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59
An extension problem related to the fractional Laplacian
 1 DEGENERATE PARABOLIC EQUATIONS 19
, 2007
"... The operator square root of the Laplacian (−△) 1/2 can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the L ..."
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Cited by 40 (7 self)
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The operator square root of the Laplacian (−△) 1/2 can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the Laplacian and other integrodifferential operators. From those characterizations we derive some properties of these integrodifferential equations from purely local arguments in the extension problems. 1
Heat kernel estimates for Dirichlet fractional Laplacian
 J. European Math. Soc
"... In this paper, we consider the fractional Laplacian −(−∆) α/2 on an open subset in R d with zero exterior condition. We establish sharp twosided estimates for the heat kernel of such Dirichlet fractional Laplacian in C 1,1 open sets. This heat kernel is also the transition density of a rotationally ..."
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Cited by 35 (19 self)
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In this paper, we consider the fractional Laplacian −(−∆) α/2 on an open subset in R d with zero exterior condition. We establish sharp twosided estimates for the heat kernel of such Dirichlet fractional Laplacian in C 1,1 open sets. This heat kernel is also the transition density of a rotationally symmetric stable process killed upon leaving a C 1,1 open set. Our results are the first sharp twosided estimates for the Dirichlet heat kernel of a nonlocal operator on open sets.
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 heat kernel estimates for censored stablelike processes
, 2008
"... In this paper we study the precise behavior of the transition density functions of censored (resurrected) αstablelike processes in C 1,1 open sets in R d, where d ≥ 1 and α ∈ (1, 2). We first show that the semigroup of the censored αstablelike process in any bounded Lipschitz open set is intrins ..."
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Cited by 24 (17 self)
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In this paper we study the precise behavior of the transition density functions of censored (resurrected) αstablelike processes in C 1,1 open sets in R d, where d ≥ 1 and α ∈ (1, 2). We first show that the semigroup of the censored αstablelike process in any bounded Lipschitz open set is intrinsically ultracontractive. We then establish sharp twosided estimates for the transition density functions of a large class of censored αstablelike processes in C 1,1 open sets. We further obtain sharp twosided estimates for the Green functions of these censored αstablelike processes in bounded C 1,1 open sets.
Estimates of Green function for some perturbations of fractional Laplacian
"... Suppose that Y (t) is a ddimensional Lévy symmetric process for which its Lévy measure differs from the Lévy measure of the isotropic αstable process (0 < α < 2) by a finite signed measure. For a bounded Lipschitz set D we compare the Green functions of the process Y and its stable counterpart. We ..."
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Cited by 23 (3 self)
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Suppose that Y (t) is a ddimensional Lévy symmetric process for which its Lévy measure differs from the Lévy measure of the isotropic αstable process (0 < α < 2) by a finite signed measure. For a bounded Lipschitz set D we compare the Green functions of the process Y and its stable counterpart. We prove a few comparability results either one sided or two sided. Assuming an additional condition about the difference of the densities of the Lévy measures, namely that it is of order of x  −d+̺ as x  → 0, where ̺> 0, we prove that the Green functions are comparable, provided D is connected. These results apply for example to αstable relativistic process. This process was studied in [R, CS3], where the bounds for its Green functions were proved for d> α and smooth sets. In the paper we also considered one dimensional case for α ≥ 1 and proved that the Green functions for an open and bounded interval are comparable. 1
Potential theory of truncated stable processes
 MATHEMATISCHE ZEITSCHRIFT
, 2007
"... For any α ∈ (0, 2), a truncated symmetric αstable process is a symmetric Lévy process in R d with a Lévy density given by cx  −d−α 1{x<1} for some constant c. In this paper we study the potential theory of truncated symmetric stable processes in detail. We prove a Harnack inequality for nonneg ..."
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Cited by 22 (18 self)
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For any α ∈ (0, 2), a truncated symmetric αstable process is a symmetric Lévy process in R d with a Lévy density given by cx  −d−α 1{x<1} for some constant c. In this paper we study the potential theory of truncated symmetric stable processes in detail. We prove a Harnack inequality for nonnegative harmonic functions of these processes. We also establish a boundary Harnack principle for nonnegative functions which are harmonic with respect to these processes in bounded convex domains. We give an example of a nonconvex domain for which the boundary Harnack principle fails.
Boundary Harnack principle for subordinate Brownian motions
"... We establish a boundary Harnack principle for a large class of subordinate Brownian motions, including mixtures of symmetric stable processes, in κfat open sets (disconnected analogue of John domains). As an application of the boundary Harnack principle, we identify the Martin boundary and the mini ..."
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Cited by 21 (18 self)
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We establish a boundary Harnack principle for a large class of subordinate Brownian motions, including mixtures of symmetric stable processes, in κfat 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 κfat open sets with respect to these processes with their Euclidean boundaries.
Intrinsic ultracontractivity of the Feynman–Kac semigroup for relativisitic stable processes
 ZBL 1112.47034 MR 2231884
, 2006
"... Let Xt be the relativistic αstable process in Rd, α ∈ (0, 2), d>α, with infinitesimal generator H (α) 0 = −((− ∆ +m2/α) α/2 − m). We study intrinsic ultracontractivity (IU) for the FeynmanKac semigroup Tt for this process with generator H (α) 0 − V, V ≥ 0, V locally bounded. We prove that if lim ..."
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Cited by 20 (3 self)
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Let Xt be the relativistic αstable process in Rd, α ∈ (0, 2), d>α, with infinitesimal generator H (α) 0 = −((− ∆ +m2/α) α/2 − m). We study intrinsic ultracontractivity (IU) for the FeynmanKac semigroup Tt for this process with generator H (α) 0 − V, V ≥ 0, V locally bounded. We prove that if lim x→ ∞ V (x) =∞, then for every t>0 the operator Tt is compact. We consider the class V of potentials V such that V ≥ 0, lim x→ ∞ V (x) = ∞ and V is comparable to the function which is radial, radially nondecreasing and comparable on unit balls. For V in the class V we show that the semigroup Tt is IU if and only if lim x→ ∞ V (x)/x  = ∞. If this condition is satisfied we also obtain sharp estimates of the first eigenfunction φ1 for Tt. Inparticular, when V (x) =x  β, β>0, then the semigroup Tt is IU if and only if β>1. For β>1 the first eigenfunction φ1(x) is comparable to exp(−m 1/α x)(x  +1) (−d−α−2β−1)/2.
Generalized 3G theorem and application to relativistic stable process on nonsmooth open sets
, 2006
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Censored Stable Processes
 Probab. Theory Relat. Fields
"... We present several constructions of a \censored stable process" in an open set D R n , i.e., a symmetric stable process which is not allowed to jump outside D. ..."
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Cited by 19 (11 self)
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We present several constructions of a \censored stable process" in an open set D R n , i.e., a symmetric stable process which is not allowed to jump outside D.