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Part of the book series: Mathematics and Its Applications ((MAIA,volume 82))

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Abstract

The mathematical modelling of stochastic systems in continuum physics and mechanics can be realized in such a fashion that the time and space behaviour of the dependent variable is defined by the superposition on a deterministic evolution (in space and time) of an additional weighted noise. This type of modelling was already announced in Chapter 1 when we presented Eq.(1.14).

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References to Chapter 4

  1. Nelson J., Quantum Fluctuations, Princeton University Press, Princeton, (1985).

    MATH  Google Scholar 

  2. Arnold L., Stochastic Differential Equations: Theory and Applications, Wiley, New York, (1974).

    MATH  Google Scholar 

  3. Soong T.T., Random Differential Equations in Science and Engineering, Academic Press, New York, (1973).

    MATH  Google Scholar 

  4. McShane E.J., Stochastic Calculus and Stochastic Models, Academic Press, New York, (1974).

    MATH  Google Scholar 

  5. Kazimierczyk P., Identification of parametric noises in physical systems via equations for moments, Arch. Mech., 37, (1984), 49–74.

    MathSciNet  Google Scholar 

  6. Kotulski Z. and Sobczyk K., Characteristic functionals of randomly excited physical systems, Physica, 123A, (1984), 261–278.

    Google Scholar 

  7. Bellomo N. and Flandoli F., Stochastic partial differential equations in continuum physics: On the foundation of the stochastic interpolation method for Ito’s type equations, Math. Comp. Simul., 31, (1989), 3–17.

    Article  MathSciNet  MATH  Google Scholar 

  8. Flandoli F., Dirichlet boundary value problem for stochastic parabolic equations: Compatibility relations and regularity of solutions, Stochastics 29, 331–357.

    Google Scholar 

  9. Sobczyk K., Stochastic Differential Equations and Applications, Kluwer, Amsterdam, (1991).

    Book  MATH  Google Scholar 

  10. Krener A.J. and Lobry C, The complexity of stochastic differential equations, Stochastics, 4, (1981), 193–203.

    Article  MathSciNet  MATH  Google Scholar 

  11. De Blasi F.S. and Myjak J., Random differential equations on closed subset of a Banach space, J. Math. Analysis Appl., 90, (1982), 273–285.

    Article  MATH  Google Scholar 

  12. Ito S., Random fixed point theorems with an application to random differential equations in Banach spaces, J. Math. Analysis Appl., 67, (1979), 261–273.

    Article  Google Scholar 

  13. Papageorgiu N.S., Random differential inclusions in Banach spaces, J. Differential Equations, 65, (1986), 287–303.

    Article  MathSciNet  Google Scholar 

  14. Rybinski L.E., Random fixed points and viable random solutions of functional differential inclusions, J. Math. Analysis Appl., 142, (1989), 53–61.

    Article  MathSciNet  MATH  Google Scholar 

  15. Sussman H., On the gap between deterministic and stochastic ordinary differential equations, Annals of Probability, 6, (1978), 19–41.

    Article  Google Scholar 

  16. Schuss Z., Theory and Applications of Stochastic Differential Equations, Wiley, New York, (1980).

    MATH  Google Scholar 

  17. Soize C, Steady solutions of Fokker-Plank equations in higher dymensions, Prob. Engineering Mechanics, 3, (1988), 196–206.

    Article  Google Scholar 

  18. Bernard P., Mèthodes Mathèmatique d’Ètude des Vibrations Aléatoires et Analyse sur les Èspaces Gaussiens, These, Universite Blaise Pascal, Clermont-Ferrant, (1990).

    Google Scholar 

  19. Roozen H., Equilibrium and extinction in stochastic population dynamics, Bull. Math. Biol., 49, (1987), 671–696.

    MathSciNet  MATH  Google Scholar 

  20. Roozen H., An asymptotic solution of a two-dimensional exit problem arising in population dynamics, SIAM J. AppL Math., 49, (1989), 1793–1810.

    Article  MathSciNet  MATH  Google Scholar 

  21. Elworthy K., Stochastic Differential Equations on Manifolds, Cambridge University Press, Cambridge, (1982).

    MATH  Google Scholar 

  22. Friedman A., Stochastic Differential Equations and Applications, Academic Press, New York, (1975).

    MATH  Google Scholar 

  23. Brzezniak Z., Stochastic partial differential equations in M-type 2 Banach spaces, preprint.

    Google Scholar 

  24. Da Prato G., Some results on linear stochastic evolution equations in Hilbert spaces by the semigroup method, Stochastic Analysis and Appl., 1, (1983), 57–83.

    Article  MathSciNet  MATH  Google Scholar 

  25. Flandoli F., Dirichlet boundary problem for stochastic parabolic equation: Compatibility relations and regularity of solutions, Stochastics, 29, (1990), 331–357.

    MathSciNet  MATH  Google Scholar 

  26. Ichikawa A., Stability of of semilinear stochastic evolution equation,. Math. Analysis Appl., 90, (1982), 12–44.

    Article  MathSciNet  MATH  Google Scholar 

  27. Lions J.L., Quelques Méthodes de Résolution des Problèmes aux Limités Non Linéaires, Dunod, Paris, (1969).

    MATH  Google Scholar 

  28. Pardoux E., Equations aux Dérivées Partielles Stochastiques Non-linéaires Monotones, Thèse, Université de Paris XI, (1975).

    Google Scholar 

  29. Pardoux E., Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 3, (1979), 127–167.

    Article  MathSciNet  MATH  Google Scholar 

  30. Bensoussan A. and Temam R., Equations stochastique du type Navier-Stokes, J. Func. Anal. 13, (1973), 195–222.

    Article  MathSciNet  MATH  Google Scholar 

  31. Brzezniak Z., Capinski M. and Flandoli F., A convergence result for stochastic partial differential equations, Stochastics, 24, (1988), 423–445.

    Article  MathSciNet  MATH  Google Scholar 

  32. Brzezniak Z., Capinski M. and Flandoli F., Approximation for diffusion in random fields, Stochastic Analysis and Appl., 8, (1990), 293–313.

    Article  MathSciNet  MATH  Google Scholar 

  33. Brzezniak Z., Capinski M. and Flandoli F., Stochastic partial differential equations and turbulence, M 3 AS: Mathematical Models and Meth. in Appl. Sci., 1, (1991), 41–60.

    Article  MathSciNet  MATH  Google Scholar 

  34. Brzezniak Z., Capinski M. and Flandoli F., Stochastic Navier-Stokes equations with multiplicative noise, Stochastic Analysis and Appl., (1992), to appear.

    Google Scholar 

  35. Chorin A.J. and Marsden J.E., A Mathematical Introduction to Fluid Mechanics, Springer Verlag, New York (1979).

    Book  MATH  Google Scholar 

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Bellomo, N., Brzezniak, Z., de Socio, L.M. (1992). Stochastic Systems with Addional Weighted Noise. In: Nonlinear Stochastic Evolution Problems in Applied Sciences. Mathematics and Its Applications, vol 82. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-1820-0_4

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  • DOI: https://doi.org/10.1007/978-94-011-1820-0_4

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4803-3

  • Online ISBN: 978-94-011-1820-0

  • eBook Packages: Springer Book Archive

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