一类 Markov 模算子半群与相应的算子值 Dirichlet 型刻画

本文基于Ⅱ_1-型因子把非交换对称Dirichlet型理论推广到算子值情形.在此框架下建立了算子值Dirichlet型,Markov模算子半群及Markov预解集之间的一一对应关系.     

Small time asymptotics of diffusion processes

We establish the short-time asymptotic behaviour of the Markovian semigroups associated with strongly local Dirichlet forms under very general hypotheses. Our results apply to a wide class of strongly elliptic, subelliptic and degenerate elliptic operators. In the degenerate case the asymptotics incorporate possible non-ergodicity.     

An Integrated Version of Varadhan’s Asymptotics for Lower-Order Perturbations of Strong Local Dirichlet Forms

The studies of J. A. Ramírez, Hino–Ramírez, and Ariyoshi–Hino showed that an integrated version of Varadhan’s asymptotics holds for Markovian semigroups associated with arbitrary strong local symmetric Dirichlet forms. In this paper, we consider non-symmetric bilinear forms that are the sum of strong local symmetric Dirichlet forms and lower-order perturbed terms. We give sufficient conditions for the associated semigroups to have asymptotics of the same type.     

Uniqueness of Dirichlet forms associated with systems of infinitely many Brownian balls in ℝ d

Summary. Dirichlet forms associated with systems of infinitely many Brownian balls in ℝ ^d are studied. Introducing a linear operator L ₀ defined on a space of smooth local functions, we show the uniqueness of Dirichlet forms associated with self adjoint Markovian extensions of L ₀. We also discuss the ergodicity of the reversible process associated with the Dirichlet form. Received: 18 July 1996/In revised form: 13 February 1997     

Markovian extensions of symmetric second order elliptic differential operators

Let be bounded with a smooth boundary Γ and let S be the symmetric operator in given by the minimal realization of a second order elliptic differential operator. We give a complete classification of the Markovian self‐adjoint extensions of S by providing an explicit one‐to‐one correspondence between such extensions and the class of Dirichlet forms in which are additively decomposable by the bilinear form of the Dirichlet‐to‐Neumann operator plus a Markovian form. By such a result two further equivalent classifications are provided: the first one is expressed in terms of an additive decomposition of the bilinear forms associated to the extensions, the second one uses the additive decomposition of the resolvents provided by Kre?n's formula. The Markovian part of the decomposition allows to characterize the operator domain of the corresponding extension in terms of Wentzell‐type boundary conditions. Some properties of the extensions, and of the corresponding Dirichlet forms, semigroups and heat kernels, like locality, regularity, irreducibility, recurrence, transience, ultracontractivity and Gaussian bounds are also discussed.     

On the Existence of Feller Semigroups with Discontinuous Coefficients

This paper is devoted to the functional analytic approach to the problem of the existence of Markov processes in probability theory. More precisely, we construct Feller semigroups with Dirichlet conditions for second-order, uniformly elliptic integro-differential operators with discontinuous coefficients. In other words, we prove that there exists a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space until it dies at the time when it reaches the boundary.     

Derivations as square roots of Dirichlet forms

The aim of this work is to analyze the structure of a tracially symmetric Dirichlet form on a -algebra, in terms of a killing weight and a closable derivation taking values in a Hilbert space with a bimodule structure. It is shown that the generator of the associate Markovian semigroup always appears, in a natural way, as the divergence of a closable derivation. Applications are shown to the decomposition of Dirichlet forms and to the construction of differential calculus on metric spaces.     

Gradient estimates for diffusion semigroups with singular coefficients

Uniform gradient estimates are derived for diffusion semigroups, possibly with potential, generated by second order elliptic operators having irregular and unbounded coefficients. We first consider the Rd-case, by using the coupling method. Due to the singularity of the coefficients, the coupling process we construct is not strongly Markovian, so that additional difficulties arise in the study. Then, more generally, we treat the case of a possibly unbounded smooth domain of Rd with Dirichlet boundary conditions. We stress that the resulting estimates are new even in the Rd-case and that the coefficients can be Hölder continuous. Our results also imply a new Liouville theorem for space-time bounded harmonic functions with respect to the underlying diffusion semigroup.     

On Quadratic g-Evaluations/Expectations and Related Analysis

This work is concerned with coupling and exponential convergence rate for a class of Markovian switching jump-diffusion processes. The processes under consideration can be thought of as a number of jump-diffusion processes modulated by a Markovian switching device. For this class of processes, we construct some order-preserving couplings. Furthermore, by virtue of the coupling results, we also provide an estimate of exponential convergence rate for the Markovian switching jump-diffusion processes without Gaussian noise.     

Mosco convergence of quasi-regular dirichlet forms

1.IntroductionactEbeaHa~tOPologicalspaceandmamdritemeasureonitsBorela-algebraB(E).In[11,MOSCointroducedthefollowingnow-calledMoscoconvergenceofsyllUnetricac~l~:tlf621i.t,..2.1].AsequenceofsynUnetricbilinearforms(En,D(en)),nEN,issaidtoMO8coconvergetoa...