For any $N\ge 2$ and $\alpha=(\alpha_1,\cdots, \alpha_{ N+1})\in (0,\infty)^{ N+1}$, let $\mu^{ (N)}_{ \alpha}$ 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 $${ \scr E}_\alpha^{ (N)}(f,f):= \sum_{ n=1}^N \int_{ \Delta^{ (N)}} \bigg(1-\sum_{ 1\le i\le N}x_i\bigg) x_n(\partial_n f)^2(x)\,\mu^{ (N)}_\alpha(d x)$$ with Domain ${ \scr D}({ \scr E}_\alpha^{ (N)})$ being the closure of $C^1(\Delta^{ (N)})$. We prove the Nash inequality $$\mu_\alpha^{ (N)}(f^2)\le C { \scr E}_\alpha^{ (N)}(f,f)^{ \frac p{ p+1} }\mu_\alpha^{ (N)} (|f|)^{ \frac 2 { p+1}},\ \ f\in { \scr D}({ \scr E}_\alpha^{ (N)}), \mu_\alpha^{ (N)}(f)=0$$ for some constant $C>0$ and $p= (\alpha_{ N+1}-1)^+ +\sum_{ i=1}^N 1\lor (2\alpha_i),$ 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.
Categories: math.PR