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2012 Harnack inequalities for subordinate Brownian motions
Ante Mimica, Panki Kim
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Electron. J. Probab. 17: 1-23 (2012). DOI: 10.1214/EJP.v17-1930


In this paper, we consider subordinate Brownian motion $X$ in $\mathbb{R}^d$, $d \ge 1$, where the Laplace exponent $\phi$ of the corresponding subordinator satisfies some mild conditions. The scale invariant Harnack inequality is proved for $X$. We first give new forms of asymptotical properties of the Lévy and potential density of the subordinator near zero. Using these results we find asymptotics of the Lévy density and potential density of $X$ near the origin, which is essential to our approach. The examples which are covered byour results include geometric stable processes and relativistic geometric stable processes, i.e. the cases when the subordinator has the Laplace exponent\[\phi(\lambda)=\log(1+\lambda^{\alpha/2})\ (0<\alpha\leq 2)\]and\[\phi(\lambda)=\log(1+(\lambda+m^{\alpha/2})^{2/\alpha}-m)\ (0<\alpha<2,\,m>0)\,.\]


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Ante Mimica. Panki Kim. "Harnack inequalities for subordinate Brownian motions." Electron. J. Probab. 17 1 - 23, 2012.


Accepted: 27 May 2012; Published: 2012
First available in Project Euclid: 4 June 2016

zbMATH: 1248.60092
MathSciNet: MR2928720
Digital Object Identifier: 10.1214/EJP.v17-1930

Primary: 60J45
Secondary: 60G50 , 60G51 , 60J25 , 60J27

Keywords: Geometric stable process , Green function , Harmonic function , Harnack inequality , Poisson kernel , potential , Subordinate Brownian motion , subordinator


Vol.17 • 2012
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