Chapter One. Symmetric Markovian Semigroups and Dirichlet Forms

2011 ◽  
pp. 1-36
Keyword(s):  
Dirichlet Forms ◽  
Markovian Semigroups

DIRICHLET FORMS AND SYMMETRIC MARKOVIAN SEMIGROUPS ON ℤ2-GRADED VON NEUMANN ALGEBRAS

2003 ◽  
Vol 15 (08) ◽  
pp. 823-845 ◽  
Author(s):  
CHANGSOO BAHN ◽  
CHUL KI KO ◽  
YONG MOON PARK
Keyword(s):  
Von Neumann Algebras ◽  
Dirichlet Forms ◽  
Gauge Invariant ◽  
Von Neumann ◽  
Markovian Semigroups

We extend the construction of Dirichlet forms and symmetric Markovian semigroups on standard forms of von Neumann algebras given in [1] to the case of ℤ2-graded von Neumann algebras. As an application of the extension, we construct symmetric Markovian semigroups on CAR algebras with respect to gauge invariant quasi-free states and also investigate detailed properties such as ergodicity of the semigroups.

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.

CONSTRUCTION OF DIRICHLET FORMS ON STANDARD FORMS OF VON NEUMANN ALGEBRAS

2000 ◽  
Vol 03 (01) ◽  
pp. 1-14 ◽  
Author(s):  
YONG MOON PARK
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Admissible Function ◽  
Standard Form ◽  
Dirichlet Forms ◽  
Quantum Spin ◽  
Quantum Spin Systems ◽  
Translation Invariant ◽  
Von Neumann ◽  
Markovian Semigroups

For a von Neumann algebra ℳ acting on a Hilbert space ℋ with a cyclic and separating vector ξ0, we give an explicit expression of Dirichlet forms on the natural standard form [Formula: see text] associated with the pair (ℳ, ξ0). For any self-adjoint analytic element x of ℳ and an admissible function f, we construct a (bounded) Dirichlet form which generates a symmetric Markovian semigroup on ℋ. We then apply our result to construct translation invariant, symmetric, Markovian semigroups for quantum spin systems with finite range interactions.

scholarly journals Lower estimates of heat kernels for non-local Dirichlet forms on metric measure spaces

2017 ◽  
Vol 272 (8) ◽  
pp. 3311-3346 ◽  
Author(s):  
Alexander Grigor'yan ◽  
Eryan Hu ◽  
Jiaxin Hu
Keyword(s):  
Dirichlet Forms ◽  
Heat Kernels ◽  
Metric Measure Spaces ◽  
Measure Spaces ◽  
Non Local

scholarly journals ORLICZ–POINCARÉ INEQUALITIES

2008 ◽  
Vol 51 (2) ◽  
pp. 529-543 ◽  
Author(s):  
Feng-Yu Wang
Keyword(s):  
Porous Media ◽  
Probability Measure ◽  
Poincaré Inequality ◽  
Dirichlet Forms ◽  
Sobolev Inequalities ◽  
Young Function ◽  
Poincare Inequality ◽  
Poincaré Inequalities ◽  
Poincare Inequalities ◽  
Porous Media Equations

AbstractCorresponding to known results on Orlicz–Sobolev inequalities which are stronger than the Poincaré inequality, this paper studies the weaker Orlicz–Poincaré inequality. More precisely, for any Young function $\varPhi$ whose growth is slower than quadric, the Orlicz–Poincaré inequality$$ \|f\|_\varPhi^2\le C\E(f,f),\qquad\mu(f):=\int f\,\mathrm{d}\mu=0 $$is studied by using the well-developed weak Poincaré inequalities, where $\E$ is a conservative Dirichlet form on $L^2(\mu)$ for some probability measure $\mu$. In particular, criteria and concrete sharp examples of this inequality are presented for $\varPhi(r)=r^p$ $(p\in[1,2))$ and $\varPhi(r)= r^2\log^{-\delta}(\mathrm{e} +r^2)$ $(\delta>0)$. Concentration of measures and analogous results for non-conservative Dirichlet forms are also obtained. As an application, the convergence rate of porous media equations is described.

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