On Gauss-Weierstrass semigroups and inversion of potentials associated with the Bessel differential operator

In this paper, we define the generalized Gauss Weierstrass semigroups with Weierstrass kernel, and give some of their properties. Using them, we study the inversion formulas for the generalized Riesz and Bessel potentials, generated by the generalized shift operators and associated with the Laplace Bessel differential operator.     

Nonlinear filtering of semi-Dirichlet processes

Herein, we consider the nonlinear filtering problem for general right continuous Markov processes, which are assumed to be associated with semi-Dirichlet forms. First, we derive the filtering equations in the semi-Dirichlet form setting. Then, we study the uniqueness of solutions of the filtering equations via the Wiener chaos expansions. Our results on the Wiener chaos expansions for nonlinear filters with possibly unbounded observation functions are novel and have their own interests. Furthermore, we investigate the absolute continuity of the filtering processes with respect to the reference measures and derive the density equations for the filtering processes.     

Optimal convergence rate for Yosida approximations of bounded holomorphic semigroups

In this paper, we obtain optimal bounds for convergence rate for Yosida approximations of bounded holomorphic semigroups. We also provide asymptotic expansions for semigroups in terms of Yosida approximations and obtain optimal error bounds for these expansions.     

Laguerre polynomials as Jensen polynomials of Laguerre-Pólya entire functions

We prove that the only Jensen polynomials associated with an entire function in the Laguerre-Pólya class that are orthogonal are the Laguerre polynomials.     

Neumann Bessel Heat Kernel Monotonicity

We prove that the diagonal of the transition probabilities for the d-dimensional Bessel processes on (0, 1], reflected at 1, which we denote by , is an increasing function of r for d > 2 and that this is false for d = 2. The first and third authors were supported in part by NSF grant # 0603701-DMS. The second author was supported in part by KBN Grant 1 P03A 020 28.     

Time-dependent perturbation theory for nonequilibrium lattice models

We develop a time-dependent perturbation theory for nonequilibrium interacting particle systems. We focus on models such as the contact process which evolve via destruction and autocatalytic creation of particles. At a critical value of the destruction rate there is a continuous phase transition between an active steady state and the vacuum state, which is absorbing. We present several methods for deriving series for the evolution starting from a single seed particle, including expansions for the ultimate survival probability in the super- and subcritical regions, expansions for the average number of particles in the subcritical region, and short-time expansions. Algorithms for computer generation of the various expansions are presented. Rather long series (24 terms or more) and precise estimates of critical parameters are presented.     

On the mean field theory bound on the magnetization

The mean field bound on magnetization is proved for a class of one-component ferromagnetic systems and forD components systems with arbitraryD.