## The Annals of Probability

### Occupation time large deviations for critical branching Brownian motion, super-Brownian motion and related processes

#### Abstract

We derive a large deviation principle for the occupation time func-tional, acting on functions with zero Lebesgue integral, for both super-Brownian motion and critical branching Brownian motion in three dimensions. Our technique, based on a moment formula of Dynkin, allows us to compute the exact rate functions, which differ for the two processes. Obtaining the exact rate function for the super-Brownian motion solves a conjecture of Lee and Remillard. We also show the corresponding CLT and obtain similar results for the superprocesses and critical branching process built over the symmetric stable process of index $\beta$ in $R^d$, with $d < 2\beta < 2 + d$ .

#### Article information

Source
Ann. Probab., Volume 26, Number 2 (1998), 602-643.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855645

Digital Object Identifier
doi:10.1214/aop/1022855645

Mathematical Reviews number (MathSciNet)
MR1626503

Zentralblatt MATH identifier
0962.60013

#### Citation

Deuschel, Jean-Dominique; Rosen, Jay. Occupation time large deviations for critical branching Brownian motion, super-Brownian motion and related processes. Ann. Probab. 26 (1998), no. 2, 602--643. doi:10.1214/aop/1022855645. https://projecteuclid.org/euclid.aop/1022855645

#### References

• [1] Rodemich, E., Garsia, A. and Rumsey, H. (1970). A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Univ. Math. J. 20 565-578.
• [2] Cox, J. and Griffeath, D. (1985). Occupation times for critical branching Brownian motions. Ann. Probab. 13 1108-1132.
• [3] Gorostiza, L., Dawson, D. and Wakolbinger, A. (1997). Occupation time fluctuations in branching systems. Unpublished manuscript.
• [4] Dawson, D. (1993). Measure-valued Markov processes. Ecole d'et´e de probabilit´es de St. Flour XXI. Lecture Notes in Math. 1541. Springer, Berlin.
• [5] Dembo, A. and Zeitouni, O. (1993). Large Deviations Techniques and Applications. Jones and Bartlett, Boston.
• [6] Deuschel, J.-D. and Stroock, D. (1989). Large Deviations. Academic Press, Boston.
• [7] Deuschel, J.-D. and Wang, K. (1997). Large deviations for occupation time functionals of branching Brownian particles and super-Brownian motion. Preprint.
• [8] Dynkin, E. B. (1989). Superprocesses and their linear additive functionals. Trans. Amer. Math. Soc. 314 255-282.
• [9] Iscoe, I. (1986). A weighted occupation time for a class of measure-valued branching processes.Wahrsch. Verw. Gebiete 71 85-116.
• [10] Iscoe, I. and Lee, T. (1993). Large deviations for occupation times of measure-valued branching Brownian motions. Stochastics Stochastics Rep. 45 177-209.
• [11] Lee, T.-Y. and Remillard, B. (1995). Large deviations for three dimensional superBrownian motion. Ann. Probab. 23 1755-1771.
• [12] Rosen, J. (1992). Renormalization and limit theorems for self-intersections of superprocesses. Ann. Probab. 20 1341-1368.