scholarly journals Dirichlet Forms Constructed from Annihilation Operators on Bernoulli Functionals

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Caishi Wang ◽  
Beiping Wang
Keyword(s):  
Dirichlet Form ◽  
Nonnegative Function ◽  
Mathematical Structure ◽  
Dirichlet Forms ◽  
Symmetric Bilinear Form ◽  
Dense Subspace ◽  
Markov Semigroup ◽  
Semigroup Of Operators ◽  
Canonical Anticommutation Relation ◽  
Adjoint Operators

The annihilation operators on Bernoulli functionals (Bernoulli annihilators, for short) and their adjoint operators satisfy a canonical anticommutation relation (CAR) in equal-time. As a mathematical structure, Dirichlet forms play an important role in many fields in mathematical physics. In this paper, we apply the Bernoulli annihilators to constructing Dirichlet forms on Bernoulli functionals. Let w be a nonnegative function on N. By using the Bernoulli annihilators, we first define in a dense subspace of L2-space of Bernoulli functionals a positive, symmetric, bilinear form Ew associated with w. And then we prove that Ew is closed and has the contraction property; hence, it is a Dirichlet form. Finally, we consider an interesting semigroup of operators associated with w on L2-space of Bernoulli functionals, which we call the w-Ornstein-Uhlenbeck semigroup, and, by using the Dirichlet form, Ew we show that the w-Ornstein-Uhlenbeck semigroup is a Markov semigroup.

Equivalence between logarithmic Sobolev inequality and hypercontractivity in a probability gage space

2020 ◽  
pp. 1-13
Author(s):  
Zhang Lunchuan
Keyword(s):  
Dirichlet Form ◽  
Sobolev Inequality ◽  
Logarithmic Sobolev Inequality ◽  
Markov Semigroup ◽  
Quantum Markov Semigroup

Abstract In this paper, we prove the equivalence between logarithmic Sobolev inequality and hypercontractivity of a class of quantum Markov semigroup and its associated Dirichlet form based on a probability gage space.

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.

Spectral Multipliers for Laplacians Associated with some Dirichlet Forms

2008 ◽  
Vol 51 (3) ◽  
pp. 581-607 ◽  
Author(s):  
Andrea Carbonaro ◽  
Giancarlo Mauceri ◽  
Stefano Meda
Keyword(s):  
Lebesgue Measure ◽  
Essential Spectrum ◽  
Dirichlet Form ◽  
Adjoint Operator ◽  
Dirichlet Forms ◽  
The Self ◽  
Spectral Multipliers ◽  
Curvature Condition ◽  
Image Position ◽  
Conditions At Infinity

AbstractLet be the self-adjoint operator associated with the Dirichlet formwhere ϕ is a positive C2 function, dλϕ = ϕdλ and λ denotes Lebesgue measure on ℝd. We study the boundedness on Lp(λϕ) of spectral multipliers of . We prove that if ϕ grows or decays at most exponentially at infinity and satisfies a suitable ‘curvature condition’, then functions which are bounded and holomorphic in the intersection of a parabolic region and a sector and satisfy Mihlin-type conditions at infinity are spectral multipliers of Lp(λϕ). The parabolic region depends on ϕ, on p and on the infimum of the essential spectrum of the operator on L2(λϕ). The sector depends on the angle of holomorphy of the semigroup generated by on Lp(λϕ).

scholarly journals Quasi-homeomorphisms of Dirichlet forms

1994 ◽  
Vol 136 ◽  
pp. 1-15 ◽  
Author(s):  
Zhen-Qing Chen ◽  
Zhi-Ming Ma ◽  
Michael Röckner
Keyword(s):  
Dirichlet Form ◽  
Dirichlet Forms ◽  
European Symposium ◽  
Locally Compact ◽  
The Third ◽  
Gelfand Transform ◽  
General State Space ◽  
Irregular Systems ◽  
Separable Metric Space ◽  
Mathematics And Physics

Extending fundamental work of M. Fukushima, M. L. Silverstein, S. Carrillo Menendez, and Y. Le Jan (cf. [F71a, 80], [Si74], [Ca-Me75], [Le77]) it was recently discovered that there is a one-to-one correspondence between (equivalent classes of) all pairs of sectorial right processes and quasi-regular Dirichlet forms (see [AM91], [AM92], [AMR90], [AMR92a], [AMR92b], [MR92]). Based on the potential theory for quasi-regular Dirichlet forms, it was shown that any quasi-regular Dirichlet form on a general state space can be considered as a regular Dirichlet form on a locally compact separable metric space by “local compactification”. There are several ways to implement this local compactification. One relies on h-transformation which was mentioned in [AMR90, Remark 1.4]. A direct way using a modified Ray-Knight compactification was announced on the “5th French-German meeting: Bielefeld Encounters in Mathematics and Physics IX. Dynamics in Complex and Irregular Systems”, Bielefeld, December 16 to 21, 1991, and the “Third European Symposium on Analysis and Probability”, Paris, January 3-10, 1992, and appeared in [MR92, Chap. VI] and [AMR92b] (see also the proof of Theorem 3.7 below). One can also do this by Gelfand-transform. This way was found by the first named author independetly and announced in the “12th Seminar on Stochastic Processes”, Seattle, March 26-28, 1992. It will be discussed in Section 4.

scholarly journals The Davies method revisited for heat kernel upper bounds of regular Dirichlet forms on metric measure spaces

2018 ◽  
Vol 30 (5) ◽  
pp. 1129-1155 ◽  
Author(s):  
Jiaxin Hu ◽  
Xuliang Li
Keyword(s):  
Heat Kernel ◽  
Upper Bound ◽  
Dirichlet Form ◽  
Measure Space ◽  
Dirichlet Forms ◽  
Upper Bounds ◽  
Metric Measure Spaces ◽  
Unified Approach ◽  
Measure Spaces ◽  
Non Local

AbstractWe apply the Davies method to prove that for any regular Dirichlet form on a metric measure space, an off-diagonal stable-like upper bound of the heat kernel is equivalent to the conjunction of the on-diagonal upper bound, a cutoff inequality on any two concentric balls, and the jump kernel upper bound, for any walk dimension. If in addition the jump kernel vanishes, that is, if the Dirichlet form is strongly local, we obtain a sub-Gaussian upper bound. This gives a unified approach to obtaining heat kernel upper bounds for both the non-local and the local Dirichlet forms.

Upper estimates of heat kernels for non-local Dirichlet forms on doubling spaces

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiaxin Hu ◽  
Guanhua Liu
Keyword(s):  
Dirichlet Form ◽  
Survival Function ◽  
Dirichlet Forms ◽  
Mean Value ◽  
Heat Kernels ◽  
Heat Semigroup ◽  
New Approach ◽  
Mean Value Inequality ◽  
Non Local ◽  
Upper Estimate

Abstract In this paper, we present a new approach to obtaining the off-diagonal upper estimate of the heat kernel for any regular Dirichlet form without a killing part on the doubling space. One of the novelties is that we have obtained the weighted L 2 {L^{2}} -norm estimate of the survival function 1 - P t B ⁢ 1 B {1-P_{t}^{B}1_{B}} for any metric ball B, which yields a nice tail estimate of the heat semigroup associated with the Dirichlet form. The parabolic L 2 {L^{2}} mean-value inequality is borrowed to use.

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.

LÉVY–DIRICHLET FORMS

2006 ◽  
Vol 09 (03) ◽  
pp. 435-449
Author(s):  
S. ALBEVERIO ◽  
Ya. BELOPOLSKAYA ◽  
M. FELLER
Keyword(s):  
Markov Process ◽  
Laplace Operator ◽  
Dirichlet Form ◽  
Dirichlet Forms ◽  
Infinite Dimensional ◽  
Natural Connection

We construct a Dirichlet form associated with the infinite dimensional Lévy–Laplace operator. We show that there exists a natural connection between this form and a Markov process.

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