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

Let J be the Lévy density of a symmetric Lévy process in (mathbb {R}^{d}) with its Lévy 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 .