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One dimensional stochastic partial differential equations and the branching measure diffusion
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  • Published: April 1989

One dimensional stochastic partial differential equations and the branching measure diffusion

  • Mark Reimers1 

Probability Theory and Related Fields volume 81, pages 319–340 (1989)Cite this article

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Summary

An existence theorem in the spirit of Keisler [Ke], is proved for the simple one-dimensional diffusion equation driven by white noise modulated by a non-linear function of the solution. This is used to obtain a density and a Stochastic Partial Differential Equation in one dimension for the critical branching diffusion studied by Dawson et al.

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References

  1. Anderson, R.M.: A non-standard representation for brownian motion and Ito integration. Israel J. Math. 25, 15–46 (1976)

    Google Scholar 

  2. Bhattacharya, R.N., Rao, R.R.: Normal approximations and asymptotic expansions. New York: Wiley 1976

    Google Scholar 

  3. Burkholder, D.L.: Distribution function inequalities for martingales. Ann. Probab. 1, 19–42 (1973)

    Google Scholar 

  4. Cutler, C.: Some measure-theoretic and topological results for measure-valued and set-valued processes. Technical Report No. 49, Laboratory for research in statistics and probability, Carleton University, 1985

  5. Cutland, N.J.: Nonstandard measure theory and its applications. Bull. London Math. Soc. 15, 529–589 (1983)

    Google Scholar 

  6. Dawson, D.A.: Stochastic evolution equations and related measure processes. J. Multivariate Anal. 3, 1–52 (1975)

    Google Scholar 

  7. Dawson, D.A., Hochberg, K.J.: The carrying dimension of a stochastic measure diffusion. Ann. Probab. 7, 693–703 (1979)

    Google Scholar 

  8. Dawson, D.A.: Measure-valued processes: construction, qualitative behaviour, and stochastic geometry. Technical report No. 53, Laboratory for research in statistics and probability. Carleton University 1985

  9. Hoover, D.N., Perkins, E.: Non-standard construction of the stochastic integral and applications to stochastic differential equations. I and II. Trans. Amer. Math. Soc. 275, 1–58 (1983)

    Google Scholar 

  10. Keisler, H.J.: An infinitesimal approach to stochastic analysis. Mem. Am. Math. Soc. 297, (1984)

  11. Konno, N., Shiga, T.: Stochastic differential equations for some measure-valued diffusions (1980)

  12. Perkins, E.: Weak invariance principles for local time. Z. Wahrscheinlichkeitstheor. Verw. Geb. 60, 437–451 (1982)

    Google Scholar 

  13. Reimers, M.: Hyperfinite methods applied to the critical branching diffusion. Probab. Th. Rel. Fields 81, 11–27 (1989)

    Google Scholar 

  14. Roelly-Coppoletta, S.: A criterion of convergence of measure-valued processes: application to measure branching processes. Stochastics 17, 43–65 (1986)

    Google Scholar 

  15. Stoll, A.: A non-standard construction of levy brownian motion. Technical report, Universität Bochum 1984

  16. Stoll, A.: A nonstandard construction of levy brownian motion. Probab. Th. Rel. Fields 71, 321–334 (1986)

    Google Scholar 

  17. Walsh, J.B.: A stochastic model of neural response. Adv. Appl. Prob. 13, 231–281 (1981)

    Google Scholar 

  18. Walsh, J.B.: Martingales with a multidimensional parameter and stochastic integrals in the plane. In: Hennquin, P.L. (ed.) Ecole d'Eté de St. Flour. (Lect. Notes Math., vol. 1180) Berlin Heidelberg New York: Springer 1986

    Google Scholar 

  19. Watanabe, S.: A limit theory of branching processes and continuous state branching processes. J. Math. Kyoto Univ. 8, 141–167 (1968)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Psychology, University of Utah, 84112, Salt Lake City, Utah, USA

    Mark Reimers

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  1. Mark Reimers
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Additional information

This research was carried out while the author was at the Department of Mathematics, University of British Columbia

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Reimers, M. One dimensional stochastic partial differential equations and the branching measure diffusion. Probab. Th. Rel. Fields 81, 319–340 (1989). https://doi.org/10.1007/BF00340057

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  • Received: 28 April 1987

  • Revised: 15 June 1988

  • Issue Date: April 1989

  • DOI: https://doi.org/10.1007/BF00340057

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Keywords

  • Differential Equation
  • Partial Differential Equation
  • Stochastic Process
  • White Noise
  • Probability Theory
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