Heat kernel upper estimates for symmetric jump processes with small jumps of high intensity (Q663502)

Starting with a (non-local) operator \[ \mathcal{A}f(x) = \lim_{\varepsilon\to 0} \int_{\{y\in {\mathbb{R}}^d : |x-y|>\varepsilon\}} (f(y)-f(x))n(x,y)\,dh \] where \[ n(x,y) \asymp \frac{1}{|x-y|^{d+2}\left(\ln\frac{2}{|x-y|}\right)^{1+\beta}} \] (for \(|x-y| \leq 1 \) and \(\beta \in (0,1]\)), the author proves an upper estimate for the transition density of the associated symmetric Markov jump process: \[ p(t, x , y) \leq C \min \left \{t^ {-d /2} \left ( \ln \frac{2}{t}\right )^ {\beta d / 2} , \frac{t}{|x-y|^{d+2} \left (\ln \frac{2}{|x-y|}\right )^ {\beta }} \right \} \] During the proof, examples of Lévy processes with generator of the type above are investigated.