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.

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 -