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.

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.

On Unsymmetric Dirichlet Forms

1973 ◽  
Vol 25 (2) ◽  
pp. 252-260 ◽  
Author(s):  
Joanne Elliott
Keyword(s):  
Compact Space ◽  
Measure Space ◽  
Closed Subspace ◽  
Finite Measure ◽  
Dirichlet Forms ◽  
Locally Compact ◽  
Locally Compact Space ◽  
Finite Measure Space

Let F be a linear, but not necessarily closed, subspace of L2[X, dm], where (X,,m) is a σ-finite measure space with the Borel subsets of the locally compact space X. If u and v are measureable functions, then v is called a normalized contraction of u if and Assume that F is stable under normalized contractions, that is, if u ∈ F and v is a normalized contraction of u, then v ∈ F.

scholarly journals Essential supremum norm differentiability

1985 ◽  
Vol 8 (3) ◽  
pp. 433-439
Author(s):  
I. E. Leonard ◽  
K. F. Taylor
Keyword(s):  
Banach Space ◽  
Measure Space ◽  
Real Banach Space ◽  
Finite Measure ◽  
Linear Operators ◽  
Supremum Norm ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Fréchet Differentiability ◽  
Finite Measure Space

The points of Gateaux and Fréchet differentiability inL∞(μ,X)are obtained, where(Ω,∑,μ)is a finite measure space andXis a real Banach space. An application of these results is given to the spaceB(L1(μ,ℝ),X)of all bounded linear operators fromL1(μ,ℝ)intoX.

scholarly journals Integration in Orlicz-Bochner Spaces

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Marian Nowak
Keyword(s):  
Integral Representation ◽  
Measure Space ◽  
Finite Measure ◽  
Linear Operators ◽  
Continuous Linear ◽  
Topological Properties ◽  
Young Function ◽  
Finite Measure Space ◽  
Bochner Space ◽  
Continuous Linear Operators

Let (Ω,Σ,μ) be a complete σ-finite measure space, φ be a Young function, and X and Y be Banach spaces. Let Lφ(X) denote the Orlicz-Bochner space, and Tφ∧ denote the finest Lebesgue topology on Lφ(X). We study the problem of integral representation of (Tφ∧,·Y)-continuous linear operators T:Lφ(X)→Y with respect to the representing operator-valued measures. The relationships between (Tφ∧,·Y)-continuous linear operators T:Lφ(X)→Y and the topological properties of their representing operator measures are established.

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.

A Multiple Sequence Ergodic Theorem

1983 ◽  
Vol 26 (4) ◽  
pp. 493-497 ◽  
Author(s):  
James H. Olsen
Keyword(s):  
Ergodic Theorem ◽  
Measure Space ◽  
Finite Measure ◽  
Linear Operators ◽  
Multiple Sequence ◽  
Finite Measure Space ◽  
Image Position

AbstractLet be a σ-finite measure space, {T1, …, Tk} a set of linear operators of , some p, 1≤p≤∞.Ifexists a.e. for all f ∊ Lp, we say that the multiple sequence ergodic theorem holds for {T1, …, Tk}. If f≥0 implies Tf≥0, we say that T is positive. If there exists an operator S such that |Tf(x)|≥S |f|(x) a.e., we say that T is dominated by S. In this paper we prove that if T1, …, Tk are dominated by positive contractions of , p fixed, 1<p<∞, then the multiple sequence ergodic theorem holds for {T1, …, Tk}.

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.

Korovkin Theorems for Integral Operators with Kernels of Finite Oscillation

1974 ◽  
Vol 26 (6) ◽  
pp. 1390-1404 ◽  
Author(s):  
M. J. Marsden ◽  
S. D. Riemenschneider
Keyword(s):  
Banach Space ◽  
Banach Lattice ◽  
Measure Space ◽  
Integral Operators ◽  
Finite Measure ◽  
Bounded Sequence ◽  
Positive Operators ◽  
Linear Operators ◽  
Finite Measure Space

There has been considerable interest recently in the investigation of "Korovkin sets". Briefly, for X a Banach space and a family of linear operators on X, a subset K ⊂ X is a Korovkin set relative to if for any bounded sequence {Tn} ⊂ , Tnk → k in X for each k ∊ K implies Tnx → x for each x ∊ X. A large portion of these investigations have been carried out for X being one of the spaces C(S), S compact Hausdorff, the usual Lp spaces of functions on some finite measure space, or some Banach lattice; while is one of the classes +-positive operators, 1-contractions (i.e., ||T|| 1), or + ⋂1

On the Integrability of the Ergodic Hilbert Transform for A Class of Functions with Equal Absolute Values

1998 ◽  
Vol 5 (2) ◽  
pp. 101-106
Author(s):  
L. Ephremidze
Keyword(s):  
Hilbert Transform ◽  
Measure Space ◽  
Integrable Function ◽  
Finite Measure ◽  
Ergodic Transformation ◽  
Finite Measure Space ◽  
Class Of Functions

Abstract It is proved that for an arbitrary non-atomic finite measure space with a measure-preserving ergodic transformation there exists an integrable function f such that the ergodic Hilbert transform of any function equal in absolute values to f is non-integrable.

Lacunary ideal convergence in measure for sequences of fuzzy valued functions

2021 ◽  
Vol 40 (3) ◽  
pp. 5517-5526
Author(s):  
Ömer Kişi
Keyword(s):  
Measure Space ◽  
Finite Measure ◽  
Ideal Convergence ◽  
Convergence In Measure ◽  
Ideal Form ◽  
Measurable Functions ◽  
Finite Measure Space ◽  
Convergence Of Sequences

We investigate the concepts of pointwise and uniform I θ -convergence and type of convergence lying between mentioned convergence methods, that is, equi-ideally lacunary convergence of sequences of fuzzy valued functions and acquire several results. We give the lacunary ideal form of Egorov’s theorem for sequences of fuzzy valued measurable functions defined on a finite measure space ( X , M , μ ) . We also introduce the concept of I θ -convergence in measure for sequences of fuzzy valued functions and proved some significant results.

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