Properties of States of Super-α-Stable Motion with Branching of Index 1 + β

Conference paper
First Online:
Part of the Springer Proceedings in Mathematics book series (PROM, volume 11)

Abstract

It has been well known for a long time that the measure states of the process in the title are absolutely continuous at any fixed time provided that the dimension of space is small enough. However, besides the very special case of one-dimensional continuous super-Brownian motion, properties of the related density functions were not well understood. Only in 2003, Mytnik and Perkins 21 revealed that in the Brownian motion case and if the branching is discontinuous, there is a dichotomy for the densities: Either there are continuous versions of them or they are locally unbounded. We recently showed that the same type of fixed time dichotomy holds also in the case of discontinuous motion. Moreover, the continuous versions are locally Hölder continuous, and we determined the optimal index for them. Finally, we determine the optimal index of Hölder continuity at given space points which is strictly larger than the optimal index of local Hölder continuity.

AMS 2010 Subject Classification. Primary 60J80; Secondary 60G57.

Keywords

Open Domain Martingale Measure Stochastic Partial Differential Equation Multifractal Spectrum Boundary Case 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Notes

Acknowledgements

We thank an anonymous referee for a careful reading of our manuscript. This work was supported by the German Israeli Foundation for Scientific Research and Development, Grant No. G-807-227.6/2003. Moreover, Mytnik was partially supported by ISF grant and Wachtel by GIF Young Scientists Program Grant No. G-2241-2114.6/2009.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Klaus Fleischmann
    • 1
  • Leonid Mytnik
    • 2
  • Vitali Wachtel
    • 3
  1. 1.Weierstrass Institute for Applied Analysis and StochasticsLeibniz Institute in Forschungsverbund Berlin e.V.BerlinGermany
  2. 2.Faculty of Industrial Engineering and ManagementTechnion Israel Institute of TechnologyHaifaIsrael
  3. 3.Mathematical InstituteUniversity of MunichMunichGermany

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