Two Remarks on the Wasserstein Dirichlet Form

2013 ◽  
pp. 235-255 ◽  
Author(s):  
Wilhelm Stannat
Keyword(s):  
Dirichlet Form

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.

scholarly journals CONSTRUCTION OF ELLIPTIC DIFFUSIONS WITH REFLECTING BOUNDARY CONDITION AND AN APPLICATION TO CONTINUOUS N-PARTICLE SYSTEMS WITH SINGULAR INTERACTIONS

2008 ◽  
Vol 51 (2) ◽  
pp. 337-362 ◽  
Author(s):  
Torben Fattler ◽  
Martin Grothaus
Keyword(s):  
Boundary Condition ◽  
Markov Process ◽  
Diffusion Process ◽  
Lebesgue Measure ◽  
Dirichlet Form ◽  
Measure Zero ◽  
Particle Systems ◽  
Lebesgue Measure Zero ◽  
Compact Set ◽  
Singular Interactions

AbstractWe give a Dirichlet form approach for the construction and analysis of elliptic diffusions in $\bar{\varOmega}\subset\mathbb{R}^n$ with reflecting boundary condition. The problem is formulated in an $L^2$-setting with respect to a reference measure $\mu$ on $\bar{\varOmega}$ having an integrable, $\mathrm{d} x$-almost everywhere (a.e.) positive density $\varrho$ with respect to the Lebesgue measure. The symmetric Dirichlet forms $(\mathcal{E}^{\varrho,a},D(\mathcal{E}^{\varrho,a}))$ we consider are the closure of the symmetric bilinear forms\begin{gather*} \mathcal{E}^{\varrho,a}(f,g)=\sum_{i,j=1}^n\int_{\varOmega}\partial_ifa_{ij} \partial_jg\,\mathrm{d}\mu,\quad f,g\in\mathcal{D}, \\ \mathcal{D}=\{f\in C(\bar{\varOmega})\mid f\in W^{1,1}_{\mathrm{loc}}(\varOmega),\ \mathcal{E}^{\varrho,a}(f,f)\lt\infty\}, \end{gather*}in $L^2(\bar{\varOmega},\mu)$, where $a$ is a symmetric, elliptic, $n\times n$-matrix-valued measurable function on $\bar{\varOmega}$. Assuming that $\varOmega$ is an open, relatively compact set with boundary $\partial\varOmega$ of Lebesgue measure zero and that $\varrho$ satisfies the Hamza condition, we can show that $(\mathcal{E}^{\varrho,a},D(\mathcal{E}^{\varrho,a}))$ is a local, quasi-regular Dirichlet form. Hence, it has an associated self-adjoint generator $(L^{\varrho,a},D(L^{\varrho,a}))$ and diffusion process $\bm{M}^{\varrho,a}$ (i.e. an associated strong Markov process with continuous sample paths). Furthermore, since $1\in D(\mathcal{E}^{\varrho,a})$ (due to the Neumann boundary condition) and $\mathcal{E}^{\varrho,a}(1,1)=0$, we obtain a conservative process $\bm{M}^{\varrho,a}$ (i.e. $\bm{M}^{\varrho,a}$ has infinite lifetime). Additionally, assuming that $\sqrt{\varrho}\in W^{1,2}(\varOmega)\cap C(\bar{\varOmega})$ or that $\varrho$ is bounded, $\varOmega$ is convex and $\{\varrho=0\}$ has codimension at least 2, we can show that the set $\{\varrho=0\}$ has $\mathcal{E}^{\varrho,a}$-capacity zero. Therefore, in this case we can even construct an associated conservative diffusion process in $\{\varrho>0\}$. This is essential for our application to continuous $N$-particle systems with singular interactions. Note that for the construction of the self-adjoint generator $(L^{\varrho,a},D(L^{\varrho,a}))$ and the Markov process $\bm{M}^{\varrho,a}$ we do not need to assume any differentiability condition on $\varrho$ and $a$. We obtain the following explicit representation of the generator for $\sqrt{\varrho}\in W^{1,2}(\varOmega)$ and $a\in W^{1,\infty}(\varOmega)$:$$ L^{\varrho,a}=\sum_{i,j=1}^n\partial_i(a_{ij}\partial_j)+\partial_i(\log\varrho)a_{ij}\partial_j. $$Note that the drift term can be singular, because we allow $\varrho$ to be zero on a set of Lebesgue measure zero. Our assumptions in this paper even allow a drift that is not integrable with respect to the Lebesgue measure.

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.

Girsanov Transformations for Non-Symmetric Diffusions

2009 ◽  
Vol 61 (3) ◽  
pp. 534-547 ◽  
Author(s):  
Chuan-Zhong Chen ◽  
Wei Sun
Keyword(s):  
Diffusion Process ◽  
Dirichlet Form ◽  
Dirichlet Space ◽  
Explicit Representation ◽  
Dual Process ◽  
Sufficient Condition ◽  
Dual Processes ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Girsanov Transformations

Abstract.Let X be a diffusion process, which is assumed to be associated with a (non-symmetric) strongly local Dirichlet form (ℰ, 𝓓 (ℰ)) on L2(E ;m). For u ∈ 𝓓(ℰ)e, the extended Dirichlet space, we investigate some properties of the Girsanov transformed process Y of X . First, let be the dual process of X and Ŷ the Girsanov transformed process of . We give a necessary and sufficient condition for (Y , Ŷ to be in duality with respect to the measure e2um. We also construct a counterexample, which shows that this condition may not be satisfied and hence (Y , Ŷ ) may not be dual processes. Then we present a sufficient condition under which Y is associated with a semi-Dirichlet form. Moreover, we give an explicit representation of the semi-Dirichlet form.

Reverse Hypercontractivity for Subharmonic Functions

2005 ◽  
Vol 57 (3) ◽  
pp. 506-534 ◽  
Author(s):  
Leonard Gross ◽  
Martin Grothaus
Keyword(s):  
Holomorphic Function ◽  
Dirichlet Form ◽  
Function Spaces ◽  
Subharmonic Functions ◽  
Lower Boundedness

AbstractContractivity and hypercontractivity properties of semigroups are now well understood when the generator, A, is a Dirichlet form operator. It has been shown that in some holomorphic function spaces the semigroup operators, e−tA, can be bounded below from Lp to Lq when p, q and t are suitably related. We will show that such lower boundedness occurs also in spaces of subharmonic functions.

Dirichlet Form Theory and its Applications

2014 ◽  
Vol 11 (4) ◽  
pp. 2667-2756
Author(s):  
Sergio Albeverio ◽  
Zhen-Qing Chen ◽  
Masatoshi Fukushima ◽  
Michael Röckner
Keyword(s):  
Dirichlet Form ◽  
Form Theory
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