Bernstein Functions: Theory and Applications

By René L. Schilling, Renming Song, Zoran Vondraček. 2010

559 citations recorded by the source corpus.

Subfields: Computational Mechanics, Statistics and Probability

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Citations in the corpus

On the Schoenberg Transformations in Data Analysis: Theory and Illustrations (2010, background): “…al. 2008): where µ is a non-negative measure on (0 , ∞ ) such that There exists an important literature about Bernstein functions (see e.g. Berg et al. 2008; Schilling et al. 2010; and references therein), deﬁned as the smooth non-negative functions whose ﬁrst derivatives are completely monotonic.”
Two‐sided Green function estimates for killed subordinate Brownian motions (2010, background): “12 of [27] to come up with infinitely many examples of such functions, e.”
Suprema of Lévy processes (2011, background): “Furthermore, πc̃1δ0(dζ) + μ̃(dζ) is the limit of measures − Im(ψ̃(−ζ + iε))dζ as ε → 0+ ([34], Corollary 6.”
Relaxation patterns and semi-Markov dynamics (2015, methodology): “herefore the L´evy density of ν is also completely monotone. The corresponding Bernstein function f is thus a complete Bernstein function. Thereforethe adjoint f⋆ is also complete (Proposition 7.1 in [55]) and has a L´evy density which is completely monotone with tail ν¯⋆(s) =a⋆ + Z∞ s Z∞ 0 e−twm⋆(d”
Stability of heat kernel estimates for symmetric non-local Dirichlet forms (2016, background): “See for instance [SSV] for the general theory of subordinations.”
Some properties of Prabhakar-type fractional calculus operators (2015, background): “The function eγα ,β ,1(t) is completely monotone whenever α ∈ (0,1] , 0 αγ β 1 [5, 31], and therefore by the Bernstein Theorem [26] the spectral function Kγα ,β (r) is non-negative for the same range of the parameters.”
Transition Density Estimates for a Class of Lévy and Lévy-Type Processes (2009, unknown): “This is a classic result by Schönberg and Bochner, see [33] or [25].”
A comment on a controversial issue: A generalized fractional derivative cannot have a regular kernel (2020, background): “The function g is completely monotone because (1) tβ, 0 < β < 1, is a Bernstein function; (2) the composition of the completely monotone function exp(−x) with the Bernstein function tβ is CM [20], (3) the product of two completely monotone functions is completely monotone [20], (4) g(t) is integrabl”
Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise (2011, unknown): “4], [47].”
Dirichlet heat kernel estimates for rotationally symmetric Lévy processes (2013, background): “Using the tables at the end of [23], one can come up plenty of explicit examples of complete Bernstein functions satisfying condition (A).”