Asymptotic behavior of densities of unimodal convolution semigroups

Cygan, Wojciech; Grzywny, Tomasz; Trojan, Bartosz

doi:10.1090/tran/6830

Cited by 25 publications

(25 citation statements)

References 27 publications

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“…The present article further develops the potential theory of isotropic unimodal Lévy process in R d , extending previous works in this area ( [5,6,7,14,16,22,23,24,26,27,29]). Our main results are: estimates of hitting probabilities of balls, estimates of the Green function of the full space (the potential kernel), half-spaces and balls; supremum estimate and regularity result for harmonic functions; and the boundary Harnack inequality.…”

Section: Introductionsupporting

confidence: 73%

Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes

Grzywny

Kwaśnicki

2018

Stochastic Processes and their Applications

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Abstract. In the first part of this article, we prove two-sided estimates of hitting probabilities of balls, the potential kernel and the Green function for a ball for general isotropic unimodal Lévy processes. Our bounds are sharp under the absence of the Gaussian component and a mild regularity condition on the density of the Lévy measure: its radial profile needs to satisfy a scaling-type condition, which is equivalent to O-regular variation at zero and at infinity with lower indices greater than −d − 2. We also prove a supremum estimate and a regularity result for functions harmonic with respect to a general isotropic unimodal Lévy process.In the second part we apply the recent results on the boundary Harnack inequality and Martin representation of harmonic functions for the class of isotropic unimodal Lévy processes characterised by a localised version of the scaling-type condition mentioned above. As a sample application, we provide sharp two-sided estimates of the Green function of a half-space.Our results are expressed in terms of Pruitt's functions K(r) and L(r), measuring local activity and the amount of large jumps of the Lévy process, respectively.

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Section: Introductionsupporting

confidence: 73%

Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes

Grzywny

Kwaśnicki

2018

Stochastic Processes and their Applications

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“…We note in passing that the spatial asymptotics of the heat kernel at infinity were given in the works of Blumenthal and Getoor [11,12] (see also [37]), who showed that p t (x) ∼ tν(x) as t|x| −α → 0. More results of this type for unimodal Lévy processes can be found in recent works of Tomasz Grzywny et al, including [37], however, the above papers only concern Γ = R d .…”

Section: Full Picturementioning

confidence: 95%

Yaglom limit for stable processes in cones

Bogdan¹,

Palmowski²,

Wang³

2018

Electron. J. Probab.

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We give the asymptotics of the tail of the distribution of the first exit time of the isotropic αstable Lévy process from the Lipschitz cone in R d . We obtain the Yaglom limit for the killed stable process in the cone. We construct and estimate entrance laws for the process from the vertex into the cone. For the symmetric Cauchy process and the positive half-line we give a spectral representation of the Yaglom limit.Our approach relies on the scalings of the stable process and the cone, which allow us to express the temporal asymptotics of the distribution of the process at infinity by means of the spatial asymptotics of harmonic functions of the process at the vertex; on the representation of the probability of survival of the process in the cone as a Green potential; and on the approximate factorization of the heat kernel of the cone, which secures compactness and yields a limiting (Yaglom) measure by means of Prokhorov's theorem.Keywords. Yaglom limit ⋆ stable process ⋆ Lipschitz cone ⋆ quasi-stationary measure ⋆ Green function ⋆ Martin kernel ⋆ excursions.

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“…With the above notation, under the assumption (D), for every r > 0 and t ∈ T we have the following: (11) and…”

Section: Preliminariesmentioning

confidence: 99%

Spatial asymptotics at infinity for heat kernels of integro-differential operators

Kaleta

Sztonyk

2018

Trans. Amer. Math. Soc.

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We study a spatial asymptotic behaviour at infinity of kernels p t (x) for convolution semigroups of nonlocal pseudo-differential operators. We give general and sharp sufficient conditions under which the limitsexist and can be effectively computed. Here ν is the corresponding Lévy density, T ⊂ (0, ∞) is a bounded time-set and E is a subset of the unit sphere in R d , d ≥ 1. Our results are local on the unit sphere. They apply to a wide class of convolution semigroups, including those corresponding to highly asymmetric (finite and infinite) Lévy measures. Key examples include fairly general families of stable, tempered stable, jump-diffusion and compound Poisson semigroups. A main emphasis is put on the semigroups with Lévy measures that are exponentially localized at infinity, for which our assumptions and results are strongly related to the existence of the multidimensional exponential moments. Here a key example is the evolution semigroup corresponding to the so-called quasi-relativistic Hamiltonian √ −∆ + m 2 − m, m > 0. As a byproduct, we also obtain sharp two-sided estimates of the kernels p t in generalized cones, away from the origin.

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Asymptotic behavior of densities of unimodal convolution semigroups

Abstract: Abstract. We prove the asymptotic formulas for the densities of isotropic unimodal convolution semigroups of probability measures on R d under the assumption that its Lévy-Khintchine exponent is regularly varying of index between 0 and 2.

Cited by 25 publications

References 27 publications

Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes

Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes

Yaglom limit for stable processes in cones

Spatial asymptotics at infinity for heat kernels of integro-differential operators

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