Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-24T03:27:47.319Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniqueness and non-uniqueness of solutions of Vlasov-McKean equations

Part of: Stochastic analysis Equations and systems with randomness

Published online by Cambridge University Press:  09 April 2009

Michael Scheutzow
Michael Scheutzow
Affiliation:
Fachbereich MathematikUniversität KaiserslauternD-6750 Kaiserslautern West, Germany
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the equation dY(t)/dt = f(Y(t), Eh(Y(t))) for random initial conditions, where E denotes the expected value. It turns out that in contrast to the deterministic case local Lipschitz continuity of f and h are not sufficient to ensure uniqueness of the solutions. Finally we also state some sufficient conditions for uniqueness.

MSC classification

Secondary: 34F05: Equations and systems with randomness 60H10: Stochastic ordinary differential equations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 43 , Issue 2 , October 1987 , pp. 246 - 256
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Dawson, D. A., ‘Critical dynamics and fluctuations for a mean-field model of cooperative behavior’, J. Statist. Phys. 31 (1983), 29–85.CrossRefGoogle Scholar
[2]Funaki, T., ‘A certain class of diffusion processes associated with nonlinear parabolic equations’, Z. Wahrsch. Verw. Gebiete 67 (1984), 331–348.CrossRefGoogle Scholar
[3]Liptser, R. S. and Shiryayev, A. N., Statistics of random processes I (Springer Verlag, Berlin, 1977).CrossRefGoogle Scholar
[4]McKean, H. P. Jr, ‘Propagation of chaos for a class of nonlinear parabolic equations’, Lecture series in differential equations, Vol. 2, pp. 177–194 (Van Nostrand Reinhold, New York, 1969).Google Scholar
[5]Oelschläger, K., ‘A martingale approach to the law of large numbers for weakly interacting stochastic processes’, Ann. Probab. 12 (1984), 458–479.CrossRefGoogle Scholar
[6]Scheutzow, M., ‘Periodic behavior of the stochastic Brusselator in the mean-field limit’, Probab. Theory Rel. Fields 72 (1986), 425–462.CrossRefGoogle Scholar
[7]Shiga, T. and Tanaka, H., ‘Central limit theorem for a system of Markovian particles with mean field interactions’, Z. Wahrsch. Verw. Gebiete 69 (1985), 439–459.CrossRefGoogle Scholar
[8]Sznitman, A. S., ‘An example of nonlinear diffusion process with normal reflecting boundary conditions and some related limit theorems’, preprint, Laboratoire de Probabilités, Paris, 02 1983.Google Scholar
[9]Walter, W., Gewöhnliche Differentialgleichungen (Springer Verlag, Berlin, 1976).CrossRefGoogle Scholar