Heat kernel upper estimates for symmetric jump processes with small jumps of high intensity. (English) Zbl 1239.60077
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.
Reviewer: Liliana Popa (Iaşi)
MSC:
| 60J35 | Transition functions, generators and resolvents |
| 60J75 | Jump processes (MSC2010) |
| 60G51 | Processes with independent increments; Lévy processes |
Keywords:
Dirichlet form; heat kernel estimates; jump process; Nash inequality; subordinate Brownian motionReferences:
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