Bernstein Functions

Schilling, René L.; Song, Renming; Vondraček, Zoran

doi:10.1515/9783110269338

Cited by 391 publications

(275 citation statements)

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“…We refer to the monograph [29] for details of subordinate stochastic processes. Let (X t ) t≥0 be a subordinator on [0, ∞) independent of B, i.e.…”

Section: Subordinate Brownian Motion On a Closed Manifoldmentioning

confidence: 99%

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Off-Diagonal Heat Kernel Asymptotics of Pseudodifferential Operators on Closed Manifolds and Subordinate Brownian Motion

Fahrenwaldt

2017

Integr. Equ. Oper. Theory

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Abstract.We derive the off-diagonal short-time asymptotics of the heat kernels of functions of generalised Laplacians on a closed manifold. As an intermediate step we give an explicit asymptotic series for the kernels of the complex powers of generalised Laplacians. Each asymptotic series is formulated in terms of the geodesic distance. The key application concerns upper bounds for the transition density of subordinate Brownian motion. The approach is highly explicit and tractable.Mathematics Subject Classification. Primary 35K08; Secondary 58J65, 35S05.

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“…We refer to the monograph [29] for details of subordinate stochastic processes. Let (X t ) t≥0 be a subordinator on [0, ∞) independent of B, i.e.…”

Section: Subordinate Brownian Motion On a Closed Manifoldmentioning

confidence: 99%

“…[24] for an early presentation and for example [27,28] for a general approach. We refer to [22,29] for a comprehensive and historically exhaustive presentation.…”

Section: Introductionmentioning

confidence: 99%

Off-Diagonal Heat Kernel Asymptotics of Pseudodifferential Operators on Closed Manifolds and Subordinate Brownian Motion

Fahrenwaldt

2017

Integr. Equ. Oper. Theory

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“…Nevertheless, they may be easily checked with the help of the following family of functions. We say that a C 8 -function φ : p0, 8q Ñ p0, 8q is a Bernstein function if p´1q n`1 φ pnq pλq ě 0 for all λ ą 0 and n P N .The class of Bernstein functions will be denoted by BF and it is known that every φ P BF can be uniquely represented in the following way (see [SSV12,Theorem 3…”

Section: Introductionmentioning

confidence: 99%

Exponential decay of measures and Tauberian theorems

Mimica

2016

Journal of Mathematical Analysis and Applications

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Abstract. We study behavior of a measure on r0, 8q by considering its Laplace transform. If it is possible to extend the Laplace transform to a complex half-plane containing the imaginary axis, then the exponential decay of the tail of the measure occurs and under certain assumptions we show that the rate of the decay is given by the so called abscissa of convergence and extend the result of Nakagawa from [Nak05]. Under stronger assumptions we give behavior of density of the measure by considering its Laplace transform. In situations when there is no exponential decay we study occurrence of heavy tails and give an application in the theory of non-local equations.

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“…We refers to L as a Laplacian if the semigroup e −tL is Markovian. In general, by Bochner's theorem, for any Laplacian L, the operator ψ(L) is again a Laplacian, provided ψ is a Bernstein function (see, for example, Schilling, Song and Vondraček [54]). It is known that ψ(λ) = λ α is a Bernstein function if and only if 0 < α ≤ 1.…”

Section: Subordinationmentioning

confidence: 99%

Изотропные Марковские Полугруппы На Ультраметрических Пространствах

et al. 2014

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Let (X, d) be a separable ultra-metric space with compact balls. Given a reference measure µ on X and a distance distribution function σ on [0 , ∞), we construct a symmetric Markov semigroup {P t } t≥0 acting in L 2 (X, µ). Let {X t } be the corresponding Markov process. We obtain upper and lower bounds of its transition density and its Green function, give a transience criterion, estimate its moments and describe the Markov generator L and its spectrum which is pure point. In the particular case when X = Q n p , where Q p is the field of p-adic numbers, our construction recovers the Taibleson Laplacian (spectral multiplier), and we can also apply our theory to the study of the Vladimirov Laplacian. Even in this well established setting, several of our results are new. We also elaborate the relation between our processes and Kigami's jump processes on the boundary of a tree which are induced by a random walk. In conclusion, we provide examples illustrating the interplay between the fractional derivatives and random walks.

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Bernstein Functions

Cited by 391 publications

References 0 publications

Off-Diagonal Heat Kernel Asymptotics of Pseudodifferential Operators on Closed Manifolds and Subordinate Brownian Motion

Off-Diagonal Heat Kernel Asymptotics of Pseudodifferential Operators on Closed Manifolds and Subordinate Brownian Motion

Exponential decay of measures and Tauberian theorems

Изотропные Марковские Полугруппы На Ультраметрических Пространствах

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