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Small-time sharp bounds for kernels of convolution semigroups

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Abstract

We study small-time bounds for transition densities of convolution semigroups corresponding to pure jump Lévy processes in Rd, d ≥ 1, including the processes with jump measures which are exponentially and subexponentially localized at ∞. For a large class of Lévy measures, not necessarily symmetric or absolutely continuous with respect to Lebesgue measure, we find the optimal upper bound in both time and space for the corresponding heat kernels at ∞. In case of Lévy measures that are symmetric and absolutely continuous with densities g such that g(x) ≍ f(|x|) for non-increasing profile functions f, we also prove the full characterization of the sharp two-sided transition densities bounds of the form

$${p_t}\left( x \right) \asymp h{\left( t \right)^{ - d}} \cdot {1_{\left\{ {\left| x \right| \leqslant \theta h\left( t \right)} \right\}}} + tg\left( x \right) \cdot {1_{\left\{ {\left| x \right| \geqslant \theta h\left( t \right)} \right\}}},t \in \left( {0,{t_0}} \right),{t_0} > 0,x \in {\mathbb{R}^d}.$$

This is done for small and large x separately. Mainly, our argument is based on new precise upper bounds for convolutions of Lévy measures. Our investigations lead to a surprising dichotomy correspondence of the decay properties at ∞ for transition densities of pure jump Lévy processes. All results are obtained solely by analytic methods, without use of probabilistic arguments.

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Correspondence to Paweł Sztonyk.

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K. Kaleta was supported by the National Science Center (Poland) post-doctoral internship grant on the basis of the decision No. DEC-2012/04/S/ST1/00093.

P. Sztonyk was supported by the National Science Center (Poland) grant on the basis of the decision No. DEC-2012/07/B/ST1/03356.

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Kaleta, K., Sztonyk, P. Small-time sharp bounds for kernels of convolution semigroups. JAMA 132, 355–394 (2017). https://doi.org/10.1007/s11854-017-0023-6

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  • DOI: https://doi.org/10.1007/s11854-017-0023-6

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