Results 11  20
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353
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 36 (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.
Gaussian Upper Bounds For The Heat Kernel On Arbitrary Manifolds
, 1997
"... In this paper, we develop a universal way of obtaining Gaussian upper bounds of the heat kernel on Riemannian manifolds. By the word "Gaussian" we mean those estimates which contain a Gaussian exponential factor similar to one which enters the explicit formula for the heat kernel of the co ..."
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Cited by 34 (2 self)
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In this paper, we develop a universal way of obtaining Gaussian upper bounds of the heat kernel on Riemannian manifolds. By the word "Gaussian" we mean those estimates which contain a Gaussian exponential factor similar to one which enters the explicit formula for the heat kernel of the conventional Laplace operator in R...
Functional inequalities for empty essential spectrum
 J. Funct. Anal
, 2000
"... In terms of the equivalence of Poincare ́ inequality and the existence of spectral gap, the superPoincare ́ inequality is suggested in the paper for the study of essential spectrum. It is proved for symmetric diffusions that, such an inequality is equivalent to empty essential spectrum of the corre ..."
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Cited by 33 (13 self)
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In terms of the equivalence of Poincare ́ inequality and the existence of spectral gap, the superPoincare ́ inequality is suggested in the paper for the study of essential spectrum. It is proved for symmetric diffusions that, such an inequality is equivalent to empty essential spectrum of the corresponding diffusion operator. This inequality recovers known Sobolev and Nash type ones. It is also equivalent to an isoperimetric inequality provided the curvature of the operator is bounded from below. Some results are also proved for a more general setting including symmetric jump processes. Moreover, estimates of inequality constants are also presented, which lead to a proof of a result on ultracontractivity suggested recently by D. Stroock. Finally, concentration of reference measures for superPoincare ́ inequalities is studied, the resulting estimates extend previous ones for Poincare ́ and logSobolev inequalities.
The Theory Of Generalized Dirichlet Forms And Its Applications In Analysis And Stochastics
, 1996
"... We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers b ..."
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Cited by 29 (1 self)
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We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers both the elliptic and the parabolic case within one approach. To this end we introduce a new class of bilinear forms, socalled generalized Dirichlet forms, which are in general neither symmetric nor coercive, but still generate associated C0 semigroups. Particular examples of generalized Dirichlet forms are symmetric and coercive Dirichlet forms (cf. [FOT], [MR1]) as well as time dependent Dirichlet forms (cf. [O1]). We discuss many applications to differential operators that can be treated within the new framework only, e.g. parabolic differential operators with unbounded drifts satisfying no L p conditions, singular and fractional diffusion operators. Subsequently, we analyz...
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 27 (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 25 (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.
Heatkernel approach to UV/IR mixing on isospectral deformation manifolds
"... We work out the general features of perturbative field theory on noncommutative manifolds defined by isospectral deformation. These (in general curved) ‘quantum spaces’, generalizing Moyal planes and noncommutative tori, are constructed using Rieffel’s theory of deformation quantization by actions o ..."
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Cited by 25 (3 self)
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We work out the general features of perturbative field theory on noncommutative manifolds defined by isospectral deformation. These (in general curved) ‘quantum spaces’, generalizing Moyal planes and noncommutative tori, are constructed using Rieffel’s theory of deformation quantization by actions of R l. Our framework, incorporating background field methods and tools of QFT in curved spaces, allows to deal both with compact and noncompact spaces, as well as with periodic and nonperiodic deformations, essentially in the same way. We compute the quantum effective action up to one loop for a scalar theory, showing the different UV/IR mixing phenomena for different kinds of isospectral deformations. The presence and behavior of the nonplanar parts of the Green functions is understood simply in terms of offdiagonal heat kernel contributions. For periodic deformations, a Diophantine condition on the noncommutivity parameters is found to play a role in the analytical nature of the nonplanar part of the oneloop reduced effective action. Existence of fixed points for the action may give rise to a new kind of UV/IR mixing. Keywords: noncommutative field theory, isospectral deformation, UV/IR mixing, heat kernel, Diophantine approximation.
Hypercontractivity for perturbed diffusion semigroups
 ANN. FAC. DES SC. DE TOULOUSE
, 2005
"... µ being a nonnegative measure satisfying some LogSobolev inequality, we give conditions on F for the Boltzmann measure ν = e −2F µ to also satisfy some LogSobolev inequality. This paper improves and completes the final section in [6]. A general sufficient condition and a general necessary conditio ..."
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Cited by 22 (15 self)
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µ being a nonnegative measure satisfying some LogSobolev inequality, we give conditions on F for the Boltzmann measure ν = e −2F µ to also satisfy some LogSobolev inequality. This paper improves and completes the final section in [6]. A general sufficient condition and a general necessary condition are given and examples are explicitly studied.