Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part

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Abstract

We consider the Dirichlet form given by E(f,f)=12∫Rd∑i,j=1daij(x)∂f(x)∂xi∂f(x)∂xjdx E(f,f)=12∫Rd∑i,j=1daij(x)∂f(x)∂xi∂f(x)∂xjdx +∫Rd×Rd(f(y)−f(x))2J(x,y)dxdy. +∫Rd×Rd(f(y)−f(x))2J(x,y)dxdy. Under the assumption that the aij are symmetric and uniformly elliptic and with suitable conditions on J, the nonlocal part, we obtain upper and lower bounds on the heat kernel of the Dirichlet form. We also prove a Harnack inequality and a regularity theorem for functions that are harmonic with respect to E.

Original languageEnglish
Pages (from-to)314-340
Number of pages27
JournalElectronic Journal of Probability
Volume14
DOIs
Publication statusPublished - 7 Jan 2009

Keywords

  • Harnack inequality
  • heat kernel
  • Hölder continuity
  • integro-differential operators

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