Reflected Dirichlet Spaces

2011 ◽  
Author(s):  
Zhen-Qing Chen ◽  
Masatoshi Fukushima
Keyword(s):  
Dirichlet Form ◽  
Harmonic Functions ◽  
Dirichlet Space ◽  
Random Variables ◽  
Finite Energy ◽  
Dirichlet Forms ◽  
Dirichlet Spaces ◽  
Hunt Process ◽  
Equivalent Definition ◽  
Definition Of

This chapter turns to reflected Dirichlet spaces. It first introduces the notion of terminal random variables and harmonic functions of finite energy for a Hunt process associated with a transient regular Dirichlet form. The chapter next establishes several equivalent notions of reflected Dirichlet space (ℰ ref,ℱ ref) for a regular transient Dirichlet form (E,F). One of these equivalent notions is then used to define reflected Dirichlet space for a regular recurrent Dirichlet form. Moreover, the chapter gives yet another equivalent definition of reflected Dirichlet space that is invariant under quasi-homeomorphism of Dirichlet forms. Various concrete examples of reflected Dirichlet spaces are also exhibited for regular Dirichlet forms. Finally, the chapter defines a Silverstein extension of a quasi-regular Dirichlet form (E,F) on L²(E; m) and investigates the equivalence of analytic and probabilistic concepts of harmonicity.

Basic Properties and Examples of Dirichlet Forms

2011 ◽  
Author(s):  
Zhen-Qing Chen ◽  
Masatoshi Fukushima
Keyword(s):  
Markov Process ◽  
Potential Theory ◽  
Dirichlet Form ◽  
Dirichlet Space ◽  
Dirichlet Forms ◽  
Dirichlet Spaces ◽  
Basic Properties ◽  
Time Changes ◽  
Symmetric Operators ◽  
Analytic Potential

This chapter introduces the concepts of the transience, recurrence, and irreducibility of the semigroup for general Markovian symmetric operators and presents their characterizations by means of the associated Dirichlet form as well as the associated extended Dirichlet space. These notions are invariant under the time changes of the associated Markov process. The chapter then presents some basic examples of Dirichlet forms, with special attention paid to their basic properties as well as explicit expressions of the corresponding extended Dirichlet spaces. Hereafter the chapter discusses the analytic potential theory for regular Dirichlet forms, and presents some conditions for the demonstrated Dirichlet form (E,F) to be local.

Symmetric Markovian Semigroups and Dirichlet Forms

2011 ◽  
Author(s):  
Zhen-Qing Chen ◽  
Masatoshi Fukushima
Keyword(s):  
Markov Process ◽  
Dirichlet Form ◽  
Measure Space ◽  
Dirichlet Space ◽  
Finite Measure ◽  
Dirichlet Forms ◽  
Linear Operators ◽  
Dirichlet Spaces ◽  
Finite Measure Space ◽  
Markovian Semigroups

This chapter studies the concepts of Dirichlet form and Dirichlet space by first working with a σ‎-finite measure space (E,B(E),m) without any topological assumption on E and establish the correspondence of the above-mentioned notions to the semigroups of symmetric Markovian linear operators. Later on the chapter assumes that E is a Hausdorff topological space and considers the semigroups and Dirichlet forms generated by symmetric Markovian transition kernels on E. The chapter also considers quasi-regular Dirichlet forms and the quasi-homeomorphism of Dirichlet spaces. From here, the chapter shows that there is a nice Markov process called an m-tight special Borel standard process associated with every quasi-regular Dirichlet form.

scholarly journals Ergodic Decomposition of Dirichlet Forms via Direct Integrals and Applications

2021 ◽  
Author(s):  
Lorenzo Dello Schiavo
Keyword(s):  
Radon Measure ◽  
Measure Space ◽  
Probability Space ◽  
Dirichlet Space ◽  
Dirichlet Forms ◽  
Dirichlet Spaces ◽  
Ergodic Decomposition ◽  
Direct Integral ◽  
Unique Representation ◽  
Carré Du Champ

AbstractWe study direct integrals of quadratic and Dirichlet forms. We show that each quasi-regular Dirichlet space over a probability space admits a unique representation as a direct integral of irreducible Dirichlet spaces, quasi-regular for the same underlying topology. The same holds for each quasi-regular strongly local Dirichlet space over a metrizable Luzin σ-finite Radon measure space, and admitting carré du champ operator. In this case, the representation is only projectively unique.

Symmetric Hunt Processes and Regular Dirichlet Forms

2011 ◽  
Author(s):  
Zhen-Qing Chen ◽  
Masatoshi Fukushima
Keyword(s):  
Dirichlet Form ◽  
Dirichlet Forms ◽  
Hunt Process ◽  
Numerical Function ◽  
Locally Compact ◽  
Positive Radon Measure ◽  
The One ◽  
One Point Compactification ◽  
Separable Metric Space

This chapter studies a symmetric Hunt process associated with a regular Dirichlet form. Without loss of generality, the majority of the chapter assumes that E is a locally compact separable metric space, m is a positive Radon measure on E with supp[m] = E, and X = (Xₜ, Pₓ) is an m-symmetric Hunt process on (E,B(E)) whose Dirichlet form (E,F) is regular on L²(E; m). It adopts without any specific notices those potential theoretic terminologies and notations that are formulated in the previous chapter for the regular Dirichlet form (E,F). Furthermore, throughout this chapter, the convention that any numerical function on E is extended to the one-point compactification E ∂ = E ∪ {∂} by setting its value at δ‎ to be zero is adopted.

Dirichlet forms on partial *-algebras

1988 ◽  
Vol 104 (1) ◽  
pp. 129-140 ◽  
Author(s):  
G. O. S. Ekhaguere
Keyword(s):  
Markov Processes ◽  
Dirichlet Form ◽  
Dirichlet Forms ◽  
Function Spaces ◽  
Hunt Process ◽  
Partial Algebras ◽  
Markov Fields ◽  
Associated Function ◽  
Number Of Authors

Dirichlet forms and their associated function spaces have been studied by a number of authors [4, 6, 7, 12, 15–18, 22, 25, 26]. Important motivation for the study has been the connection of Dirichlet forms with Markov processes [16–18, 25, 26]: for example, to every regular symmetric Dirichlet form, there is an associated Hunt process [13, 20]. This makes the theory of Dirichlet forms a convenient source of examples of Hunt processes. In the non-commutative setting, Markov fields have been studied by several authors [1–3, 14, 19, 24, 28]. It is therefore interesting to develop a non-commutative extension of the theory of Dirichlet forms and to study their connection with non-commutative Markov processes.

scholarly journals EXISTENCE AND APPROXIMATION OF HUNT PROCESSES ASSOCIATED WITH GENERALIZED DIRICHLET FORMS

2011 ◽  
Vol 14 (04) ◽  
pp. 613-628 ◽  
Author(s):  
VITALI PEIL ◽  
GERALD TRUTNAU
Keyword(s):  
Dirichlet Form ◽  
Structural Condition ◽  
Dirichlet Forms ◽  
Poisson Processes ◽  
Hunt Process ◽  
Generalized Dirichlet Forms ◽  
Generalized Dirichlet

We show that any strictly quasi-regular generalized Dirichlet form that satisfies the mild structural condition D3 is associated to a Hunt process, and that the associated Hunt process can be approximated by a sequence of multivariate Poisson processes. This also gives a new proof for the existence of a Hunt process associated to a strictly quasi-regular generalized Dirichlet form that satisfies SD3 and extends all previous results.

ON HUNT PROCESSES AND STRICT CAPACITIES ASSOCIATED WITH GENERALIZED DIRICHLET FORMS

2005 ◽  
Vol 08 (03) ◽  
pp. 357-382 ◽  
Author(s):  
GERALD TRUTNAU
Keyword(s):  
Dirichlet Form ◽  
Sufficient Conditions ◽  
Dirichlet Forms ◽  
Hunt Process ◽  
Capacity Zero ◽  
Smooth Measures ◽  
Generalized Dirichlet Forms ◽  
Necessary And Sufficient ◽  
Borel Measurable ◽  
Generalized Dirichlet

Introducing the corresponding strict capacity, we give necessary and sufficient conditions for a generalized Dirichlet form to be associated with a Hunt process. We also show that Borel measurable sets with strict capacity zero can be checked-out by an appropriate subclass of smooth measures. In the last part of this paper we present applications to three classes of examples.

Remarks on some generalization of the notion of microscopic sets

2020 ◽  
Vol 70 (6) ◽  
pp. 1349-1356
Author(s):  
Aleksandra Karasińska
Keyword(s):  
General Sense ◽  
Equivalent Definition ◽  
Definition Of ◽  
Null Set

AbstractWe consider properties of defined earlier families of sets which are microscopic (small) in some sense. An equivalent definition of considered families is given, which is helpful in simplifying a proof of the fact that each Lebesgue null set belongs to one of these families. It is shown that families of sets microscopic in more general sense have properties analogous to the properties of the σ-ideal of classic microscopic sets.

Sign in / Sign up

Export Citation Format

Share Document