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Comparison methods for a class of function valued stochastic partial differential equations
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  • Published: March 1992

Comparison methods for a class of function valued stochastic partial differential equations

  • Peter Kotelenez1 

Probability Theory and Related Fields volume 93, pages 1–19 (1992)Cite this article

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Summary

A comparison theorem is derived for a class of function valued stochastic partial differential equations (SPDE's) with Lipschitz coefficients driven by cylindrical and regular Hilbert space valued Brownian motions. Moreover, we obtain necessary and sufficient conditions for the positivity of the mild solutions of the SPDE's where the sufficiency follows from the comparison theorem. Thereby it is, e.g., possible to identify a class of SPDE's, which can serve as stochastic space-time models for the density of particles. As a consequence we can construct unique mild solutions of SPDE's on the cone of positive functions with non-Lipschitz drift parts including the case of arbitrary polynomialsR(x) withR(O)≧O and leading negative coefficient.

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References

  1. Arnold, L., Curtain, R.F., Kotelenez, P.: Nonlinear evolution equations in Hilbert space. Forschungsschwerpunkt Dynamische Systeme, Universität Bremen,. Report Nr. 17 (1980).

  2. Curtain, R.F., Pritchard, A.J.: Infinite dimensional linear systems theory. (Lect. Notes. Control. Sci., vol. 8). Berlin Heidelberg New York: Springer 1978.

    Google Scholar 

  3. Daletskij, Yu. L., Fomin, S.V.: Measures and differential equations in infinite dimensional spaces (in Russian). Moscow: Nauka 1983

    Google Scholar 

  4. Dawson, D.A.: Stochastic evolution equations. Math Biosci.15, 287–316 (1972)

    Google Scholar 

  5. Dawson, D.A., Gorostiza, L.G.: Generalized solutions of nuclear space valued stochastic evolution equations. (Preprint 1989); Appl. Math. Optimization (to appear)

  6. Ethier, N.E., Kurtz, T.G.: Markov processes: Characterization and convergence. New York: Wiley 1986

    Google Scholar 

  7. Faris, W.G., Jona-Lasinio, G.: Large fluctuations for a nonlinear heat equation with noise. J. Phys. A. Math. Gen15, 3025–3055 (1982)

    Google Scholar 

  8. Funaki, T.: Random, motions of strings and related stochastic evolution equations Nagoya Math. J.89, 129–193 (1983)

    Google Scholar 

  9. Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam Oxford New York: North Holland 1981

    Google Scholar 

  10. Kotelenez P.: High density limit theorems for nonlinear chemical reactions with diffusion. Probab. Theory Relat. Fields78, 11–37 (1988)

    Google Scholar 

  11. Kotelenez, P.: Fluctuations in a nonlinear reaction-diffusion model. Ann. Appl. Probab. (to appear)

  12. Kotelenez, P.: Existence, Uniqueness and Smoothness for a Class of Function Valued Stochastic Partial Differential Equations. (Case Western Reserve, University, Department of Mathematics, Preprint #91-111)

  13. Lang, L.S.: Differentiable manifolds. Reading, Mass: Addison-Wesley 1972

    Google Scholar 

  14. Manthey, R.: On the Cauchy problem for reaction-diffusion equations with white noise. Math. Nachr.136, 209–228 (1988)

    Google Scholar 

  15. Tanabe, H.: Equations of evolution. London San Francisco Melbourne: Pitman 1979

    Google Scholar 

  16. Walsh, J.B.: An introduction to stochastic partial differential equations. In: Hennequin, P.L. (ed.), Ecole d'Ete de Probabilites de Saint-Flour XIV-1984 (Lect. Notes Math., vol. 1180) Berlin Heidelberg New York: Springer 1986

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics & Statistics, Case Western Reserve University, 44106, Cleveland, OH, USA

    Peter Kotelenez

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  1. Peter Kotelenez
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This research was supported by a grant from ONR

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Cite this article

Kotelenez, P. Comparison methods for a class of function valued stochastic partial differential equations. Probab. Th. Rel. Fields 93, 1–19 (1992). https://doi.org/10.1007/BF01195385

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  • Received: 21 November 1989

  • Revised: 02 January 1992

  • Issue Date: March 1992

  • DOI: https://doi.org/10.1007/BF01195385

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Keywords

  • Differential Equation
  • Hilbert Space
  • Partial Differential Equation
  • Stochastic Process
  • Brownian Motion
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