{\mathcal{E}}_{\alpha }^{(N)}(f,f):={\displaystyle \sum _{n=\mathrm{1}}^N}{\displaystyle {\int }_{{\mathrm{\Delta }}^{(N)}}}(\mathrm{1}-{\displaystyle \sum _{\mathrm{1}\le i\le N}}{x}_i)x_n{({\partial }_{n}f)}^{\mathrm{2}}(x){\mu }_{\alpha }^{(N)}(\mathrm{d}x)

with Domain \mathcal{D}({\mathcal{E}}_{\alpha }^{(N)}) being the closure of {C^1(\mathrm{\Delta }}^{(N)}). We prove the Nash inequality

{\mu }_{\alpha }^{(N)}(f^{\mathrm{2}})\le C{\mathcal{E}}_{\alpha }^{(N)}{(f,f)}^{p/(p+\mathrm{1})}{\mu }_{\alpha }^{(N)}{(|f|)}^{\mathrm{2}/(p+\mathrm{1})},\mathrm{\hspace{0.17em}}f\in \mathcal{D}({\mathcal{E}}_{\alpha }^{(N)}),{\mu }_{\alpha }^{(N)}(f)=0,

for some constant C>0 and p={({\alpha }_{N+\mathrm{1}}-\mathrm{1})}^{+}+{\displaystyle {\sum }_{i=\mathrm{1}}^N\mathrm{1}}\vee (\mathrm{2}{\alpha }_i), where the constant p is sharp when {\max }_{\mathrm{1}\le i\le N}\mathrm{\hspace{0.17em}}{\alpha }_i\le 1/2 and {\alpha }_{N+\mathrm{1}}\ge \mathrm{1}. This Nash inequality also holds for the corresponding Fleming-Viot process.

Key words： Dirichlet distribution; Nash inequality; super Poincaré inequality; diffusion process

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