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

Let J be the Levy density of a symmetric Levy process in \(\mathbb {R}^{d}\) with its Levy exponent satisfying a weak lower scaling condition at infinity. Consider the non-symmetric and non-local operator $$\mathcal{L}^{\kappa}f(x):= \lim_{{\varepsilon} \downarrow 0} {\int}_{\{z \in \mathbb{R}^{d}: |z|>{\varepsilon}\}} (f(x+z)-f(x))\kappa(x,z)J(z)\, dz\, , $$ where κ(x, z) is a Borel function on \(\mathbb {R}^{d}\times \mathbb {R}^{d}\) satisfying 0 < κ 0 ≤ κ(x, z) ≤ κ 1, κ(x, z) = κ(x,−z) and |κ(x, z) − κ(y, z)|≤ κ 2|x − y| β for some β ∈ (0, 1]. We construct the heat kernel p κ (t, x, y) of \(\mathcal {L}^{\kappa }\), 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 .