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Potential theory of special subordinators and subordinate killed stable processes
 J. Theoret. Probab
, 2006
"... In this paper we introduce a large class of subordinators called special subordinators and study their potential theory. Then we study the potential theory of processes obtained by subordinating a killed symmetric stable process in a bounded open set D with special subordinators. We establish a one ..."
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Cited by 26 (19 self)
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In this paper we introduce a large class of subordinators called special subordinators and study their potential theory. Then we study the potential theory of processes obtained by subordinating a killed symmetric stable process in a bounded open set D with special subordinators. We establish a onetoone correspondence between the nonnegative harmonic functions of the killed symmetric stable process and the nonnegative harmonic functions of the subordinate killed symmetric stable process. We show that nonnegative harmonic functions of the subordinate killed symmetric stable process are continuous and satisfy a Harnack inequality. We then show that, when D is a bounded κfat set, both the Martin boundary and the minimal Martin boundary of the subordinate killed symmetric stable process in D coincide with the Euclidean boundary ∂D.
Twosided estimates on the density of Brownian motion with singular drift
 Ill. J. Math
, 2006
"... Abstract. Let µ = (µ 1,..., µ d) be such that each µ i is a signed measure on R d belonging to the Kato class Kd,1. The existence and uniqueness of a continuous Markov process X on R d, called a Brownian motion with drift µ, was recently established by Bass and Chen. In this paper we study the poten ..."
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Cited by 20 (19 self)
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Abstract. Let µ = (µ 1,..., µ d) be such that each µ i is a signed measure on R d belonging to the Kato class Kd,1. The existence and uniqueness of a continuous Markov process X on R d, called a Brownian motion with drift µ, was recently established by Bass and Chen. In this paper we study the potential theory of X. We show that X has a continuous density q µ and that there exist positive constants ci, i = 1, · · · , 9, such that and c1e −c2t − t d 2 e − c3 x−y2 2t ≤ q µ (t, x, y) ≤ c4e c5t − t d 2 e − c6 x−y2 2t ∇xq µ (t, x, y)  ≤ c7e c8t − t d+1 2 e − c9 x−y2 2t for all (t, x, y) ∈ (0, ∞) × R d × R d. We further show that, for any bounded C 1,1 domain D, the density q µ,D of X D, the process obtained by killing X upon exiting from D, has the following estimates: for any T> 0, there exist positive constants Ci, i = 1, · · · , 5, such that and C1(1 ∧ ρ(x) √ t)(1 ∧ ρ(y) √ t)t − d 2 e − C 2 x−y2 t ≤ q µ,D (t, x, y) ≤ C3(1 ∧ ρ(x) √)(1 ∧ t ρ(y)
Estimates on Green functions and Schrödingertype equations for nonsymmetric diffusions with measurevalued drifts
 J. MATH. ANAL. APPL.
, 2007
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Potential theory of subordinate Brownian motions revisited’, Stochastic analysis and applications to finance–essays
 in honour of Jiaan Yan, (eds
, 2012
"... The paper discusses and surveys some aspects of the potential theory of subordinate Brownian motion under the assumption that the Laplace exponent of the corresponding subordinator is comparable to a regularly varying function at infinity. This extends some results previously obtained under stronger ..."
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Cited by 13 (10 self)
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The paper discusses and surveys some aspects of the potential theory of subordinate Brownian motion under the assumption that the Laplace exponent of the corresponding subordinator is comparable to a regularly varying function at infinity. This extends some results previously obtained under stronger conditions.
On the potential theory of onedimensional subordinate Brownian motions with continuous components
, 2008
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Green Function Estimates and Harnack Inequality for Subordinate Brownian Motions
, 2004
"... Abstract. Let X be a Lévy process in R d, dU3, 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 Gr ..."
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Cited by 6 (3 self)
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Abstract. Let X be a Lévy process in R d, dU3, 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.
Higher order PDE’s and iterated processes
"... We introduce a class of stochastic processes based on symmetric αstable processes, for α ∈ (0, 2]. These are obtained by taking Markov processes and replacing the time parameter with the modulus of a symmetric αstable process. We call them αtime processes. They generalize Brownian time processes ..."
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Cited by 5 (3 self)
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We introduce a class of stochastic processes based on symmetric αstable processes, for α ∈ (0, 2]. These are obtained by taking Markov processes and replacing the time parameter with the modulus of a symmetric αstable process. We call them αtime processes. They generalize Brownian time processes studied in [1, 2, 3], and they introduce new interesting examples. We establish the connection of α−time processes to some higher order PDE’s for α rational. We also study the exit problem for αtime processes as they exit regular domains and connect them to elliptic PDE’s. We also obtain the PDE connection of subordinate killed Brownian motion in bounded domains of regular boundary.
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 esta ..."
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Cited by 3 (1 self)
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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 Harnackinequality 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