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Notation for Section 4. set B(x 0 , R) we consider the bump function ϕ provided by Assumption B. We let δ = sup x∈X
Source publication
We prove a boundary Harnack inequality for jump-type Markov processes on
metric measure state spaces, under comparability estimates of the jump kernel
and Urysohn-type property of the domain of the generator of the process. The
result holds for positive harmonic functions in arbitrary open sets. It
applies, e.g., to many subordinate Brownian motion...
Contexts in source publication
Context 1
... Figure 1. By Assumption B, m(∂V ) = 0. Note that Aϕ(x) ≤ 0 andˆAϕandˆ andˆAϕ(x) ≤ 0 if x ∈ B(x 0 , q), and δ can be arbitrarily close to (B(x 0 , q), B(x 0 , R)). ...
Context 2
... (4.14) and (4.15) can be viewed correspondingly as Dynkin's formula applied to the first exit time, and the Ikeda-Watanabe formula for X ψ t . Recall that x 0 ∈ X, 0 < r < p < q < R < R 0 , B(x 0 , q) ⊆ V ⊆ B(x 0 , R), see Figure 1, ϕ ∈ D is positive in V and vanishes in X \ V , and ϕ(x) = 1 for x ∈ B(x 0 , q). ...
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Citations
... This operator was intensively studied in the last twenty years from various perspectives. Its relation to probability and analysis, besides the groundbreaking [2], was also considered in [3,[12][13][14]. Sharp two-sided heat kernel estimates in case of a C 1,1 open set were established in [9]. ...
In this paper we consider the Dirichlet form on the half-space \({\mathbb R}^d_+\) defined by the jump kernel \(J(x,y)=|x-y|^{-d-\alpha }{{\mathcal {B}}}(x,y)\), where \({{\mathcal {B}}}(x,y)\) can be degenerate at the boundary. Unlike our previous works [16, 17] where we imposed critical killing, here we assume that the killing potential is identically zero. In case \(\alpha \in (1,2)\) we first show that the corresponding Hunt process has finite lifetime and dies at the boundary. Then, as our main contribution, we prove the boundary Harnack principle and establish sharp two-sided Green function estimates. Our results cover the case of the censored \(\alpha \)-stable process, \(\alpha \in (1,2)\), in the half-space studied in [2].
... where ( x, x d ) are the coordinates of x in CS ξ . Then, [4] and the references therein. ...
In this paper, we obtain sharp two-sided estimates of Poisson kernels for pure jump symmetric Lévy processes in C1,1 open sets. In particular, by combining Green function estimates obtained by Kim and Mimica (Electron. J. Probab., 23(64), 1–45, 2018), our result covers subordinate Brownian motions with high intensity of small jumps such as conjugate geometric stable process whose Laplace exponent is λ↦λ/(log(1+λα/2))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda \mapsto \lambda /(\log (1+\lambda ^{\alpha /2}))$\end{document} for 0 < α < 2. As an application, we show the existence of tangential limits for harmonic functions in C1,1 open sets.
... The result in [Bog97] was only valid for Lipschitz domains, but this was later extended to any bounded domain in [SW99]. Over the proceeding decade there were several papers proving Harnack inequalities for more general jump processes including [BL02;BKK15], see also [BKK08]. For fully nonlinear nonlocal PDE, a Harnack inequality was established in [CS09], see also [CS11]. ...
We prove the Harnack inequality for antisymmetric $s$-harmonic functions in a general domain. This may be used in conjunction with the method of moving planes to obtain quantitative stability results for symmetry and overdetermined problems driven by the fractional Laplacian. The proof is split into two parts: an interior Harnack inequality away from the plane of symmetry, and a boundary Harnack inequality close to the plane of symmetry. We prove the interior Harnack inequality by establishing the weak Harnack inequality for viscosity super-solutions and local boundedness for viscosity sub-solutions. To prove the boundary Harnack inequality, we introduce a new mean-value formula for antisymmetric $s$-harmonic functions.
... BHI has been widely studied for both local and non-local operators and the corresponding processes on bounded domains of R d . We refer the reader to the paper by Bogdan et al. (2015) for general results on jump Feller processes, an excellent overview of the history, references and discussion on applications of BHI. Our present estimate (1.6) can be understood as a discrete time and space variant of the inequality stated by Kwaśnicki for jump isotropic α-stable processes in R d and V ≡ 0 Kwaśnicki (2009, Corollary 3). ...
... Let us turn to the boundary Harnack inequality and its consequences for harmonic functions. A very general version of BHP was proved by Bogdan, Kumagai and Kwaśnicki [14]. It was later simplified in [28] in the case of unimodal Lévy processes. ...
... First we prove the uniform boundary Harnack inequality and generalize [42,Remark 3] to the whole family of Lévy processes X s : s 1 . To this end we closely examine assumptions imposed in [14], [28] and [42] and verify that constants appearing therein are in fact independent of s. First, let us show that the boundary Harnack inequality holds with the same constant for every X s . ...
We prove universality of the Yaglom limit of Lipschitz cones among all unimodal L\'{e}vy processes sufficiently close to the isotropic $\alpha$-stable L\'{e}vy process.
... BHI has been widely studied for both local and non-local operators and the corresponding processes on bounded domains of R d . We refer the reader to the paper by Bogdan, Kumagai and Kwaśnicki [11] for general results on jump Feller processes, an excellent overview of the history, references and discussion on applications of BHI. Our present estimate (1.6) can be understood as a discrete time and space variant of the inequality stated by Kwaśnicki for jump isotropic α-stable processes in R d and V ≡ 0 [44, Corollary 3]. ...
We propose and study a certain discrete time counterpart of the classical Feynman--Kac semigroup with a confining potential in countable infinite spaces. For a class of long range Markov chains which satisfy the direct step property we prove sharp estimates for functions which are (sub-, super-)harmonic in infinite sets with respect to the discrete Feynman--Kac operators. These results are compared with respective estimates for the case of a nearest-neighbour random walk which evolves on a graph of finite geometry. We also discuss applications to the decay rates of solutions to equations involving graph Laplacians and to eigenfunctions of the discrete Feynman--Kac operators. We include such examples as non-local discrete Schr\"odinger operators based on fractional powers of the nearest-neighbour Laplacians and related quasi-relativistic operators. Finally, we analyse various classes of Markov chains which enjoy the direct step property and illustrate the obtained results by examples.
... For example, the first one together with some regularity assumptions on the Lévy measure imply the boundary Harnack inequality, which is also an important potential theoretic tool (see e.g. Bogdan et al., 2015). ...
... for any x ∈ J n , r ∈ (0, R]. This means J n is an Ahlfors d-space in the sense of [6]. Y n is a diffusion process on J n with Gaussian bounds (5.2). ...
... Y n is a diffusion process on J n with Gaussian bounds (5.2). Thus, we can apply [6,Proposition A.3] and obtain the next lemma. Lemma 5.1 Let n ∈ N and K be a compact subset of J n , and ε > 0. Then there exists a function f n ∈ L ∞ (J n , m) such that f n (x) = 1 for x ∈ K, f n (x) = 0 when dist(x, K) ≥ ε, and f ∈ D(L n ). ...
... Proof of Theorem 2. 6 We denote by {p 0 t } t>0 the semigroup of X 0 . Recall that each p 0 t has a jointly measurable density p 0 t (x, y) defined on D × D. Therefore, the condition (H.1) stated in [21] is satisfied for {p 0 t } t>0 . ...
In this paper, we consider first order Sobolev spaces with Robin boundary condition on unbounded Lipschitz domains. Hunt processes are associated with these spaces. We prove that the semigroup of these processes are doubly Feller. As a corollary, we provide a condition for semigroups generated by these processes being compact.
... Proof. This result follows from a combination of Lemma 3.2, Theorem 3.5 and the discussion in Example 3.9 in [8]. We only need to check the assumptions (A)-(D) in that paper. ...
... for every 0 < r < R. This proves (D) from [8]. ...
We study the long-time asymptotic behaviour of semigroups generated by non-local Schrödinger operators of the form H=−L+V; the free operator L is the generator of a symmetric Lévy process in Rd, d>1 (with non-degenerate jump measure) and V is a sufficiently regular confining potential. We establish sharp two-sided estimates of the corresponding heat kernels for large times and identify a new general regularity property, which we call progressive intrinsic ultracontractivity, to describe the large-time evolution of the corresponding Schrödinger semigroup. We discuss various examples and applications of these estimates, for instance we characterize the heat trace and heat content. Our examples cover a wide range of processes and we have to assume only mild restrictions on the growth, resp. decay, of the potential and the jump intensity of the free process. Our approach is based on a combination of probabilistic and analytic methods; our examples include fractional and quasi-relativistic Schrödinger operators.
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