A note on the existence of transition probability densities of Lévy processes

Knopova, Victoria; Schilling, René L.

doi:10.1515/form.2011.108

Cited by 101 publications

(111 citation statements)

References 23 publications

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“…A first version of this result was already proved in [32]. While (1.4) is an exact formula, we get (1.5) p t (0) ≍ λ B d ψ 0, 1/ √ t whenever the metric measure space R n , d ψ , λ has the volume doubling property.…”

Section: Introductionmentioning

confidence: 52%

“…In this case we have local volume doubling. Let us close this section with an observation from [32,Lemma 9] where it is shown that the volume doubling property for large radii entails that (1+ψ) −κ/2 ∈ L 2 (R n , λ) for some suitable exponent κ > 0. This, in turn, has implications for the smoothness of the transition densities, cf.…”

Section: Metric Measure Spaces and Negative Definite Functionsmentioning

confidence: 87%

“…The following result is essentially contained in [32]. Note that in [32] the condition e −tψ ∈ L 1 (R n ) is substituted by a more general (Hartman-Wintner type) condition on the growth of ψ.…”

Section: Understanding the Role Of P T (0)mentioning

confidence: 99%

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A geometric interpretation of the transition density of a symmetric Lévy process

et al. 2012

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Abstract. We study for a class of symmetric Lévy processes with state space R n the transition density p t (x) in terms of two one-parameter families of metrics, (d t ) t>0 and (δ t ) t>0 . The first family of metrics describes the diagonal term p t (0); it is induced by the characteristic exponent ψ of the Lévy process by d t (x, y) = tψ(x − y). The second and new family of metrics δ t relates to √ tψ through the formulawhere F denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of the transition density:t (x,0) where p t (0) corresponds to a volume term related to √ tψ and where an "exponential" decay is governed by δ 2 t . This gives a complete and new geometric, intrinsic interpretation of p t (x). MSC 2010: Primary: 60J35. Secondary: 60E07; 60E10; 60G51; 60J45; 47D07; 31E05.

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Section: Introductionmentioning

confidence: 52%

Section: Metric Measure Spaces and Negative Definite Functionsmentioning

confidence: 87%

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A geometric interpretation of the transition density of a symmetric Lévy process

et al. 2012

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“…Clearly, this property extends to all t ≥ t b by the Markov property of (X t ) t≥0 . For more details on the existence and properties of transition probability densities for Lévy processes we refer to [37] and references therein. Finally, we remark that in many cases of interest Assumption (A3) follows directly from (rough) space-time estimates of the densities p(t, x).…”

Section: Definition 21 (Symmetric Jump-paring Lévy Processes)mentioning

confidence: 99%

Fall-Off of Eigenfunctions for Non-Local Schrödinger Operators with Decaying Potentials

Kaleta

Lőrinczi

2016

Potential Anal

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We study the spatial decay of eigenfunctions of non-local Schrödinger operators whose kinetic terms are generators of symmetric jump-paring Lévy processes with Kato-class potentials decaying at infinity. This class of processes has the property that the intensity of single large jumps dominates the intensity of all multiple large jumps. We find that the decay rates of eigenfunctions depend on the process via specific preference rates in particular jump scenarios, and depend on the potential through the distance of the corresponding eigenvalue from the edge of the continuous spectrum. We prove that the conditions of the jump-paring class imply that for all eigenvalues the corresponding positive eigenfunctions decay at most as rapidly as the Lévy intensity. This condition is sharp in the sense that if the jump-paring property fails to hold, then eigenfunction decay becomes slower than the decay of the Lévy intensity. We furthermore prove that under reasonable conditions the Lévy intensity also governs the upper bounds of eigenfunctions, and ground states are comparable with it, i.e., two-sided bounds hold. As an interesting consequence, we identify a sharp regime change in the decay of eigenfunctions as the Lévy intensity is varied from subexponential to exponential order, and dependent on the location of the eigenvalue, in the sense that through the transition Lévy intensity-driven decay becomes slower than the rate of decay of the Lévy intensity. Our approach is based on path integration and probabilistic potential theory techniques, and all results are also illustrated by specific examples.

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“…One of the key observations in [48] based on [50] was that the diagonal term of p t (x − y), i.e. p t (0), is then entirely controlled by the volume of the balls B d ψ .…”

Section: Introductionmentioning

confidence: 99%

Topics on Hamiltonian Dynamics Related to Symbols of Certain Schrodinger Operators Associated with Generators of Levy Processes

Rhind¹

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A note on the existence of transition probability densities of Lévy processes

Cited by 101 publications

References 23 publications

A geometric interpretation of the transition density of a symmetric Lévy process

A geometric interpretation of the transition density of a symmetric Lévy process

Fall-Off of Eigenfunctions for Non-Local Schrödinger Operators with Decaying Potentials

Topics on Hamiltonian Dynamics Related to Symbols of Certain Schrodinger Operators Associated with Generators of Levy Processes

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