Sharp Two-sided Estimates Research Articles

Cubic spline interpolation of functions with high gradients in boundary layers

The cubic spline interpolation of grid functions with high-gradient regions is considered. Uniform meshes are proved to be inefficient for this purpose. In the case of widely applied piecewise uniform Shishkin meshes, asymptotically sharp two-sided error estimates are obtained in the class of functions with an exponential boundary layer. It is proved that the error estimates of traditional spline interpolation are not uniform with respect to a small parameter, and the error can increase indefinitely as the small parameter tends to zero, while the number of nodes N is fixed. A modified cubic interpolation spline is proposed, for which O((ln N/N)4) error estimates that are uniform with respect to the small parameter are obtained.

Hitting Times of Points and Intervals for Symmetric L\xe9vy Processes

For one-dimensional symmetric Lévy processes, which hit every point with positive probability, we give sharp bounds for the tail function Px(TB>t), where TB is the first hitting time of B which is either a single point or an interval. The estimates are obtained under some weak type scaling assumptions on the characteristic exponent of the process. We apply these results to prove sharp two-sided estimates of the transition density of the process killed after hitting B.

Global Dirichlet Heat Kernel Estimates for Symmetric Lévy Processes in Half-Space

In this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels of a large class of symmetric (but not necessarily rotationally symmetric) Levy processes on half spaces for all t>0$t>0$. These Levy processes may or may not have Gaussian component. When Levy density is comparable to a decreasing function with damping exponent β$\beta$, our estimate is explicit in terms of the distance to the boundary, the Levy exponent and the damping exponent β$\beta$ of Levy density.

Fourier–Bessel heat kernel estimates

We provide sharp two-sided estimates of the Fourier–Bessel heat kernel and we give sharp two-sided estimates of the transition probability density for the Bessel process in (0,1) killed at 1 and killed or reflected at 0.

Drift perturbation of subordinate Brownian motions with Gaussian component

Let d ≥ 1 and Z be a subordinate Brownian motion on Rd with infinitesimal generator Δ + ψ(Δ), where ψ is the Laplace exponent of a one-dimensional non-decreasing Levy process (called subordinator). We establish the existence and uniqueness of fundamental solution (also called heat kernel) pb(t, x, y) for non-local operator ℒb = Δ + ψ(Δ) + b · ∇, where b is an Rd-valued function in Kato class Kd,1. We show that pb(t, x, y) is jointly continuous and derive its sharp two-sided estimates. The kernel pb(t, x, y) determines a conservative Feller process X. We further show that the law of X is the unique solution of the martingale problem for (Lb,C∞c (Rd)) and X is a weak solution of \(X_t = X_0 + Z_t + \int_0^t {b(X_s )ds,} t \geqslant 0\). Moreover, we prove that the above stochastic differential equation has a unique weak solution.

Hausdorff operators defined by measures on p-adic linear spaces and their norms

For Hausdorff operator defined by a measure on p-adic linear space Qpn we give the exact values for its norms in power type Morrey space,BMO(Qpn) and BLO(Qpn). Also we prove the sharp two-sided estimate for its norm in Herz space. These results generalize some previous results of the author.

Heat kernels and analyticity of non-symmetric jump diffusion semigroups

Let $d\geq 1$ and $\alpha \in (0, 2)$. Consider the following non-local and non-symmetric L\'evy-type operator on $\mR^d$: $$ \sL^\kappa_{\alpha}f(x):=\mbox{p.v.}\int_{\mR^d}(f(x+z)-f(x))\frac{\kappa(x,z)}{|z|^{d+\alpha}} \dif z, $$ where $0<\kappa_0\leq \kappa(x,z)\leq \kappa_1$, $\kappa(x,z)=\kappa(x,-z)$, and $|\kappa(x,z)-\kappa(y,z)|\leq\kappa_2|x-y|^\beta$ for some $\beta\in(0,1)$. Using Levi's method, we construct the fundamental solution (also called heat kernel) $p^\kappa_\alpha (t, x, y)$ of $\sL^\kappa_\alpha $, and establish its sharp two-sided estimates as well as its fractional derivative and gradient estimates of the heat kernel. We also show that $p^\kappa_\alpha (t, x, y)$ is jointly H\"older continuous in $(t, x)$. The lower bound heat kernel estimate is obtained by using a probabilistic argument. The fundamental solution of $\sL^\kappa_{\alpha}$ gives rise a Feller process $\{X, \mP_x, x\in \mR^d\}$ on $\mR^d$. We determine the L\'evy system of $X$ and show that $\mP_x$ solves the martingale problem for $(\sL^\kappa_{\alpha}, C^2_b(\mR^d))$. Furthermore, we obtain the analyticity of the non-symmetric semigroup associated with $\sL^\kappa_\alpha $ in $L^p$-spaces for every $p\in[1,\infty)$. A maximum principle for solutions of the parabolic equation $\partial_t u =\sL^\kappa_\alpha u$ is also established.

Heat kernel estimates for [formula omitted] under gradient perturbation

For α∈(0,2) and M>0, we consider a family of nonlocal operators {Δ+aαΔα/2,a∈(0,M]} on Rd under Kato class gradient perturbation. We establish the existence and uniqueness of their fundamental solutions, and derive their sharp two-sided estimates. The estimates give explicit dependence on a and recover the sharp estimates for Brownian motion with drift as a→0. Each fundamental solution determines a conservative Feller process X. We characterize X as the unique solution of the corresponding martingale problem as well as a Lévy process with singular drift.

Global Heat Kernel Estimates for Symmetric Markov Processes Dominated by Stable-Like Processes in Exterior C 1,η Open Sets

In this paper, we establish sharp two-sided heat kernel estimates for a large class of symmetric Markov processes in exterior C 1,� open sets for all t > 0. The processes are symmetric pure jump Markov processes with jumping kernel intensity

Sharp estimates of the Green function of hyperbolic Brownian motion

The main objective of the work is to provide sharp two-sided estimates of the $\lambda $-Green function, $\lambda \geq 0$, of the hyperbolic Brownian motion of a half-space. We rely on the recent results obtained by K. Bogus and J. Małecki (2015), regardi

Laplacian perturbed by non-local operators

Suppose that $d\geq 1$ and $0<\beta<2$. We establish the existence and uniqueness of the fundamental solution $q^b(t, x, y)$ to the operator $\mathcal{L}^b=\Delta+S^b$, where $$S^bf(x) := \int_{\mathbb{R}^d} \left( f(x+z) - f(x) - \nabla f(x) \cdot z\mathbb{1}_{\{|z| \leq 1\}} \right) \frac{b(x, z)}{|z|^{d+\beta}} dz$$ and $b(x, z)$ is a bounded measurable function on $\mathbb{R}^d \times \mathbb{R}^d$ with $b(x, z)=b(x, -z)$ for $x, z\in \mathbb{R}^d$. We show that if for each $x\in\mathbb{R}^d, b(x, z) \geq 0$ for a.e. $z\in\mathbb{R}^d$, then $q^b(t, x, y)$ is a strictly positive continuous function and it uniquely determines a conservative Feller process $X^b$, which has strong Feller property. Furthermore, sharp two-sided estimates on $q^b(t, x, y)$ are derived.

Sharp Two-Sided Heat Kernel Estimates of Twisted Tubes and Applications

We prove on-diagonal bounds for the heat kernel of the Dirichlet Laplacian $${-\Delta^D_\Omega}$$ in locally twisted three-dimensional tubes Ω. In particular, we show that for any fixed x the heat kernel decays for large times as $${{\rm e}^{-E_1t} t^{-3/2}}$$ , where E 1 is the fundamental eigenvalue of the Dirichlet Laplacian on the cross section of the tube. This shows that any, suitably regular, local twisting speeds up the decay of the heat kernel with respect to the case of straight (untwisted) tubes. Moreover, the above large time decay is valid for a wide class of subcritical operators defined on a straight tube. We also discuss some applications of this result, such as Sobolev inequalities and spectral estimates for Schrödinger operators $${-\Delta^D_\Omega-V}$$ .

Stable process with singular drift

Suppose that d≥2 and α∈(1,2). Let μ=(μ1,…,μd) be such that each μi is a signed measure on Rd belonging to the Kato class Kd,α−1. In this paper, we consider the stochastic differential equation dXt=dSt+dAt, where St is a symmetric α-stable process on Rd and for each j=1,…,d, the jth component Atj of At is a continuous additive functional of finite variation with respect to X whose Revuz measure is μj. The unique solution for the above stochastic differential equation is called an α-stable process with drift μ. We prove the existence and uniqueness, in the weak sense, of such an α-stable process with drift μ and establish sharp two-sided heat kernel estimates for such a process.

Dirichlet heat kernel estimates for rotationally symmetric Lévy processes

In this paper, we consider a large class of purely discontinuous rotationally symmetric Levy processes. We establish sharp two-sided estimates for the transition densities of such processes killed upon leaving an open set D. When D is a \kappa-fat open set, the sharp two-sided estimates are given in terms of surviving probabilities and the global transition density of the Levy process. When D is a C^{1, 1} open set and the Levy exponent of the process is given by \Psi(\xi)= \phi(|\xi|^2) with \phi being a complete Bernstein function satisfying a mild growth condition at infinity, our two-sided estimates are explicit in terms of \Psi, the distance function to the boundary of D and the jumping kernel of X, which give an affirmative answer to the conjecture posted in [Potential Anal., 36 (2012) 235-261]. Our results are the first sharp two-sided Dirichlet heat kernel estimates for a large class of symmetric Levy processes with general Levy exponents. We also derive an explicit lower bound estimate for symmetric Levy processes on R^d in terms of their Levy exponents.

Green function estimates for subordinate Brownian motions: Stable and beyond

A subordinate Brownian motion $X$ is a L\'evy process which can be obtained by replacing the time of the Brownian motion by an independent subordinator. In this paper, when the Laplace exponent $\phi$ of the corresponding subordinator satisfies some mild conditions, we first prove the scale invariant boundary Harnack inequality for $X$ on arbitrary open sets. Then we give an explicit form of sharp two-sided estimates on the Green functions of these subordinate Brownian motions in any bounded $C^{1,1}$ open set. As a consequence, we prove the boundary Harnack inequality for $X$ on any $C^{1,1}$ open set with explicit decay rate. Unlike {KSV2, KSV4}, our results cover geometric stable processes and relativistic geometric stable process, i.e. the cases when the subordinator has the Laplace exponent $$\phi(\lambda)=\log(1+\lambda^{\alpha/2}) (0<\alpha\leq 2, d > \alpha)$$ and $$\phi(\lambda)=\log(1+(\lambda+m^{\alpha/2})^{2/\alpha}-m) (0<\alpha<2,\, m>0, d >2) .$$

Dirichlet heat kernel estimates for subordinate Brownian motions with Gaussian components

Abstract In this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels, in C 1,1 open sets D in ℝ d , of a large class of subordinate Brownian motions with Gaussian components. When D is bounded, our sharp two-sided Dirichlet heat kernel estimates hold for all t &gt; 0. Integrating the heat kernel estimates with respect to the time variable t, we obtain sharp two-sided estimates for the Green functions, in bounded C 1,1 open sets, of such subordinate Brownian motions with Gaussian components.

Dirichlet Heat Kernel Estimates for Stable Processes with Singular Drift In Unbounded C 1,1 Open Sets

Suppose d ≥ 2 and α ∈ (1, 2). Let D be a (not necessarily bounded) C1,1 open set in ℝd and μ = (μ1, . . . , μd) where each μj is a signed measure on ℝd belonging to a certain Kato class of the rotationally symmetric α-stable process X. Let Xμ be an α-stable process with drift μ in ℝd and let Xμ,D be the subprocess of Xμ in D. In this paper, we derive sharp two-sided estimates for the transition density of Xμ,D.

ON ESTIMATES OF POISSON KERNELS FOR SYMMETRIC LÉVY PROCESSES

In this paper, using elementary calculus only, we give a simple proof that Green function estimates imply the sharp two-sided pointwise estimates for Poisson kernels for subordinate Brownian motions. In particular, by combining the recent result of Kim and Mimica [5], our result provides the sharp two-sided estimates for Poisson kernels for a large class of subordinate Brownian motions including geometric stable processes.

Global uniform boundary Harnack principle with explicit decay rate and its application

In this paper, we consider a large class of subordinate Brownian motions X via subordinators with Laplace exponents which are complete Bernstein functions satisfying some mild scaling conditions at zero and at infinity. We first discuss how such conditions govern the behavior of the subordinator and the corresponding subordinate Brownian motion for both large and small time and space. Then we establish a global uniform boundary Harnack principle in (unbounded) open sets for the subordinate Brownian motion. When the open set satisfies the interior and exterior ball conditions with radius R>0, we get a global uniform boundary Harnack principle with explicit decay rate. Our boundary Harnack principle is global in the sense that it holds for all R>0 and the comparison constant does not depend on R, and it is uniform in the sense that it holds for all balls with radii r≤R and the comparison constant depends neither on D nor on r. As an application, we give sharp two-sided estimates for the transition densities and Green functions of such subordinate Brownian motions in the half-space.

Potential theory of subordinate Brownian motions with Gaussian components

In this paper we study a subordinate Brownian motion with a Gaussian component and a rather general discontinuous part. The assumption on the subordinator is that its Laplace exponent is a complete Bernstein function with a Lévy density satisfying a certain growth condition near zero. The main result is a boundary Harnack principle with explicit boundary decay rate for non-negative harmonic functions of the process in C1,1 open sets. As a consequence of the boundary Harnack principle, we establish sharp two-sided estimates on the Green function of the subordinate Brownian motion in any bounded C1,1 open set D and identify the Martin boundary of D with respect to the subordinate Brownian motion with the Euclidean boundary.