Potential theory of geometric stable processes

Hrvoje Šikić, Renming Song, Zoran Vondraček

Research output: Contribution to journalArticlepeer-review


In this paper we study the potential theory of symmetric geometric stable processes by realizing them as subordinate Brownian motions with geometric stable subordinators. More precisely, we establish the asymptotic behaviors of the Green function and the Lévy density of symmetric geometric stable processes. The asymptotics of these functions near zero exhibit features that are very different from the ones for stable processes. The Green function behaves near zero as 1/(|x|d log2 |x|). while the Lévy density behaves like 1/|x|d. We also study the asymptotic behaviors of the Green function and Lévy density of subordinate Brownian motions with iterated geometric stable subordinators. As an application, we establish estimates on the capacity of small balls for these processes, as well as mean exit time estimates from small balls and a Harnack inequality for these processes.

Original languageEnglish (US)
Pages (from-to)547-575
Number of pages29
JournalProbability Theory and Related Fields
Issue number4
StatePublished - Aug 2006


  • Capacity
  • Geometric stable processes
  • Green function
  • Harnack inequality

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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