Nash inequality for Diffusion Processes Associated with Dirichlet Distributions (1801.09209v1)
Abstract: 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 $C1(\Delta{(N)})$. We prove the Nash inequality $$\mu\alpha{(N)}(f2)\le C {\scr E}\alpha{(N)}(f,f){\frac p{p+1} }\mu\alpha{(N)} (|f|){\frac 2 {p+1}},\ \ f\in {\scr D}({\scr E}\alpha{(N)}), \mu\alpha{(N)}(f)=0$$ 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.