Functional inequalities for the two-parameter extension of the infinitely-many-neutral-alleles diffusion
Let \[ \nabla:= \{ x=(x_1,x_2,\dots): x_1\geq x_2\geq \dots \geq 0,\;\sum_{i=1}^\infty x_i \leq 1\}, \] \[ A_{\alpha,\theta}= \frac{1}{2} \left\{\sum_{i,j=1}^\infty x_i(\delta_{ij}-x_j) \frac{\partial^2}{\partial x_i\partial x_j} -\sum_{i=1}^\infty (\theta x_i+\alpha)\frac{\partial}{\partial x_i}\right\} \] defined on some domain \({\mathcal D}(A_{\alpha, \theta})\), specified in the paper. Let \(U_k\), \(k\geq 1\), be a sequence of independent random variables having \(B(1-\alpha, \theta+k\alpha)\) distribution; \(V_1^{\alpha,\theta}=U_1\), \(V_n^{\alpha,\theta}=(1-U_1)\dots (1-U_{n-1})U_n\), \(n\geq 2\), and let \(P(\alpha,\theta)=(\rho_1,\rho_2,\dots )\) denote \((V_1^{\alpha \theta},V_2^{\alpha,\theta},\dots )\) in descending order. Denote by \(\mu_{\alpha,\theta}\) the law of \(P(\alpha,\theta)\). It is proved that \((A_{\alpha, \theta}, {\mathcal D}(A_{\alpha, \theta}))\) generates \(\nabla\)-valued diffusion \(X_{\alpha,\theta}(t)\), the law of which is reversible and absolutely continuous with respect to \(\mu_{\alpha,\theta}\). The density \(p(t,x,y)\) satisfies \[ p(t,x,y)\leq c t^{\frac{c \log t}{t}}, \quad t>0, \quad x,y\in v. \] This uniform upper bound is used in the proof of the super-Poincaré, the super-log-Sobolev and the \(F\)-Sobolev inequality.
two-parameter Poisson-Dirichlet distribution
infinitely-many-neutral-alleles diffusion
log-Sobolev inequality, super-Poincaré inequality, super-log-Sobolev inequality
scientific article; zbMATH DE number 5825486
scientific article