Global uniform boundary Harnack principle with explicit decay rate and its application (original) (raw)
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Boundary Harnack principle for subordinate Brownian motions
Stochastic Processes and their Applications, 2009
We establish a boundary Harnack principle for a large class of subordinate Brownian motions, including mixtures of symmetric stable processes, in κ-fat open sets (disconnected analogue of John domains). As an application of the boundary Harnack principle, we identify the Martin boundary and the minimal Martin boundary of bounded κ-fat open sets with respect to these processes with their Euclidean boundaries.
2017
The purpose of this note is to provide a summary of the main results of our recent paper [8], where we estabhsh scale invariant Harnack inequality (HI) and boundary Harnack principle for subordinate killed Brownian motions. For simplicity, we only present the results in the case when the dimension is greater tham or equal to 3 and the domain D is bounded. AMS 2010 Mathematics Subject Classification: Primary 60\mathrm{J}45 ; Secondary 60\mathrm{J}50, 60\mathrm{J}75.
Two-sided Green function estimates for killed subordinate Brownian motions
Proceedings of the London Mathematical Society, 2012
A subordinate Brownian motion is a Lévy process which can be obtained by replacing the time of the Brownian motion by an independent subordinator. The infinitesimal generator of a subordinate Brownian motion is −φ(−∆), where φ is the Laplace exponent of the subordinator. In this paper, we consider a large class of subordinate Brownian motions without diffusion component and with φ comparable to a regularly varying function at infinity. This class of processes includes symmetric stable processes, relativistic stable processes, sums of independent symmetric stable processes, sums of independent relativistic stable processes, and much more. We give sharp two-sided estimates on the Green functions of these subordinate Brownian motions in any bounded κ-fat open set D. When D is a bounded C 1,1 open set, we establish an explicit form of the estimates in terms of the distance to the boundary. As a consequence of such sharp Green function estimates, we obtain a boundary Harnack principle in C 1,1 open sets with explicit rate of decay.
Green function estimates and Harnack inequality for subordinate Brownian motions
Potential analysis, 2006
Let X be a Lévy process in R d , d ≥ 3, obtained by subordinating Brownian motion with a subordinator with a positive drift. Such a process has the same law as the sum of an independent Brownian motion and a Lévy process with no continuous component. We study the asymptotic behavior of the Green function of X near zero. Under the assumption that the Laplace exponent of the subordinator is a complete Bernstein function we also describe the asymptotic behavior of the Green function at infinity. With an additional assumption on the Lévy measure of the subordinator we prove that the Harnack inequality is valid for the nonnegative harmonic functions of X.
Potential theory of subordinate Brownian motions with Gaussian components
Stochastic Processes and their Applications, 2013
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 C 1,1 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 C 1,1 open set D and identify the Martin boundary of D with respect to the subordinate Brownian motion with the Euclidean boundary.
On the potential theory of one-dimensional subordinate Brownian motions with continuous components
2008
Suppose thatS is a subordinator with a nonzero drift andW is an independent 1-dimensional Brownian motion. We study the subordinate Brownian motion X dened by Xt = W (St). We give sharp bounds for the Green function of the process X killed upon exiting a bounded open interval and prove a boundary Harnack principle. In the case when S is a stable subordinator with a positive drift, we prove sharp bounds for the Green function of X in (0;1), and sharp bounds for the Poisson kernel of X in a bounded open interval.
A note on local asymptotic behaviour for Brownian motion in Banach spaces
International Journal of Mathematics and Mathematical Sciences, 1979
In this paper we obtain an integral characterization of a two-sided upper function for Brownian motion in a real separable Banach space. This characterization generalizes that of Jain and Taylor [2] whereB=ℝn. The integral test obtained involves the index of a mean zero Gaussian measure on the Banach space, which is due to Kuelbs [3]. The special case that whenBis itself a real separable Hilbert space is also illustrated.
Harmonic functions of subordinate killed Brownian motion
Journal of Functional Analysis, 2004
In this paper we study harmonic functions of subordinate killed Brownian motion in a domain D. We first prove that, when the killed Brownian semigroup in D is intrinsic ultracontractive, all nonnegative harmonic functions of the subordinate killed Brownian motion in D are continuous and then we establish a Harnack inequality for these harmonic functions. We then show that, when D is a bounded Lipschitz domain, both the Martin boundary and the minimal Martin boundary of the subordinate killed Brownian motion in D coincide with the Euclidean boundary ∂D. We also show that, when D is a bounded Lipschitz domain, a boundary Harnack principle holds for positive harmonic functions of the subordinate killed Brownian motion in D.
Minimal thinness for subordinate Brownian motion in half-space
Annales de l’institut Fourier, 2012
We study minimal thinness in the half-space H := {x = (x, x d) : x ∈ R d−1 , x d > 0} for a large class of rotationally invariant Lévy processes, including symmetric stable processes and sums of Brownian motion and independent stable processes. We show that the same test for the minimal thinness of a subset of H below the graph of a nonnegative Lipschitz function is valid for all processes in the considered class. In the classical case of Brownian motion this test was proved by Burdzy.