Harnack inequalities for subordinate Brownian motions

By Panki Kim, Ante Mimica. 2012

54 citations recorded by the source corpus.

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 Levy and potential density of the subordinator near zero. Using these results we find asymptotics of the Levy 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\[\phi(\lambda)=\log(1+\lambda^{\alpha/2})\ (0 0)\,.\]

Subfields: Applied Mathematics, Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/c590b66695fd7233e43cdb68f48e18b5ad59377a

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In Applied Mathematics

In Finance

In Management Science and Operations Research

Citations in the corpus

Barriers, exit time and survival probability for unimodal Lévy processes (2013, unknown): “(This observation was used before in [31].”
Gradient estimates of harmonic functions and transition densities for Levy processes (2013, methodology): “One can directly check that (H10) is satisfied (see also Example 1 in [17]), the fact that (H7) holds is well known, (H4) is obvious.”
On Harnack Inequality and Hölder Regularity for Isotropic Unimodal Lévy Processes (2013, background): “A class of special subordinate Brownian motions have been studied in [14], where bounds for the densities of Lévy measure and potential measure and Harnack inequalities were established.”