On the log-Sobolev constant for the simple random walk on the n-cycle: the even cases

Consider the simple random walk on the n-cycle . For this example, Diaconis and Saloff-Coste (Ann. Appl. Probab. 6 (1996) 695) have shown that the log-Sobolev constant α is of the same order as the spectral gap λ. However the exact value of α is not known for n>4. (For n=2, it is a well known result of Gross (Amer. J. Math. 97 (1975) 1061) that α is . For n=3, Diaconis and Saloff-Coste (Ann. Appl. Probab. 6 (1996) 695) showed that . For n=4, the fact that follows from n=2 by tensorization.) Based on an idea that goes back to Rothaus (J. Funct. Anal. 39 (1980) 42; 42 (1981) 110), we prove that if n?4 is even, then the log-Sobolev constant and the spectral gap satisfy . This implies that when n is even and n?4.     

A priori estimate for a family of semi-linear elliptic equations with critical nonlinearity

We consider positive solutions of on B₁ (n?5) where μ and K>0 are smooth functions on B₁. If K is very sub-harmonic at each critical point of K in B2/3 and the maximum of u in is comparable to its maximum over , then all positive solutions are uniformly bounded on . As an application, a priori estimate for solutions of equations defined on Sn is derived.     

Statistical inference of minimum BD estimators and classifiers for varying-dimensional models

Stochastic modeling for large-scale datasets usually involves a varying-dimensional model space. This paper investigates the asymptotic properties, when the number of parameters grows with the available sample size, of the minimum- estimators and classifiers under a broad and important class of Bregman divergence (), which encompasses nearly all of the commonly used loss functions in the regression analysis, classification procedures and machine learning literature. Unlike the maximum likelihood estimators which require the joint likelihood of observations, the minimum-BD estimators are useful for a range of models where the joint likelihood is unavailable or incomplete. Statistical inference tools developed for the class of large dimensional minimum- estimators and related classifiers are evaluated via simulation studies, and are illustrated by analysis of a real dataset.     

A bounded property for gradients of diffusion semigroups on Euclidean spaces

We consider uniformly elliptic diffusion processes X(t,x) on Euclidean spaces , with some conditions in terms of the drift term (see assumptions A2 and A3). By using interpolation theory, we show a bounded property which gives an estimate of involving |x| and but not ||∇f||_∞, and a power of smaller than 1.     

The moment problem on the Wiener space

Consider an L¹-continuous functional ? on the vector space of polynomials of Brownian motion at given times, suppose ? commutes with the quadratic variation in a natural sense, and consider a finite set of polynomials of Brownian motion at rational times, , mapping the Wiener space to R.In the spirit of Schmüdgen's solution to the finite-dimensional moment problem, we give sufficient conditions under which ? can be written in the form ∫⋅dμ for some probability measure μ on the Wiener space such that μ-almost surely, all the random variables are nonnegative.