Harnack inequality and boundary Harnack principle for subordinate killed Brownian motion (Probability Symposium) (original) (raw)

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.

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.

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.

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.

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 with respect to subordinate killed Brownian motions

Stochastic Processes and their Applications, 2015

Minimal thinness is a notion that describes the smallness of a set at a boundary point. In this paper, we provide tests for minimal thinness for a large class of subordinate killed Brownian motions in bounded C 1,1 domains, C 1,1 domains with compact complements and domains above graphs of bounded C 1,1 functions.

Potential theory of subordinate killed Brownian motion in a domain

Probability Theory and Related Fields, 2003

Subordination of a killed Brownian motion in a bounded domain D ⊂ R d via an α/2stable subordinator gives a process Z t whose infinitesimal generator is −(−∆| D) α/2 , the fractional power of the negative Dirichlet Laplacian. In this paper we study the properties of the process Z t in a Lipschitz domain D by comparing the process with the rotationally invariant α-stable process killed upon exiting D. We show that these processes have comparable killing measures, prove the intrinsic ultracontractivity of the semigroup of Z t , and, in the case when D is a bounded C 1,1 domain, obtain bounds on the Green function and the jumping kernel of Z t .

Harnack Inequality and Applications for

2014

The dimension-free Harnack inequality and uniform heat kernel upper/lower bounds are derived for a class of infinite-dimensional GEM processes, which was introduced in [7] to simulate the two-parameter GEM distributions. In particular, the associated Dirichlet form satisfies the super log-Sobolev inequality which strengthens the log-Sobolev inequality derived in [7]. To prove the main results, explicit Harnack inequality and super Poincaré inequality are established for the one-dimensional Wright-Fisher diffusion processes. The main tool of the study is the coupling by change of measures.

Harnack inequality for some classes of Markov processes

Mathematische Zeitschrift, 2004

1 Department of Mathematics, University of Illinois,Urbana, IL 61801, USA (e-mail: rsong@math.uiuc.edu) 2 Department of Mathematics, University of Zagreb, Zagreb, Croatia (e-mail: vondra@math.hr) ... Received: 16 October 2002; in final form: 16 May 2003 / ...