Heat Kernels of Non-symmetric Jump Processes: Beyond the Stable Case

Panki Kim, Renming Song, Zoran Vondraček

Research output: Contribution to journalArticlepeer-review


Let J be the Lévy density of a symmetric Lévy process in ℝd with its Lévy exponent satisfying a weak lower scaling condition at infinity. Consider the non-symmetric and non-local operatorℒκf(x):=limε↓0∫{z∈ℝd:|z|>ε}(f(x+z)−f(x))κ(x,z)J(z)dz,where κ(x, z) is a Borel function on ℝd× ℝd satisfying 0 < κ0 ≤ κ(x, z) ≤ κ1, κ(x, z) = κ(x,−z) and |κ(x, z) − κ(y, z)|≤ κ2|x − y|β for some β ∈ (0, 1]. We construct the heat kernel pκ(t, x, y) of ℒκ, establish its upper bound as well as its fractional derivative and gradient estimates. Under an additional weak upper scaling condition at infinity, we also establish a lower bound for the heat kernel pκ.

Original languageEnglish (US)
Pages (from-to)37-90
Number of pages54
JournalPotential Analysis
Issue number1
StatePublished - Jul 1 2018


  • Heat kernel estimates
  • Non-symmetric Markov process
  • Non-symmetric operator
  • Subordinate Brownian motion
  • Symmetric Lévy process

ASJC Scopus subject areas

  • Analysis


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