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A conceptual approach to a path result for branching Brownian motion. (English) Zbl 1114.60065
Continuing their spine approach to branching diffusions [http:///www.bath.ac.uk/\(^\sim\)massch/ Research/Papers] the authors give an intuitive, fairly straightforward proof of a path large-deviation result for branching Brownian motions with local space-independent binary branching. In contrast to the proof of T.-Y. Lee [Ann. Probab. 20, No. 3, 1288–1309 (1992; Zbl 0759.60024)], which relied on Freidlin’s work on rescaling solutions of reaction-diffusion equations, it combines a spine change of measure with Schilder’s classical large-deviation result for Brownian motion.

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60F10 Large deviations
60J55 Local time and additive functionals
Full Text: DOI
[1] Chauvin, Brigitte, Arbres et processus de bellman – harris, Ann. inst. H. Poincaré probab. statist., 22, 2, 209-232, (1986) · Zbl 0597.60078
[2] Chauvin, Brigitte; Rouault, Alain, Supercritical branching Brownian motion and K-P-P equation in the critical speed-area, Math. nachr., 149, 41-59, (1990) · Zbl 0724.60091
[3] Chauvin, Brigitte, Product martingales and stopping lines for branching Brownian motion, Ann. probab., 19, 3, 1195-1205, (1991) · Zbl 0738.60079
[4] Dembo, Amir; Zeitouni, Ofer, Large deviations techniques and applications, (1998), Springer · Zbl 0896.60013
[5] Git, Yoav, ()
[6] Robert Hardy, Simon C. Harris, A spine approach to branching diffusions with applications to martingale convergence, 2006 (submitted for publication) · Zbl 1193.60100
[7] Robert Hardy, Simon C. Harris, A new formulation of the spine approach to branching diffusions, 2004, no. 0404, Mathematics Preprint, University of Bath. http://www.bath.ac.uk/ massch/Research/Papers/spine-foundations.pdf
[8] Robert Hardy, Simon C. Harris, Spine proofs for \(\mathcal{L}^p\)-convergence of branching-diffusion martingales, 2004, no. 0405, Mathematics Preprint, University of Bath. http://www.bath.ac.uk/ massch/Research/Papers/spine-Lp-cgce.pdf
[9] Robert Hardy, Simon C. Harris, A spine proof of a lower-bound for a typed branching diffusion, 2004, no. 0408, Mathematics Preprint, University of Bath. http://www.bath.ac.uk/ massch/Research/Papers/spine-oubbm.pdf
[10] Harris, Simon C.; Williams, David, Large deviations and martingales for a typed branching diffusion. I, Astérisque, 236, 133-154, (1996), Hommage à P. A. Meyer et J. Neveu · Zbl 0857.60088
[11] Kurtz, Thomas; Lyons, Russell; Pemantle, Robin; Peres, Yuval, A conceptual proof of the kesten – stigum theorem for multi-type branching processes, (), 181-185 · Zbl 0868.60068
[12] Kyprianou, Andreas, Travelling wave solutions to the K-P-P equation: alternatives to Simon harris’s probabilistic analysis, Ann. inst. H. Poincaré probab. statist., 40, 1, 53-72, (2004) · Zbl 1042.60057
[13] Lee, Tzong-Yow, Some large-deviation theorems for branching diffusions, Ann. probab., 20, 3, 1288-1309, (1992) · Zbl 0759.60024
[14] Liu, Quansheng; Rouault, Alain, On two measures defined on the boundary of a branching tree, (), 187-201 · Zbl 0867.60065
[15] Lyons, Russell, A simple path to biggins’ martingale convergence for branching random walk, (), 217-221 · Zbl 0897.60086
[16] Lyons, Russell; Pemantle, Robin; Peres, Yuval, Conceptual proofs of \(L \log L\) criteria for Mean behavior of branching processes, Ann. probab., 23, 3, 1125-1138, (1995) · Zbl 0840.60077
[17] McKean, Henry P., Application of Brownian motion to the equation of kolmogorov – petrovskii – piskounov, Comm. pure appl. math., 28, 323-331, (1975) · Zbl 0316.35053
[18] Neveu, Jacques, Arbres et processus de galton – watson, Ann. inst. H. Poincaré probab. statist., 22, 2, 199-207, (1986) · Zbl 0601.60082
[19] Neveu, Jacques, Multiplicative martingales for spatial branching processes, (), 223-241
[20] Varadhan, Srinivasa R.S., Large deviations and applications, (1984), SIAM · Zbl 0661.60040
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