Heat Kernel Estimates for Operators with Boundary Conditions

We prove Gaussian upper bounds for kernels associated with non – symmetric, non – autonomous second order parabolic operators of divergence form subject to various boundary conditions. The growth of the kernel in time is determined by theboundary conditions and the geometric properties of the domain. The theory gives a unified treatment for Dirichlet, Neumann and Robin boundary conditions, and the existence of a Gaussian type bound is essentially reduced to verifying some properties of the Hilbert space in the weak formulation of the problem.     

Heat Kernel Estimates on Weighted Graphs

We prove upper and lower heat kernel bounds for the Laplacianon weighted graphs which include the case that the weights haveno strictly positive lower bound. Our estimates give rise toa very explicit probabilistic interpretation, and can be formulatedin terms of a weighted metric. Interestingly, this metric isnot equivalent to the intrinsic metric. 1991 Mathematics SubjectClassification 39A12.     

完备流形上带位势的热核估计

在完备非紧流形上获得了关于带位势热方程正解的梯度估计；接着，利用测地线的技巧获得了Harnack不等式；进一步，建立了两个积分不等式，综合Harnack不等式获得了热核的上下界；最后，利用函数的结果来控制p形式的热核。     

Logarithmic Gradient of the Heat Kernel on Riemannian Manifolds

Mathematical Notes -     

Hardy Inequality and Heat Semigroup Estimates for Riemannian Manifolds with Singular Data

Upper bounds are obtained for the heat content of an open set D in a geodesically complete Riemannian manifold M with Dirichlet boundary condition on ?D, and non-negative initial condition. We show that these upper bounds are close to being sharp if (i) the Dirichlet-Laplace-Beltrami operator acting in L ²(D) satisfies a strong Hardy inequality with weight δ², (ii) the initial temperature distribution, and the specific heat of D are given by δ^?α and δ^?β respectively, where δ is the distance to ?D, and 1 < α <2, 1 < β <2.     

Heat Kernel Estimates for Non-symmetric Finite Range Jump Processes

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 ■,where ■ 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.     

Two-Sided Estimates of the Heat Kernel of the Schrodinger Operator

Two-sided bounds for the heat kernel of the Schrödingeroperator with a singular potential are obtained with explicitconstants. 1991 Mathematics Subject Classification 35B05, 35J10.     

Heat Kernel Estimates and LP Spectral Independence of Elliptic Operators

Let be an open subset of R^d, and let T_p for p[1, ) be consistentC₀-semigroups given by kernels that satisfy an upper heat kernelestimate. Denoting their generators by A_p, we show that thespectrum (A_p) is independent of p[1, ). We also treat the caseof weighted L^p-spaces for weights that satisfy a subexponentialgrowth condition. An example shows that independence of thespectrum may fail for an exponential weight. 1991 MathematicsSubject Classification 47D06, 47A10, 35P05.     

Curvature Dimension Inequalities and Subelliptic Heat Kernel Gradient Bounds on Contact Manifolds

We study curvature dimension inequalities for the sub-Laplacian on contact Riemannian manifolds. This new curvature dimension condition is then used to obtain:

• Geometric conditions ensuring the compactness of the underlying manifold (Bonnet–Myers type results);
• Volume estimates of metric balls;
• Gradient bounds and stochastic completeness for the heat semigroup generated by the sub-Laplacian;
• Spectral gap estimates.

     

Endp oint Estimates for Commutators of Fractional Integrals Asso ciated to Op erators with Heat Kernel Bounds

Let L be the infinitesimal generator of an analytic semigroup on L~2(R~n)with pointwise upper bounds on heat kernel,and denote by L~(-α/2)the fractional integrals of L.For a BMO function b(x),we show a weak type Llog L estimate of the commutators [b,L~(-α/2)](f)(x) = b(x)L~(-α/2)(f)(x)-L~(-α/2)(bf)(x).We give applications to large classes of differential operators such as the Schr¨odinger operators and second-order elliptic operators of divergence form.     

Heat Kernel Upper Estimates for Symmetric Jump Processes with Small Jumps of High Intensity

We consider the following non-local operator

Uniform Estimates and the Whole Asymptotic Series of the Heat Content on Manifolds

We give a recursive algorithm for the computation of the complete asymptotic series, for small time, of the amount of heat inside a domain with smooth boundary in a Riemannian manifold; we consider arbitrary smooth initial data, and we impose Dirichlet condition on the boundary. When the Ricci curvature of the domain and the mean curvature of its boundary are both nonnegative, we also give sharp upper and lower bounds of the heat content which hold for all values of time. These estimates extend to convex sets of the Euclidean space having arbitrary boundary.     

Heat Kernel Estimates for Strongly Recurrent Random Walk on Random Media

We establish general estimates for simple random walk on an arbitrary infinite random graph, assuming suitable bounds on volume and effective resistance for the graph. These are generalizations of the results in Barlow et al. (Commun. Math. Phys. 278:385–431, 2008, Sects. 1, 2) and in particular imply the spectral dimension of the random graph. We will also give an application of the results to random walk on a long-range percolation cluster. J. Misumi research partially supported by the 21 century COE program at Graduate School of Mathematical Sciences, the University of Tokyo.     

变量核奇异积分算子交换子的紧性

证明了在一定条件下,带变量核的奇异积分箅子交换子[b,T]是Lp上的紧算子,也证明了,如核函数满足一定的条件,并且带变量核的奇异积分算子的交换子[b,T]是Lp上的有界算子或紧算子,那么6∈BMO(Rn)或6∈CM0(Rn).     

Ricci曲率平行的一类Riemann流形的C∞紧性

本文证明,在Gromov-Hausdorff拓扑下,Ricci曲率平行,截面曲率和单一半径有下界,体积有上界的Riemann流形的集合是c^∞紧的.作为应用,我们证明一个pinching结果,即在某些条件下,Ricci平坦的流形必定平坦.     

Global Heat Kernel Estimates for Relativistic Stable Processes in Half-space-like Open Sets

In this paper, by using probabilistic methods, we establish sharp two-sided large time estimates for the transition densities of relativistic α-stable processes with mass m ∈ (0, 1] (i.e., for the Dirichlet heat kernels of m − (m ^2/α  − Δ) ^α/2 with m ∈ (0, 1]) in half-space-like C ^{1, 1} open sets. The estimates are uniform in m in the sense that the constants are independent of m ∈ (0, 1]. Combining with the sharp two-sided small time estimates, established in Chen et al. (Ann Probab, 2011), valid for all C ^{1, 1} open sets, we have now sharp two-sided estimates for the transition densities of relativistic α-stable processes with mass m ∈ (0, 1] in half-space-like C ^{1, 1} open sets for all times. Integrating the heat kernel estimates with respect to the time variable, one can recover the sharp two-sided Green function estimates for relativistic α-stable processes with mass m ∈ (0, 1] in half-space-like C ^{1, 1} open sets established recently in Chen et al. (Stoch Process their Appl, 2011).     

带权流形上的纽曼特征值估计

讨论一类光滑紧致带权黎曼流形上的纽曼特征值估计问题,假定这类流形具有光滑边界,边界是凸的,而且流形上的Bakery-Emery Ricci曲率具有正的下界.利用了极大模原理去证明热方程解的梯度估计,然后得到热核上界估计.再利用热核与特征值的关系,得到了特征值的下界估计.     

Expected Volume of Intersection of Pinned Wiener Sausages and Heat Kernel Norms on Compact Riemannian Manifolds with Boundary

Estimates are obtained for the expected volume of intersection of independent pinned Wiener sausages in Euclidean space in the limit of small pinning time.