TY - JOUR
T1 - Properties of States of Super-α-Stable Motion with Branching of Index 1 + β
AU - Fleischmann, Klaus
AU - Mytnik, Leonid
AU - Wachtel, Vitali
N1 - Funding Information:
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.
PY - 2012
Y1 - 2012
N2 - 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 onedimensional continuous super-Brownian motion, properties of the related density functionswere 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.
AB - 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 onedimensional continuous super-Brownian motion, properties of the related density functionswere 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.
UR - http://www.scopus.com/inward/record.url?scp=84904099777&partnerID=8YFLogxK
U2 - 10.1007/978-3-642-23811-6_16
DO - 10.1007/978-3-642-23811-6_16
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AN - SCOPUS:84904099777
SN - 2190-5614
VL - 11
SP - 409
EP - 421
JO - Springer Proceedings in Mathematics
JF - Springer Proceedings in Mathematics
ER -