## Electronic Journal of Probability

### Hydrodynamic Limit Fluctuations of Super-Brownian Motion with a Stable Catalyst

#### Abstract

We consider the behaviour of a continuous super-Brownian motion catalysed by a random medium with infinite overall density under the hydrodynamic scaling of mass, time, and space. We show that, in supercritical dimensions, the scaled process converges to a macroscopic heat flow, and the appropriately rescaled random fluctuations around this macroscopic flow are asymptotically bounded, in the sense of log-Laplace transforms, by generalised stable Ornstein-Uhlenbeck processes. The most interesting new effect we observe is the occurrence of an index-jump from a Gaussian situation to stable fluctuations of index $1+\gamma$, where $\gamma \in (0,1)$ is an index associated to the medium.

#### Article information

Source
Electron. J. Probab., Volume 11 (2006), paper no. 29, 723-767.

Dates
Accepted: 27 August 2006
First available in Project Euclid: 31 May 2016

https://projecteuclid.org/euclid.ejp/1464730564

Digital Object Identifier
doi:10.1214/EJP.v11-348

Mathematical Reviews number (MathSciNet)
MR2242662

Zentralblatt MATH identifier
1110.60082

Rights

#### Citation

Fleischmann, Klaus; Mörters, Peter; Wachtel, Vitali. Hydrodynamic Limit Fluctuations of Super-Brownian Motion with a Stable Catalyst. Electron. J. Probab. 11 (2006), paper no. 29, 723--767. doi:10.1214/EJP.v11-348. https://projecteuclid.org/euclid.ejp/1464730564

#### References

• D.A. Dawson. Limit theorems for interaction free geostochastic systems. In Point Processes and Queuing Problems, volume 24 of Coll. Math. Soc. János Bolyai, pages 27-47, Keszthely (Hungary), 1978.
• D.A. Dawson. Measure-valued Markov processes. In P.L. Hennequin, editor, École d'Été de Probabilités de Saint Flour XXI-1991, volume 1541 of Lecture Notes Math., pages 1-260. Springer-Verlag, Berlin, 1993.
• D.A. Dawson and K. Fleischmann. On spatially homogeneous branching processes in a random environment. Math. Nachr., 113:249-257, 1983.
• D.A. Dawson and K. Fleischmann. Critical dimension for a model of branching in a random medium. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 70:315-334, 1985.
• D.A. Dawson and K. Fleischmann. Diffusion and reaction caused by point catalysts. SIAM J. Appl. Math., 52:163-180, 1992.
• D.A. Dawson and K. Fleischmann. Catalytic and mutually catalytic super-Brownian motions. In Ascona 1999 Conference, volume 52 of Progress in Probability, pages 89-110. Birkhäuser Verlag, 2002.
• D.A. Dawson, K. Fleischmann, and L. Gorostiza. Stable hydrodynamic limit fluctuations of a critical branching particle system in a random medium. Ann. Probab., 17:1083-1117, 1989.
• D.A. Dawson, K. Fleischmann, and P. Mörters. Strong clumping of super-Brownian motion in a stable catalytic medium. Ann. Probab., 30(4):1990-2045, 2002.
• A. De Masi and E. Presutti. Mathematical Methods for Hydrodynamic Limits, volume 1501 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1991.
• P. Dittrich. Limit theorems for branching diffusions in hydrodynamical rescaling. Math. Nachr., 131:59-72, 1987.
• A.M. Etheridge. An Introduction to Superprocesses, volume 20 of Univ. Lecture Series. AMS, Rhode Island, 2000.
• R.A. Holley and D.W. Stroock. Generalized Ornstein-Uhlenbeck processes and infinite particle branching Brownian motions. Publ. Res. Inst. Math. Sci., 14(3):741-788, 1978.
• I. Karatzas and S.E. Shreve. Brownian Motion and Stochastic Calculus, volume 113 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2nd edition, 1991.
• C. Kipnis and C. Landim. Scaling Limits of Interacting Particle Systems, volume 320 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999.
• A. Klenke. A review on spatial catalytic branching. In Luis G. Gorostiza and B. Gail Ivanoff, editors, Stochastic Models, volume 26 of CMS Conference Proceedings, pages 245-263. Amer. Math. Soc., Providence, 2000.
• J.-F. Le Gall. Sur la saucisse de Wiener et les points multiples du mouvement Brownien. Ann. Probab., 14(4):1219-1244, 1986.
• A. Müller and D. Stoyan. Comparison Methods for Stochastic Models and Risks. Wiley, Chichester, 2002.
• E.A. Perkins. Dawson-Watanabe superprocesses and measure-valued diffusions. In P. Bernard, editor, École d'Été de Probabilités de Saint Flour XXIX-1999, Lecture Notes Math., pages 125-329, Springer-Verlag, Berlin, 2002.
• D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer-Verlag, Berlin, 1991.
• H. Spohn. Large Scale Dynamics of Interacting Particles. Texts and Monographs in Physics No. 342, Springer-Verlag, Heidelberg, 1991.