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

Let J be the Lévy density of a symmetric Lévy process in ℝ^d 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 ℝ^d× ℝ^d 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 p^κ(t, x, y) of ℒ^κ, 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 p^κ.