The Annals of Probability

Optimal local Hölder index for density states of superprocesses with (1+β)-branching mechanism

Klaus Fleischmann, Leonid Mytnik, and Vitali Wachtel

Full-text: Open access

Abstract

For 0<α≤2, a super-α-stable motion X in $\mathsf{R}^{d}$ with branching of index 1+β∈(1, 2) is considered. Fix arbitrary t>0. If d<α/β, a dichotomy for the density function of the measure Xt holds: the density function is locally Hölder continuous if d=1 and α>1+β but locally unbounded otherwise. Moreover, in the case of continuity, we determine the optimal local Hölder index.

Article information

Source
Ann. Probab., Volume 38, Number 3 (2010), 1180-1220.

Dates
First available in Project Euclid: 2 June 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1275486191

Digital Object Identifier
doi:10.1214/09-AOP501

Mathematical Reviews number (MathSciNet)
MR2674997

Zentralblatt MATH identifier
1207.60055

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G57: Random measures

Keywords
Dichotomy for density of superprocess states Hölder continuity optimal exponent critical index local unboundedness multifractal spectrum Hausdorff dimension

Citation

Fleischmann, Klaus; Mytnik, Leonid; Wachtel, Vitali. Optimal local Hölder index for density states of superprocesses with (1+ β )-branching mechanism. Ann. Probab. 38 (2010), no. 3, 1180--1220. doi:10.1214/09-AOP501. https://projecteuclid.org/euclid.aop/1275486191


Export citation

References

  • [1] Dawson, D. A. (1993). Measure-valued Markov processes. In École D’Été de Probabilités de Saint-Flour XXI—1991. Lecture Notes in Math. 1541 1–260. Springer, Berlin.
  • [2] Feller, W. (1971). An Introduction to Probability Theory and Its Applications II, 2nd ed. Wiley, New York.
  • [3] Fleischmann, K. (1988). Critical behavior of some measure-valued processes. Math. Nachr. 135 131–147.
  • [4] Fleischmann, K., Mytnik, L. and Wachtel, V. (2009). Hölder index for density states of (α, 1, β)-superprocesses at a given point. Preprint. Available at arXiv:0901.2315v1.
  • [5] Fuk, D. K. and Nagaev, S. V. (1971). Probability inequalities for sums of independent random variables. Theory Probab. Appl. 16 643–660.
  • [6] Le Gall, J.-F. and Mytnik, L. (2005). Stochastic integral representation and regularity of the density for the exit measure of super-Brownian motion. Ann. Probab. 33 194–222.
  • [7] Gikhman, I. I. and Skorokhod, A. V. (1969). Introduction to the Theory of Random Processes. W. B. Saunders Co., Philadelphia, PA.
  • [8] Gikhman, I. I. and Skorokhod, A. V. (1980). The Theory of Stochastic Processes. I, English ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 210. Springer, Berlin.
  • [9] Hausenblas, E. (2007). SPDEs driven by Poisson random measure with non Lipschitz coefficients: Existence results. Probab. Theory Related Fields 137 161–200.
  • [10] Jaffard, S. (1999). The multifractal nature of Lévy processes. Probab. Theory Related Fields 114 207–227.
  • [11] Jakod, J. (1979). Calcul Stochastique et Problèmes de Martigales. Lecture Notes in Math. 714 Springer, Berlin.
  • [12] Konno, N. and Shiga, T. (1988). Stochastic partial differential equations for some measure-valued diffusions. Probab. Theory Related Fields 79 201–225.
  • [13] Mueller, C., Mytnik, L. and Stan, A. (2006). The heat equation with time-independent multiplicative stable Lévy noise. Stochastic Process. Appl. 116 70–100.
  • [14] Mytnik, L. (2002). Stochastic partial differential equation driven by stable noise. Probab. Theory Related Fields 123 157–201.
  • [15] Mytnik, L. and Perkins, E. (2003). Regularity and irregularity of (1+β)-stable super-Brownian motion. Ann. Probab. 31 1413–1440.
  • [16] Reimers, M. (1989). One-dimensional stochastic partial differential equations and the branching measure diffusion. Probab. Theory Related Fields 81 319–340.
  • [17] Rosen, J. (1987). Joint continuity of the intersection local times of Markov processes. Ann. Probab. 15 659–675.
  • [18] Saint Loubert Bié, E. (1998). Étude d’une EDPS conduite par un bruit poissonnien. Probab. Theory Related Fields 111 287–321.
  • [19] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École D’Été de Probabilités de Saint-Flour, XIV—1984. Lecture Notes in Math. 1180 265–439. Springer, Berlin.