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

Let J be the Lévy density of a symmetric Lévy process in ℝd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}^{d}$\end{document} with its Lévy exponent satisfying a weak lower scaling condition at infinity. Consider the non-symmetric and non-local operator ℒκf(x):=limε↓0∫{z∈ℝd:|z|>ε}(f(x+z)−f(x))κ(x,z)J(z)dz,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{L}^{\kappa}f(x):= \lim_{{\varepsilon} \downarrow 0} {\int}_{\{z \in \mathbb{R}^{d}: |z|>{\varepsilon}\}} (f(x+z)-f(x))\kappa(x,z)J(z)\, dz\, , $$\end{document} where κ(x, z) is a Borel function on ℝd×ℝd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}^{d}\times \mathbb {R}^{d}$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {L}^{\kappa }$\end{document}, 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.