Summary
For a critical binary branching Bessel process starting fromx≫1 and stopped atx=1, we prove some conditional limit laws of the number of particles arriving atx=1 before a scaled large time. Five regions of the dimensional index of a Bessel process: −∞<d<2,d=2, 2<d<4,d=4 and 4<d<∞ are showed to have somewhat different behaviors. Our probabilistic results are proved by analyzing differential equations satisfied by generating functions. A salient theme is a comparison principle technique deliberately used to estimate solutions of\(u_t - \left( {D^2 + \frac{{d - 1}}{x}D} \right)u + u^p = 0\) inR +×R + wherep is greater than 1. The casep=2 corresponds to the process considered.
References
Sawyer, S.A., Fleischman, J.: Maximum geographical range of a mutant allele considered as a subtype of a Brownian branching random field. Proc. Natl. Acad. Sci. USA76, 872–875 (1979)
Lee, T.-Y.: Conditioned limit theorems of stopped critical branching Bessel processes. Ann. Probab.18 (1990)
Sawyer, S.A.: A formula for semigroups with an application to branching diffusion processes. Trans. Am. Math. Soc.152, 1–38 (1970)
Fowler, R.H.: Further studies on Emden's and similar differential equations. Q. J. Math.2, 259–288 (1931)
Brezis, H., Pelitier, L.A., Terman, D.: A very singular solution of the heat equation with absorption. Arch. Ration. Mech. Anal.95, 185–209 (1986)
Giga, Y., Kohn, R.V.: Asymptotically self-similar blow-up of semilinear heat equations. Commun. Pure Appl. Math.38, 297–319 (1985)
Kavian, O.: Remarks on the large time behavior of a nonlinear diffusion equation. Ann. Inst. Henri Poincaré,4, 423–452 (1987)
Kamin, S., Peletier, L.A.: Large time behavior of solutions of the heat equation with absorption. Ann. Sc. Norm. Super. Pisa, Cl. Sci. IV12, 393–408 (1985)
Iscoe, I.: Ergodic theory and a local occupation time for measure-valued critical branching Brownian motion. Stochastics18, 197–243 (1986)
Feller, W.: An introduction to probability theory and its applications, vol. 2. New York: Wiley 1971
Fleischmann, K., Gärtner, J.: Occupation time processes at a critical point. Math. Nachr.125, 275–290 (1986)
Fleischman, K.: Critical behavior of some measure valued processes. Math. Nachr.135, 131–147 (1988)
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Lee, T.Y. Some limit theorems for critical branching Bessel processes, and related semilinear differential equations. Probab. Th. Rel. Fields 84, 505–520 (1990). https://doi.org/10.1007/BF01198317
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DOI: https://doi.org/10.1007/BF01198317
Keywords
- Differential Equation
- Stochastic Process
- Probability Theory
- Limit Theorem
- Large Time