Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-06T02:07:34.508Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniqueness of the null solution to a nonlinear partial differential equation satisfied by the explosion probability of a branching diffusion

Part of: Special processes Markov processes Parabolic equations and systems Partial differential equations

Published online by Cambridge University Press:  24 October 2016

K. Bruce Erickson
K. Bruce Erickson*
Affiliation:
University of Washington
*
* Postal address: Department of Mathematics, University of Washington, Seattle, WA 98195, USA. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The explosion probability before time t of a branching diffusion satisfies a nonlinear parabolic partial differential equation. This equation, along with the natural boundary and initial conditions, has only the trivial solution, i.e. explosion in finite time does not occur, provided the creation rate does not grow faster than the square power at ∞.

Keywords

Quasi-linearbranching diffusionBesselprobability of explosion

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 35A02: Uniqueness problems 35K59: Quasilinear parabolic equations 60K99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 3 , September 2016 , pp. 938 - 945
Copyright
Copyright © Applied Probability Trust 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Fujita, H. and Watanabe, S. (1968).On the uniqueness and non-uniqueness of solutions of initial value problems for some quasi-linear parabolic equations.Commun. Pure Appl. Math. 21,631–652.CrossRefGoogle Scholar
[2] Ikeda, N. Nagasawa, M. and Watanabe, S. (1969).Branching Markov processes. III.J. Math. Kyoto Univ. 9,95–160.Google Scholar
[3] Itô, K. and Mckean, H. P., Jr. (1974).Diffusion Processes and Their Sample Paths,2nd edn.Springer ,Berlin.Google Scholar
[4] Polyanin, A. D. and Zaitsev, V. F. (2012).Handbook of Nonlinear Partial Differential Equations, 2nd edn.CRC,Boca Raton, FL.Google Scholar
[5] Savits, T. H. (1969).The explosion problem for branching Markov process.Osaka J. Math. 6,375–395.Google Scholar