NONTRIVIAL SOLUTIONS FOR SEMILINEAR DIRICHLET FORMS VIA MORSE THEORY

Using variational methods and Morse theory, we obtain some existence results of multiple solutions for certain semilinear problems associated with general Dirichlet forms.

A NOTE ON PERTURBATION OF NON-SYMMETRIC DIRICHLET FORMS BY SIGNED SMOOTH MEASURES

This article discusses the perturbation of a non-symmetric Dirichlet form, （ε, D（ε））, by a signed smooth measure u, where u=u1 -u2 with u1 and u2 being smooth measures. It gives a sufficient condition for the perturbed form （ε^u ,D（ε^u）） （for some a0 ≥ 0） to be a coercive closed form.

The characterization of a class of quantum Markov semigroups and the associated operator-valued Dirichlet forms based on Hilbert C~*-module l_2(A)

We characterize A-linear symmetric and contraction module operator semigroup{Tt}t∈R+L(l2(A)),where A is a finite-dimensional C-algebra,and L(l2(A))is the C-algebra of all adjointable module maps on l2(A).Next,we introduce the concept of operator-valued quadratic forms,and give a one to one correspondence between the set of non-positive definite self-adjoint regular module operators on l2(A)and the set of non-negative densely defined A-valued quadratic forms.In the end,we obtain that a real and strongly continuous symmetric semigroup{Tt}t∈R+L(l2(A))being Markovian if and only if the associated closed densely defined A-valued quadratic form is a Dirichlet form.

The Characterization of a Class of Quantum Markov Semigroups and the Associated Operator-Valued Dirichlet Forms Based on Hilbert W＊-Module

In this paper, we introduce the concept of operator-valued quadratic form based on Hilbert W＊-module l2 A, and give a one to one correspondence between the set of positive self-adjoint regular module operators on l2 A and the set of regular quadratic forms, where A is a finite and a-finite von Neumann algebra. Furthermore, we obtain that a strict continuous symmetric regular module operator semigroup （Tt）t∈R＋ C L（l2 A） is Markovian if and only if the associated A-valued quadratic form is a Dirichlet form, where L（l2 A） is the yon Neumann algebra of all adjointable module maps on l2 A.

Stochastic calculus for Markov processes associated with non-symmetric Dirichlet forms

Nakao＇s stochastic integrals for continuous additive functionals of zero energy are extended from the symmetric Dirichlet forms setting to the non-symmetric Dirichlet forms setting. ItS＇s formula in terms of the extended stochastic integrals is obtained.

STRONG SUBORDINATION OF SYMMETRIC DIRICHLET FORMS

The author introduces a notion of subordination for symmetric Dirichlet forms and proves that the subordination is actually equivalent to the killing transformation by multiplicative functionals in the theory of symmetric Markov processes. This also gives a way to characterize bivariate smooth measures.

Quasi-Homeomorphisms and Measures of Finite Energy Integrals of Generalized Dirichlet Forms

We introduce the quasi-homeomorphisms of generalized Dirichlet forms and prove that any quasi-regular generalized Dirichlet form is quasi-homeomorphic to a semi-regular generalized Dirichlet form. Moreover. we apply this quasi-homeomorphism method to study the measures of finite energy integrals of generalized Dirichlet forms. We show that any 1-coexcessive function which is dominated by a function in is associated with a measure of finite energy integral. Consequently, we prove that a Borel set B is-exceptional if and only if μ(B)=0 for any measure μ of finite energy integral.

MOSCO CONVERGENCE OFQUASI-REGULAR DIRICHLET FORMS

A sufficient condition for the Mosco limit of a sequence of quasi-regular Dirichlet forms to be quasi-regular is given. In particular, a Dirichlet form is a quasi-regular Dirchlet form if and only if its Yosida approximation sequency satisfies the conditon. Furthermore, conditions for the Mosco limit of a sequence of symmetric (strictly strong) local quasi-regular Dirichlet forms to be (strictly strong) local are also presented. This paper extends the results of [1] from regular Dirichlet space to quasi-regular Dirichlet space.

TRANSFORMATION OF SYMMETRIC MULTIPLICATIVE FUNCTIONALS

In the present paper the transformation of symmetric Markov processes by symmetric martingale multiplicative functionals is studied and the corresponding Dirichlet form is formulated.

LOWER INEQUALITIES OF HEAT SEMIGROUPS BY USING PARABOLIC MAXIMUM PRINCIPLE

Using parabolic maximum principle, we apply the analytic method to obtain lower comparison inequalities for non-negative weak supersolutions of the heat equation associated with a regular strongly p-local Dirichlet form on the abstract metric measure space. As an application we obtain lower estimates for heat kernels on some Riemannian manifolds.

HARMONIC MAPPINGS OF THE HEXAGASKET TO THE CIRCLE

Harmonic mappings from the hexagasket to the circle are described in terms of boundary values and topological data. Explicit formulas are also given for the energy of the mapping. We have generalized the results in [10].

The α-orthogonal complements of regular Dirichlet subspaces for one-dimensional Brownian motion

Roughly speaking a regular Dirichlet subspace of a Dirichlet form is a subspace which is also a regular Dirichlet form on the same state space. In particular, the domain of regular Dirichlet subspace is a closed subspace of the Hilbert space induced by the domain and a-inner product of original Dirichlet form. We investigate the orthogonal complement of regular Dirichlet subspace for one-dimensional Brownian motion in this paper. Our main results indicate that this orthogonal complement has a very close connection with the a-harmonic equation under Neumann type condition.

Algebraic Structure on Dirichlet Spaces

In this short note, we shall give a few equivalent conditions for a closed form to be Markovian, and prove that the closure of a sub-algebra of bounded functions in a Dirichlet space must be Markovian. We also study the regular representation of Dirichlet spaces and the classification of Dirichlet subspaces.

The Dirichlet Form of a Gradient-type Drift Transformation of a Symmetric Diffusion

In the context of a symmetric diffusion process X, we give a precise description of the Dirichlet form of the process obtained by subjecting X to a drift transformation of gradient type. This description relies on boundary-type conditions restricting an associated reflecting Dirichlet form.

The Heat Equation for the Dirichlet Fractional Laplacian with Negative Potentials:Existence and Blow-up of Nonnegative Solutions

We establish conditions ensuring either existence or blow-up of nonnegative solutions for the heat equation generated by the Dirichlet fractional Laplacian perturbed by negative potentials on bounded sets. The elaborated theory is supplied by some examples.

Weak convergence of Dirichlet processes

Weak convergence of Markov processes is studied by means of Dirichlet forms and two theorems for weak convergence of Hunt processes on general metric spaces are established.As applications,examples for weak convergence of symmetric or non symmetric Dirichlet processes on finite and infinite spaces are given.

Dirichlet Form of Product of Variational Fractals

Much effort has gone into constructing Dirichlet forms to define Laplacians on self-similar sets. However, the results have only been successful on p.c.f. (post critical finite) fractals. We prove the existence of a Dirichlet form on a class of non- p.c.f. sets that are the product of variational fractals.

Stability of elliptic Harnack inequalities Dedicated to Professor Jia-an Yan on the Occasion of His 80th Birthday

We survey some recent progress in the study of stability of elliptic Harnack inequalities under form-bounded perturbations for strongly local Dirichlet forms on complete locally compact separable metric spaces.

Convergence of Symmetric Diffusions on Wiener Spaces

In this paper, we study the distorted Ornstein-Uhlenbeck processes associated with given densities on an abstract Wiener space. We prove that the laws of distorted Ornstein-Uhlenbeck processes converge in total variation norm if the densities converge in Sobolev space .

THE FEYNMAN-KAC FORMULA FOR SYMMETRIC MARKOV PROCESSES

LetXbe an m s ymmetric Markov process andMa multiplicative functional ofXsuch that theMsubprocess ofXis alsom-symmetric.The author characterizes the Dirichlet form associated with the subprocess in terms of that associated withXand the bivariate Revuz measure ofM.