TY - JOUR
T1 - Heat Kernels of Non-symmetric Jump Processes
T2 - Beyond the Stable Case
AU - Kim, Panki
AU - Song, Renming
AU - Vondraček, Zoran
N1 - Funding Information:
Panki Kim was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (No. 2016R1E1A1A01941893) Renming Song supported in part by a grant from the Simons Foundation (208236) Zoran Vondracˇek was supported in part by the Croatian Science Foundation under the project 3526.
Publisher Copyright:
© 2017, Springer Science+Business Media B.V.
PY - 2018/7/1
Y1 - 2018/7/1
N2 - 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.
AB - 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.
KW - Heat kernel estimates
KW - Non-symmetric Markov process
KW - Non-symmetric operator
KW - Subordinate Brownian motion
KW - Symmetric Lévy process
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U2 - 10.1007/s11118-017-9648-4
DO - 10.1007/s11118-017-9648-4
M3 - Article
AN - SCOPUS:85027078259
VL - 49
SP - 37
EP - 90
JO - Potential Analysis
JF - Potential Analysis
SN - 0926-2601
IS - 1
ER -