Random Walk, Green Function, and Equilibrium Measure

  • Alexander Drewitz
  • Balázs Ráth
  • Artëm Sapozhnikov
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


In this chapter we collect some preliminary facts that we will need in the sequel. In Sect. 1.1 we introduce our basic notation related to subsets of \({\mathbb{Z}}^{d}\) and functions on \({\mathbb{Z}}^{d}\). In Sect. 1.2 we introduce simple random walk on \({\mathbb{Z}}^{d}\) and discuss properties of the Green function in the transient case d ≥ 3. In Sect. 1.3 we discuss the basics of potential theory, introduce the notion of equilibrium measure and capacity, and derive some of their properties.


Probability Measure Harmonic Function Green Function Harmonic Measure Electric Network 
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Copyright information

© The Author(s) 2014

Authors and Affiliations

  • Alexander Drewitz
    • 1
  • Balázs Ráth
    • 2
  • Artëm Sapozhnikov
    • 3
  1. 1.Department of MathematicsColumbia UniversityNew York CityUSA
  2. 2.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  3. 3.Max-Planck Institute of Mathematics in the SciencesLeipzigGermany

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