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Minimal thinness with respect to subordinate killed Brownian motions
Kim, P.,Song, R.,Vondracek, Z. North-Holland Pub. Co ; Elsevier Science Ltd 2016 Stochastic processes and their applications Vol.126 No.4
<P>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,C-1 domains, C-1,C-1 domains with compact complements and domains above graphs of bounded C-1,C-1 functions. (C) 2015 Elsevier B.V. All rights reserved.</P>
Potential theory of subordinate Brownian motions with Gaussian components
Kim, P.,Song, R.,Vondracek, Z. North-Holland Pub. Co ; Elsevier Science Ltd 2013 Stochastic processes and their applications Vol.123 No.3
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 Levy 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<SUP>1,1</SUP> 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<SUP>1,1</SUP> open set D and identify the Martin boundary of D with respect to the subordinate Brownian motion with the Euclidean boundary.
Boundary Harnack principle for subordinate Brownian motions
Kim, P.,Song, R.,Vondracek, Z. North-Holland Pub. Co ; Elsevier Science Ltd 2009 Stochastic processes and their applications Vol.119 No.5
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.
Global uniform boundary Harnack principle with explicit decay rate and its application
Kim, P.,Song, R.,Vondracek, Z. North-Holland Pub. Co ; Elsevier Science Ltd 2014 Stochastic processes and their applications Vol.124 No.1
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.