Conditioning super-Brownian motion on its boundary statistics, and fragmentation. (English) Zbl 1296.60224

Salisbury, Thomas S.; Sezer, A. Deniz

Ann. Probab. 41, No. 5, 3617-3657 (2013).

Authors’ abstract: We condition super-Brownian motion on ”boundary statistics” of the exit measure \(X_{D}\) from a bounded domain \(D\). These are random variables defined on an auxiliary probability space generated by sampling from the exit measure \( {X}_{D}\). Two particular examples are: conditioning on a Poisson random measure with intensity \(\beta X_{D}\) and conditioning on \(X_{D}\) itself. We find the conditional laws as \(h\)-transforms of the original SBM law using Dynkin’s formulation of \(X\)-harmonic functions considered. We also obtain explicit constructions of these conditional laws in terms of branching particle systems. For example, we give a fragmentation system description of the law of SBM conditioned on \(X_{D}=\nu \), in terms of a particle system, called the backbone. Each particle in the backbone is labeled by a measure \(\tilde{\nu}\), representing its descendants’ total contribution to the exit measure. The particle’s spatial motion is an \(h\) -transform of Brownian motion, where \(h\) depends on \(\tilde{\nu}\). At the particle’s death two new particles are born, and \(\tilde{\nu}\) is passed to the newborns by fragmentation.

Reviewer: Marius Iosifescu (Bucureşti)

MSC:

60J68	Superprocesses
60J25	Continuous-time Markov processes on general state spaces
60J60	Diffusion processes
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords:

measure valued processes; diffusion; conditioning super-Brownian motion; X-harmonic functions; fragmentation; extreme X-harmonic functions; Poisson random measure; branching backbone system; Martin boundary

References:

[1]	Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102 . Cambridge Univ. Press, Cambridge. · Zbl 1107.60002
[2]	Doob, J. L. (1959). Discrete potential theory and boundaries. J. Math. Mech. 8 433-458; erratum 993. · Zbl 0101.11503
[3]	Dynkin, E. B. (2002). Diffusions , Superdiffusions and Partial Differential Equations. American Mathematical Society Colloquium Publications 50 . Amer. Math. Soc., Providence, RI. · Zbl 0999.60003
[4]	Dynkin, E. B. (2004). Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations. University Lecture Series 34 . Amer. Math. Soc., Providence, RI. · Zbl 1079.60006
[5]	Dynkin, E. B. (2006a). A note on \(X\)-harmonic functions. Illinois J. Math. 50 385-394 (electronic). · Zbl 1109.60064
[6]	Dynkin, E. B. (2006b). On extreme \(X\)-harmonic functions. Math. Res. Lett. 13 59-69. · Zbl 1095.31002
[7]	Etheridge, A. M. (1993). Conditioned superprocesses and a semilinear heat equation. In Seminar on Stochastic Processes , 1992 ( Seattle , WA , 1992). Progress in Probability 33 89-99. Birkhäuser, Boston, MA. · Zbl 0791.60076
[8]	Etheridge, A. M. and Williams, D. R. E. (2003). A decomposition of the \((1+\beta)\)-superprocess conditioned on survival. Proc. Roy. Soc. Edinburgh Sect. A 133 829-847. · Zbl 1040.60072
[9]	Evans, S. N. (1993). Two representations of a conditioned superprocess. Proc. Roy. Soc. Edinburgh Sect. A 123 959-971. · Zbl 0784.60052
[10]	Evans, S. N. and Perkins, E. (1990). Measure-valued Markov branching processes conditioned on nonextinction. Israel J. Math. 71 329-337. · Zbl 0717.60099
[11]	Le Gall, J.-F. (1999). Spatial Branching Processes , Random Snakes and Partial Differential Equations . Birkhäuser, Basel. · Zbl 0938.60003
[12]	Mselati, B. (2004). Classification and Probabilistic Representation of the Positive Solutions of a Semilinear Elliptic Equation. Mem. Amer. Math. Soc. 168 xvi+121. · Zbl 1274.35139
[13]	Overbeck, L. (1993). Conditioned super-Brownian motion. Probab. Theory Related Fields 96 545-570. · Zbl 0792.60037
[14]	Overbeck, L. (1994). Pathwise construction of additive \(H\)-transforms of super-Brownian motion. Probab. Theory Related Fields 100 429-437. · Zbl 0813.60046
[15]	Roelly-Coppoletta, S. and Rouault, A. (1989). Processus de Dawson-Watanabe conditionné par le futur lointain. C. R. Acad. Sci. Paris Sér. I Math. 309 867-872. · Zbl 0684.60062
[16]	Salisbury, T. S. and Verzani, J. (1999). On the conditioned exit measures of super Brownian motion. Probab. Theory Related Fields 115 237-285. · Zbl 0953.60078