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A stochastic simulation for solving scalar reaction–diffusion equations

Published online by Cambridge University Press:  01 July 2016

B. Chauvin  and
Rouault
B. Chauvin*
Affiliation:
Université Paris VI
Rouault*
Affiliation:
Université Paris XI
*
∗Postal address: Université Paris VI, Laboratoire de Probabilités, 4, Place Jussieu, tour 56, 3 ème étage ‐ 75230 Paris Cedex 05, France.
∗∗Postal address: UA-CNRS 743, Statistique Appliquée, Université Paris Sud Mathématiques, Bat. 425, 91405 Orsay Cedex, France.
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Abstract

A recent Monte Carlo method for solving one-dimensional reaction–diffusion equations is considered here as a convergence problem for a sequence of spatial branching processes with interaction. The martingale problem is studied and a limit theorem is proved by embedding spaces of measures in Sobolev spaces.

Keywords

REACTION–DIFFUSION EQUATIONSPATIAL BRANCHING PROCESSESMARTINGALE PROBLEMMONTE CARLO METHODSOBOLEV SPACES
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 1 , March 1990 , pp. 88 - 100
Copyright
Copyright © Applied Probability Trust 1990 

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References

Adams, R. A. (1975) Sobolev Spaces. Academic Press, New York.Google Scholar
Borde, A. M. (1990) Stochastic demographic models. Age of a population. Stoch. Proc. Appl. Google Scholar
Chauvin, B. and Rouault, A. (1988) KPP equation and supercritical branching Brownian motion in the subcritical speed-area; application to spatial trees. Probab. Theory Rel. Fields 80, 299–314.Google Scholar
Dieudonne, J. (1970) Fondements de l'analyse moderne, Tome 2. Gauthier-Villars, Paris.Google Scholar
Ethier, S. and Kurtz, T. (1986) Markov Processes, Characterisation and Convergence. Wiley, New York.Google Scholar
Ghoniem, A. F. and Sherman, F. S. (1985) Grid-free simulation of diffusion using random walk method. J. Comput. Phys. 61, 1–37.Google Scholar
Joffe, A. and Metivier, M. (1986) Weak convergence of sequences of semimartingales with applications to multitype branching processes. Adv. Appl. Prob. 18, 20–65.CrossRefGoogle Scholar
Kolmogorov, A., Petrovski, I. and Piscounov, N. (1937) Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Mosc. Univer. Bull. Math. 1, 1–25.Google Scholar
Mckean, H. P. (1975) Application of Brownian motion to the equation of Kolmogorov–Petrovski–Piscounov. Comm. Pure Appl. Math. 28, 323–331 and 29, 553–554.Google Scholar
Metivier, M. (1982) Semimartingales. de Gruyter, Berlin.Google Scholar
Metivier, M. (1984) Quelques problèmes liés aux systèmes infinis de particules et leurs limites. In Lecture Notes in Mathematics 1204, Springer-Verlag, New York, 426–446.Google Scholar
Metivier, M. (1985) Weak convergence of measure valued processes using Sobolev-imbedding techniques. In Lecture Notes in Mathematics 1236, Springer-Verlag, New York, 172–183.Google Scholar
Oelschlager, K. (1985) A law of large numbers for moderately interacting diffusion processes. Z. Wahrscheinlichkeitsth. 69, 279–322.Google Scholar
Roelly-Coppoletta, S. and Rouault, A. (1989) Construction et propriétés des processus de branchement à valeurs mesures. Preprint de l'Université Paris VI.Google Scholar
Sherman, A. S. and Peskin, C. S. (1986) A Monte-Carlo method for scalar reaction–diffusion equations. SIAM J. Sci. Stat. Comp. 7, 1360–1372.Google Scholar
Uchiyama, K. (1978) The behaviour of solutions of some non-linear diffusion equations for large time. J. Math. Kyoto Univ. 18, 453–508.Google Scholar