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heat kernel Estimates for Non-symmetric Finite Range Jump Processes

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Acta Mathematica Sinica,English Series 2021年第2期37卷 229-248页

作者： Jie Ming WANG Department of Mathematics and Statistics Beijing Institute of TechnologyBeijing 100081P.R.China

In this paper,we first establish the sharp two-sided heat kernel estimates and the gradient estimate for the truncated fractional Laplacian under gradient perturbation S^(b):=△^(α/2)+b·▽,where △^(α/2) is the truncated fractional Laplacian,α∈(1,2) and b ∈ K_(d)^(α-1).In the second part,for a more general finite range jump process,we present some sufficient conditions to allow that the two sided estimates of the heat kernel are comparable to the Poisson type function for large distance |x-y|in short time.

关键词： heat kernel transition density function gradient estimate finite range jump process truncated fractional Laplacian martingale problem

heat kernel AND HARDY'S THEOREM FOR JACOBI TRANSFORM

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Chinese Annals of Mathematics,Series B 2003年第3期24卷 359-366页

作者： T.KAWAZOE LIUJIANMING DepartmentofMathematics KeioUniversityatFujisawa5322EndoFujisawaKanagawa252-8520Japan. SchoolofMathematicalSciences PekingUniversityBeijing100871China.

In this paper, the authors obtain sharp upper and lower bounds for the heat kernel associatedwith Jacobi transform, and get some analogues of Hardy's Theorem for Jacobi transform byusing the sharp estimate of the heat... 详细信息

关键词： heat kernel Jacobi transform Uncertainty principle

heat kernel ESTIMATES ON JULIA SETS

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Acta Mathematica Scientia 2017年第5期37卷 1399-1414页

作者：杨甍 -Department of Mathematical Sciences Tsinghua University

We give heat kernel estimates on Julia sets J（f;） for quadratic polynomials f c（z） = z;+ c for c in the main cardioid or the ±1/k-bulbs where k ≥ 2. First we use external ray parametrization to construct a regular, strongly local and conservative Dirichlet form on Julia set. Then we show that this Dirichlet form is a resistance form and the corresponding resistance metric induces the same topology as Euclidean metric. Finally, we give heat kernel estimates under the resistance metric.

关键词： Julia sets heat kernel resistance metric

heat kernel estimates on effective resistance metric spaces

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Progress in Natural Science:Materials International 2007年第7期17卷 775-783页

作者： Hu Jiaxin Ji Yuan Wen Zhiying (Department of Mathematical Sciences,Tsinghua University,Beijing 100084,China) [a]Department of Mathematical Sciences Tsinghua University Beijing China

We obtain two-sided estimates of heat kernels on effective-resistance metric spaces by using an analytic *** applications,such as on post-critically finite self-similar sets and on products of effective resistance spa... 详细信息

关键词： heat kernel effective resistance Dirichlet form Green’s function

Asymptotic Expansions of the heat kernel of the Laplacian for General Annular Bounded Domains with Robin Boundary Conditions：Further Results

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Acta Mathematica Sinica,English Series 2003年第4期19卷 679-694页

作者： E.M.E.ZAYED DepartmentofMathematics FacultyofScienceZagazigUniversityZagazigEgypt

The asymptotic expansions of the trace of the heat kernel Θ(t) = Σ_(v =1)~∞exp(-tλ_v) for small positive t, where {λ_v} are the eigenvalues of the negative Laplacian-△_n = -Σ_(i = 1)~n(partial deriv/(partial deriv)x^i)~2 in R^n (n = 2 or 3), are studied for ageneral annular bounded domain Ω with a smooth inner boundary (partial deriv)Ω_1 and a smoothouter boundary (partial deriv)Ω_2, where a finite number of piecewise smooth Robin boundaryconditions (partial deriv/(partial deriv)n_j + γ_j)φ = 0 on the components Γ_j(j = 1, …, k) of(partial deriv)Ω_1 and on the components Γ_j(j = k + 1, …, m) of (partial deriv)Ω_2 areconsidered such that (partial deriv)Ω_1 = ∪_(j = 1)~kΓ_j and (partial deriv)Ω_2 = ∪_(j = k +1)~mΓ_j and where the coefficients γ_j(j = 1, …, m) are piecewise smooth positive functions. Someapplications of Θ(t) for an ideal gas enclosed in the general annular bounded domain Ω are *** results are also obtained.

关键词： inverse problem heat kernel eigenvalues Robin boundary conditions classical ideal gas

Geodesic metrics on fractals and applications to heat kernel estimates

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Science China Mathematics 2023年第5期66卷 907-934页

作者： Qingsong Gu Ka-Sing Lau Hua Qiu Huo-Jun Ruan Department of Mathematics Nanjing UniversityNanjing 210093China Department of Mathematics The Chinese University of Hong KongHong KongChina Department of Mathematics University of PittsburghPittsburghPA 15217USA School of Mathematical Sciences Zhejiang UniversityHangzhou 310027China

It is well known that for a Brownian motion, if we change the medium to be inhomogeneous by a measure μ, then the new motion(the time-changed process) will diffuse according to a different metric D(·, ·).In 2009, Kigami initiated a general scheme to construct such metrics through some self-similar weight functions g on the symbolic space. In order to provide concrete models to Kigami’s theoretical construction, in this paper,we give a thorough study of his metric on two classes of fractals of primary importance: the nested fractals and the generalized Sierpinski carpets;we further assume that the weight functions g := ga are generated by“symmetric” weights a. Let M be the domain of a such that Dgadefines a metric, and let S be the boundary of M. One of our main results is that the metrics from ga satisfy the metric chain condition if and only if a ∈ *** determine M and S, we provide a recursive weight transfer construction on the nested fractals, and a basic symmetric argument on the Sierpinski carpet. As an application, we use the metric chain condition to obtain the lower estimate of the sub-Gaussian heat kernel. This together with the upper estimate obtained by Kigami allows us to have a concrete class of metrics for the time change, and the two-sided sub-Gaussian heat kernel estimate on the fundamental fractals.

关键词： Brownian motion heat kernel metric chain condition nested fractal quasisymmetry resistance metric Sierpinski carpet weight function

On Li-Yau heat kernel Estimate

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Acta Mathematica Sinica,English Series 2021年第8期37卷 1205-1218页

作者： Lu LI Zhen Lei ZHANG School of Mathematics Capital Normal UniversityBeijing 100048P.R.China

We present some improvements of the Li-Yau heat kernel estimate on a Riemannian manifold with Ricci curvature bounded *** a consequence we prove a gradient estimate to the heat kernel with an optimal leading term.

关键词： heat kernel Li-Yau estimate gradient estimate

THE DAVIES METHOD FOR heat kernel UPPER BOUNDS OF NON-LOCAL DIRICHLET FORMS ON ULTRA-METRIC SPACES

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Acta Mathematica Scientia 2020年第5期40卷 1240-1248页

作者： Jin GAO Department of Mathematical Sciences Tsinghua UniversityBeijing 100084China

We apply the Davies method to give a quick proof for the upper estimate of the heat kernel for the non-local Dirichlet form on the ultra-metric space. The key observation is that the heat kernel of the truncated Dirichlet form vanishes when two spatial points are separated by any ball of a radius larger than the truncated range. This new phenomenon arises from the ultra-metric property of the space.

关键词： heat kernel ultra-metrie Davies method

A Note on the heat kernel for the Rescaled Harmonic Oscillator from Two Step Nilpotent Lie Groups

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Acta Mathematica Sinica,English Series 2022年第9期38卷 1597-1611页

作者： Zhi Peng YANG Department of Mathematics Yunnan Normal UniversityKunming 650500P.R.China

In this note,we use Schr?dinger representations and the Fourier transform on two step nilpotent Lie groups to compute the explicit formula of the sub-Laplacian operator and its symbol,which is associated with the rescaled harmonic *** we can give an explicit formula for the heat kernel of the rescaled harmonic oscillator for the singularity at the *** results are useful for the general two step nilpotent Lie groups,including the Heisenberg group and H-type group.

关键词： Sub-Laplacian heat kernel nilpotent Lie groups

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