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Ka-sing Lau - One of the best experts on this subject based on the ideXlab platform.
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On a recursive construction of Dirichlet Form on the Sierpiński gasket
Journal of Mathematical Analysis and Applications, 2019Co-Authors: Ka-sing Lau, Hua QiuAbstract:Abstract Let Γ n denote the n-th level Sierpinski graph of the Sierpinski gasket K. We consider, for any given conductance ( a 0 , b 0 , c 0 ) on Γ 0 , the Dirichlet Form E on K obtained from a recursive construction of compatible sequence of conductances ( a n , b n , c n ) on Γ n , n ≥ 0 . We prove that there is a dichotomy situation: either a 0 = b 0 = c 0 and E is the standard Dirichlet Form, or a 0 > b 0 = c 0 (or the two symmetric alternatives), and E is a non-self-similar Dirichlet Form independent of a 0 , b 0 . The second situation has been studied in [13] , [9] as a one-dimensional asymptotic diffusion. The analytic approach here is more direct and yields sharper results; in particular, for the spectral property, we give a precise estimate of the eigenvalue distribution of the associated Laplacian, which improves a similar result in [9] .
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on a recursive construction of Dirichlet Form on the sierpi nski gasket
arXiv: Functional Analysis, 2017Co-Authors: Ka-sing Lau, Hua QiuAbstract:Let $\Gamma_n$ denote the $n$-th level Sierpinski graph of the Sierpinski gasket $K$. We consider, for any given conductance $(a_0, b_0, c_0)$ on $\Gamma_0$, the Dirchlet Form ${\mathcal E}$ on $K$ obtained from a recursive construction of compatible sequence of conductances $(a_n, b_n, c_n)$ on $\Gamma_n, n\geq 0$. We prove that there is a dichotomy situation: either $a_0= b_0 =c_0$ and ${\mathcal E}$ is the standard Dirichlet Form, or $a_0 >b_0 =c_0$ (or the two symmetric alternatives), and ${\mathcal E}$ is a non-self-similar Dirichlet Form independent of $a_0, b_0$. The second situation has also been studied in [Hattori et al 1994][Hambley et al 2002] as a one-dimensional asymptotic diffusion process on the Sierpinski gasket. For the spectral property, we give a sharp estimate of the eigenvalue distribution of the associated Laplacian, which improves a similar result in [Hambley et al 2002].
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Obtaining Upper Bounds of Heat Kernels from Lower Bounds
Communications on Pure and Applied Mathematics, 2008Co-Authors: Alexander Grigor′yan, Ka-sing LauAbstract:We show that a near-diagonal lower bound of the heat kernel of a Dirichlet Form on a metric measure space with a regular measure implies an on-diagonal upper bound. If in addition the Dirichlet Form is local and regular, then we obtain a full off-diagonal upper bound of the heat kernel provided the Dirichlet heat kernel on any ball satisfies a near-diagonal lower estimate. This reveals a new phenomenon in the relationship between the lower and upper bounds of the heat kernel. c � 2007 Wiley Periodicals, Inc.
Wei Sun - One of the best experts on this subject based on the ideXlab platform.
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On Girsanov and generalized Feynman-Kac transForms for symmetric Markov processes
2016Co-Authors: Chuan-zhong Chen, Wei SunAbstract:Let X be a Markov process, which is assumed to be associated with a symmetric Dirichlet Form (E,D(E)). For u ∈ D(E)e, the ex-tended Dirichlet space, we have the classical Fukushima’s decomposi-tion: ũ(Xt) − ũ(X0) =Mut +Nut, where u ̃ is a quasi-continuous version of u, Mut the martingale part and N u t the zero energy part. In this pa-per, we investigate two important transFormations for X, the Girsanov transForm induced by Mut and the generalized Feynman-Kac transForm induced by Nut. For the Girsanov transForm, we present necessary and sufficient conditions for which to induce a positive supermartingale and hence to determine another Markov process X̂. Moreover, we charac-terize the symmetric Dirichlet Form associated with the Girsanov trans-Formed process X̂. For the generalized Feynman-Kac transForm, we give a necessary and sufficient condition for the generalized Feynman-Kac semigroup to be strongly continuous. Abbreviated title. Girsanov and generalized Feynman-Kac transFormation
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extensions of levy khintchine Formula and beurling deny Formula in semi Dirichlet Forms setting
Journal of Functional Analysis, 2006Co-Authors: Wei SunAbstract:Abstract The Levy–Khintchine Formula or, more generally, Courrege's theorem characterizes the infinitesimal generator of a Levy process or a Feller process on R d . For more general Markov processes, the Formula that comes closest to such a characterization is the Beurling–Deny Formula for symmetric Dirichlet Forms. In this paper, we extend these celebrated structure results to include a general right process on a metrizable Lusin space, which is supposed to be associated with a semi-Dirichlet Form. We start with decomposing a regular semi-Dirichlet Form into the diffusion, jumping and killing parts. Then, we develop a local compactification and an integral representation for quasi-regular semi-Dirichlet Forms. Finally, we extend the Formulae of Levy–Khintchine and Beurling–Deny in semi-Dirichlet Forms setting through introducing a quasi-compatible metric.
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Extensions of Lévy–Khintchine Formula and Beurling–Deny Formula in semi-Dirichlet Forms setting
Journal of Functional Analysis, 2006Co-Authors: Wei SunAbstract:Abstract The Levy–Khintchine Formula or, more generally, Courrege's theorem characterizes the infinitesimal generator of a Levy process or a Feller process on R d . For more general Markov processes, the Formula that comes closest to such a characterization is the Beurling–Deny Formula for symmetric Dirichlet Forms. In this paper, we extend these celebrated structure results to include a general right process on a metrizable Lusin space, which is supposed to be associated with a semi-Dirichlet Form. We start with decomposing a regular semi-Dirichlet Form into the diffusion, jumping and killing parts. Then, we develop a local compactification and an integral representation for quasi-regular semi-Dirichlet Forms. Finally, we extend the Formulae of Levy–Khintchine and Beurling–Deny in semi-Dirichlet Forms setting through introducing a quasi-compatible metric.
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Quasi-Homeomorphisms and Measures of Finite Energy Integrals of Generalized Dirichlet Forms
Acta Mathematica Sinica English Series, 2000Co-Authors: Wei SunAbstract:We introduce the quasi-homeomorphisms of generalized Dirichlet Forms and prove that any quasi-regular generalized Dirichlet Form is quasi-homeomorphic to a semi-regular generalized Dirichlet Form. Moreover, we apply this quasi-homeomorphism method to study the measures of finite energy integrals of generalized Dirichlet Forms. We show that any 1-coexcessive function which is dominated by a function in ${\hat{\cal F}}$ is associated with a measure of finite energy integral. Consequently, we prove that a Borel set B is ɛ-exceptional if and only if μ (B) = 0 for any measure μ of finite energy integral.
Meng Yang - One of the best experts on this subject based on the ideXlab platform.
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On the Domains of Dirichlet Forms on Metric Measure Spaces.
arXiv: Functional Analysis, 2019Co-Authors: Meng YangAbstract:We prove that the domain of the local Dirichlet Form is strictly contained in the domain of any stable-like non-local Dirichlet Form on general metric measure spaces.
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On the Domains of Dirichlet Forms on the Sierpi\'nski Carpet
arXiv: Functional Analysis, 2019Co-Authors: Meng YangAbstract:We prove that the domain of the local regular Dirichlet Form is \emph{strictly} contained in the domain of any stable-like non-local regular Dirichlet Form on the Sierpinski carpet.
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Mosco Convergence of Stable-Like Non-Local Dirichlet Forms on Metric Measure Spaces
arXiv: Functional Analysis, 2019Co-Authors: Meng YangAbstract:We prove that stable-like non-local Dirichlet Forms converge to local Dirichlet Form in the sense of Mosco on metric measure spaces. We prove that subordinated Dirichlet Forms converge to the original Dirichlet Form in the sense of Mosco on metric measure spaces.
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construction of local regular Dirichlet Form on the sierpi nski gasket using gamma convergence
arXiv: Functional Analysis, 2017Co-Authors: Meng YangAbstract:We construct a self-similar local regular Dirichlet Form on the Sierpinski gasket using $\Gamma$-convergence of stable-like non-local closed Forms. As a continuation of a recent paper by Grigor'yan and the author, we give the first \emph{unified} purely analytic construction of local regular Dirichlet Forms that works both on the Sierpinski gasket and the Sierpinski carpet.
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the local regular Dirichlet Form on the sierpi nski gasket via the averaging method
2017Co-Authors: Meng YangAbstract:We construct a self-similar local regular Dirichlet Form on the Sierpinski gasket using $\Gamma$-convergence of stable-like non-local closed Forms. As a continuation of a recent paper by Grigor'yan and the author, we give the first \emph{unified} purely analytic construction of local regular Dirichlet Forms that works both on the Sierpinski gasket and the Sierpinski carpet.
Toshihiro Uemura - One of the best experts on this subject based on the ideXlab platform.
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On Symmetric Stable-Type Processes with Degenerate/Singular Lévy Densities
Journal of Theoretical Probability, 2020Co-Authors: Haruna Okamura, Toshihiro UemuraAbstract:We consider symmetric stable-type processes with degenerate/singular Lévy densities via Dirichlet Form theory. We give conditions of some global path properties of the processes such as recurrence, transience or conservativeness. We also show the polarity of a point.
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on the structure of the domain of a symmetric jump type Dirichlet Form
Publications of The Research Institute for Mathematical Sciences, 2012Co-Authors: René L. Schilling, Toshihiro UemuraAbstract:We characterize the structure of the domain of a pure jump-type Dirichlet Form which is given by a Beurling{Deny Formula. In particular, we obtain sucient conditions in terms of the jumping kernel guaranteeing that the test functions are a core for the Dirichlet Form and that the Form is a Silverstein extension. As an application we show that for recurrent Dirichlet Forms the extended Dirichlet space can be interpreted in a natural way as a homogeneous Dirichlet space. For reected Dirichlet spaces this leads to a simple purely analytic proof that the active reected Dirichlet space (in the sense of Chen, Fukushima and Kuwae) coincides with the extended active reected Dirichlet space.
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On an extension of jump-type symmetric Dirichlet Forms
Electronic Communications in Probability, 2007Co-Authors: Toshihiro UemuraAbstract:We show that any element from the ($L^2$-)maximal domain of a jump-type symmetric Dirichlet Form can be approximated by test functions under some conditions. This gives us a direct proof of the fact that the test functions is dense in Bessel potential spaces.
Nicolas Bouleau - One of the best experts on this subject based on the ideXlab platform.
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Introduction to the Theory of Dirichlet Forms
Dirichlet Forms Methods for Poisson Point Measures and Lévy Processes, 2015Co-Authors: Nicolas Bouleau, Laurent DenisAbstract:A Dirichlet Form is a generalization of the energy Form \(f\mapsto \int _\Omega |\nabla f|^2 d\lambda \) introduced in the 1840s especially by William Thomson (Lord Kelvin) (cf. Temple, 100 Years of Mathematics, 1981, [351], Chap. 15) in order to solve by minimization the problem without second member \(\Delta f=0\) in the open set \(\Omega \) (Dirichlet principle). Riemann adopted the expression Dirichlet Form (Riemann Grundlagen fur eine allgemeine Theorie der Funktionen einer veranderlischen komplexen Grosse, 1851, [314]). The generalization now known as a Dirichlet Form keeps the notion in the same relationship with the semigroup as the energy Form holds with the heat semigroup.
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The Lent Particle Method, Application to Multiple Poisson Integrals
Bulletin of the Greek Mathematical Society, 2010Co-Authors: Nicolas BouleauAbstract:We give a extensive account of a recent new way of applying the Dirichlet Form theory to random Poisson measures. The main application is to obtain existence of density for thelaws of random functionals of Lévy processes or solutions of stochastic differential equations with jumps. As in the Wiener case the Dirichlet Form approach weakens significantly theregularity assumptions. The main novelty is an explicit Formula for the gradient or for the "carré du champ" on the Poisson space called the lent particle Formula because based on adding a new particle to the system, computing the derivative of the functional with respect to this new argument and taking back this particle before applying the Poisson measure. The article is expository in its first part and based on Bouleau-Denis [12] with several new examples, applications to multiple Poisson integrals are gathered in the last part which concerns the relation with the Fock space and some aspects of the second quantization.
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Bringing errors into focus
2007Co-Authors: Nicolas BouleauAbstract:This lecture presents recent advances in the theory of errors propagation. We first explain in which cases the propagation of errors may be perFormed with a first order differential calculus or needs a second order differential calculus. Then we point out the link between error propagation and the concept of second order vector in differential geometry, emphasizing the existence of a slight ambiguity concerning the bias operator. The third part in devoted to the powerful framework of Dirichlet Forms whose main feature is to apply easily to infinite dimensional models including the Wiener space (giving an interpretation of Malliavin calculus in terms of errors), the Poisson space and the Monte Carlo space. In the fourth part we show how an error in the usual mathematical sense, i.e. an approximate quantity, may yield a Dirichlet Form and we introduce the four bias operators. Eventually we connect the Dirichlet Form with statistics by identifying the square of field operator with the inverse of the Fisher inFormation matrix.
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When and how an error yields a Dirichlet Form
Journal of Functional Analysis, 2006Co-Authors: Nicolas BouleauAbstract:We consider a random variable $Y$ and approximations $Y_n$, defined on the same probability space with values in the same measurable space as $Y$. We are interested in situations where the approximations $Y_n$ allow to define a Dirichlet Form in the space $L^2(P_Y)$ where $P_Y$ is the law of $Y$. Our approach consists in studying both biases and variances. The article attempts to propose a general theoretical framework. It is illustrated by several examples.