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Heat Kernel Upper Estimates for Symmetric Jump Processes with Small Jumps of High Intensity

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Abstract

We consider the following non-local operator

$$ \mathcal{A}f(x)=\lim\limits_{\varepsilon\to 0}\int_{\{y\in \mathbb{R}^d\colon |x-y|>\varepsilon\}}(f(y)-f(x))n(x,y)\,dh. $$

where

$$ n(x,y)\asymp \frac{1}{|x-y|^{d+2}\left(\ln\frac{2}{|x-y|}\right)^{1+\beta}}\ \textrm{ for }\ |x-y|\leq 1 $$

and β ∈ (0, 1]. We prove upper estimates for the transition density of the associated symmetric Markov jump process X. Examples of Lévy processes with generator of the type above are studied.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, Postfach 100 131, 33501, Bielefeld, Germany

    Ante Mimica

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  1. Ante Mimica
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Correspondence to Ante Mimica.

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On leave from Department of Mathematics, University of Zagreb, Croatia.

Research supported in part by the MZOS Grant 037-0372790-2801 of the Republic of Croatia.

Research supported in part by German Science Foundation DFG via Internationales Graduiertenkolleg “Stochastics and Real World Models” and SFB 701.

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Mimica, A. Heat Kernel Upper Estimates for Symmetric Jump Processes with Small Jumps of High Intensity. Potential Anal 36, 203–222 (2012). https://doi.org/10.1007/s11118-011-9225-1

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