Warren, Jon and Williams, D. (David), 1938-. (2000) Probabilistic study of a dynamical system. Proceedings of the London Mathematical Society, Vol.81 (No.3). pp. 618-650. ISSN 0024-6115

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Abstract

This paper investigates the relation between a branching process and a non-linear dynamical system in C2. This idea has previously been fruitful in many investigations, including that of the FKPP equation by McKean, Neveu, Bramson, and others. Our concerns here are somewhat different from those in other work: we wish to elucidate those features of the dynamical system which correspond to the long-term behaviour of the random process. In particular, we are interested in how the dimension of the global attractor corresponds to that of the tail {sigma}-algebra of the process. The Poincaré–Dulac operator which (locally) intertwines the non-linear system with its linearization may sometimes be exhibited as a Fourier–Laplace transform of tail-measurable random variables; but things change markedly when parameters cross values giving the ‘primary resonance’ in the Poincaré–Dulac sense. Probability proves effective in establishing global properties amongst which is a clear description of the global convergence to the attractor. Several of our probabilistic results are analogues of ones obtained by Kesten and Stigum, and by Athreya and Ney, for discrete branching processes. Our simpler context allows the use of Itô calculus. Because the paper bridges two subjects, dynamical-system theory and probability theory, we take considerable care with the exposition of both aspects. For probabilist readers, we provide a brief guide to Poincaré–Dulac theory; and we take the view that in a paper which we hope will be read by analysts, it would be wrong to fudge any details of rigour in our probabilistic arguments.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Branching processes, Martingales (Mathematics), Stochastic processes, Probabilities, Differential equations
Journal or Publication Title:	Proceedings of the London Mathematical Society
Publisher:	Cambridge University Press
ISSN:	0024-6115
Date:	November 2000
Volume:	Vol.81
Number:	No.3
Page Range:	pp. 618-650
Identification Number:	10.1112/S0024611500012594
Status:	Peer Reviewed
Access rights to Published version:	Open Access
Funder:	Engineering and Physical Sciences Research Council (EPSRC)
Grant number:	GR/K70397 (EPSRC)
References:	1. V. I. Arnold, Geometric methods in the theory of ordinary differential equations (translated from the Russian, Springer, Berlin, 1983). 2. K. B. Athreya and P. E. Ney, Branching processes (Springer, Berlin, 1972). 3. M. D. Bramson, `Maximal displacement of branching Brownian motion', Comm. Pure. Appl. Math. 31 (1978) 531±581. 4. M. D. Bramson, `Convergence of the solutions of the Kolmogorov equation to travelling waves', Mem. Amer. Math. Soc. 285 (1983) 1±190. 5. D. A. Dawson, `Measure-valued Markov processes', E cole d'E te de ProbabiliteÂs de Saint-Flour XXI, 1993 (ed. P. L. Henniquin), Lecture Notes in Mathematics 1541 (Springer, Berlin, 1993) 2±260. 6. E. Hille, Ordinary differential equations in the complex domain (Wiley, New York, 1976). 7. L. HoÈrmander, The analysis of linear partial differential operators, III (Springer, Berlin, 1984). 8. H. P. McKean, `Application of Brownian motion to the equation of Kolmogorov±Petrovskii±Piskunov', Comm. Pure Appl. Math. 28 (1975) 323±331. 9. J. Neveu, `Multiplicative martingales for spatial branching processes', Seminar on Stochastic Processes (ed. E. CË inlar, K. L. Chung and R. K. Getoor), Progress in Probability and Statistics 15 (BirkhaÈuser, Boston, 1987) 223±241. 10. D. Revuz and M. Yor, Continuous martingales and Brownian motion (Springer, Berlin, 1991). 11. L. C. G. Rogers and D. Williams, Diffusions, Markov processes and martingales. Volume 1: foundations, 2nd edn (Wiley, Chichester, 1994). 12. L. C. G. Rogers and D. Williams, Diffusions, Markov processes and martingales. Volume 2: Itoà calculus (Wiley, Chichester, 1987). 13. J. Warren, `Some aspects of branching processes', PhD Thesis, Bath University, 1995.
URI:	http://wrap.warwick.ac.uk/id/eprint/838

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