Electronic Journal of Probability

Harnack inequalities for subordinate Brownian motions

Ante Mimica and Panki Kim

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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\[\phi(\lambda)=\log(1+(\lambda+m^{\alpha/2})^{2/\alpha}-m)\ (0<\alpha<2,\,m>0)\,.\]

Article information

Electron. J. Probab., Volume 17 (2012), paper no. 37, 23 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]
Secondary: 60G50: Sums of independent random variables; random walks 60G51: Processes with independent increments; Lévy processes 60J25: Continuous-time Markov processes on general state spaces 60J27: Continuous-time Markov processes on discrete state spaces

geometric stable process Green function Harnack inequality Poisson kernel harmonic function potential subordinator subordinate Brownian motion

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Mimica, Ante; Kim, Panki. Harnack inequalities for subordinate Brownian motions. Electron. J. Probab. 17 (2012), paper no. 37, 23 pp. doi:10.1214/EJP.v17-1930. https://projecteuclid.org/euclid.ejp/1465062359

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