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Basic Ideas

  • Sergey V. Lototsky
  • Boris L. Rozovsky
Chapter
Part of the Universitext book series (UTX)

Abstract

Given a probability space \((\varOmega,\mathcal{F}, \mathbb{P})\) and two measurable spaces, \((A,\mathcal{A})\) and \((B,\mathcal{B})\), a random function X is a (measurable) mapping from \((A\times \varOmega,\mathcal{A}\times \mathcal{F})\) to \((B,\mathcal{Y})\). In the traditional terminology, the random process corresponds to \(A,B \subset \mathbb{R}\); a random field corresponds to \(A = \mathbb{R}^{\mathrm{d}}\), \(B = \mathbb{R}\). A sample path , or sample trajectory of X is the function X(⋅ , ω) for fixed ωΩ. A modification of X is a random function \(\bar{X}\) such that, for every aA, \(\mathbb{P}\big(X(a) =\bar{ X}(a)\big) = 1\); note that this, in general, DOES NOT mean that \(\mathbb{P}\big(X(a) =\bar{ X}(a)\ \text{for all}\ a \in A\big) = 1\) (although it does if both X and \(\bar{X}\) are continuous).

References

  1. 3.
    D.G. Aronson, The porous medium equation, in Nonlinear Diffusion Problems (Montecatini Terme, 1985). Lecture Notes in Mathematics, vol. 1224 (Springer, Berlin, 1986), pp. 1–46Google Scholar
  2. 7.
    G.I. Barenblatt, On some unsteady motions of a liquid and gas in a porous medium. Akad. Nauk SSSR. Prikl. Mat. Meh. 16, 67–78 (1952)MathSciNetGoogle Scholar
  3. 29.
    J.D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics. Q. Appl. Math. 9, 225–236 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 50.
    L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, vol. 19 (American Mathematical Society, Providence, 1998)Google Scholar
  5. 68.
    M.E. Gurtin, R.C. MacCamy, On the diffusion of biological populations. Math. Biosci. 33(1–2), 35–49 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 75.
    T. Hida, N. Ikeda, Analysis on Hilbert space with reproducing kernel arising from multiple Wiener integral, in Proceedings of the Fifth Berkeley Symposium Mathematical Statistics and Probability (University of California Press, Berkeley, CA, 1965/66). Vol. II: Contributions to Probability Theory, Part 1 (University of California Press, Berkeley, CA, 1967), pp. 117–143Google Scholar
  7. 79.
    E. Hopf, The partial differential equation u t + uu x = μu xx. Commun. Pure Appl. Math. 3, 201–230 (1950)CrossRefGoogle Scholar
  8. 89.
    K. Itô, Stochastic integral. Proc. Imp. Acad. Tokyo 20(8), 519–524 (1944)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 91.
    K. Itô, On a formula concerning stochastic differentials. Nagoya Math. J. 3, 55–65 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 94.
    R. Jarrow, P. Protter, A Short History of Stochastic Integration and Mathematical Finance: The Early Years, 1880–1970. A Festschrift for Herman Rubin. IMS Lecture Notes Monograph Series, vol. 45 (Institute of Mathematical Statistics, Beachwood, 2004), pp. 75–91Google Scholar
  11. 103.
    I. Karatzas, S.E. Shreve, Brownian Motion and Stochastic Calculus, 2nd edn. Graduate Texts in Mathematics, vol. 113 (Springer, New York, 1991)Google Scholar
  12. 114.
    P. Kotelenez, Stochastic Ordinary and Stochastic Partial Differential Equations: Transition from Microscopic to Macroscopic Equations. Stochastic Modelling and Applied Probability, vol. 58 (Springer, New York, 2007)Google Scholar
  13. 120.
    N.V. Krylov, An Analytic Approach to SPDEs, Stochastic Partial Differential Equations, ed. by B.L. Rozovskii, R. Carmona. Six Perspectives, Mathematical Surveys and Monographs (American Mathematical Society, Providence, 1999), pp. 185–242Google Scholar
  14. 125.
    H. Kunita, Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics, vol. 24 (Cambridge University Press, Cambridge, 1997)Google Scholar
  15. 132.
    P. Lévy, Processus Stochastiques et Mouvement Brownien. Suivi d’une note de M. Loève (Gauthier-Villars, Paris, 1948)Google Scholar
  16. 146.
    S.V. Lototsky, A random change of variables and applications to the stochastic porous medium equation with multiplicative time noise. Commun. Stoch. Anal. 1(3), 343–355 (2007)MathSciNetzbMATHGoogle Scholar
  17. 181.
    B.K. Øksendal, Stochastic Differential Equations: An Introduction with Applications, 5th edn. (Springer, New York, 1998)CrossRefzbMATHGoogle Scholar
  18. 192.
    P. Protter, Stochastic Integration and Differential Equations, 2nd edn. (Springer, New York, 2004)zbMATHGoogle Scholar
  19. 199.
    B.L. Rozovskii, Stochastic Evolution Systems. Mathematics and Its Applications (Soviet Series), vol. 35 (Kluwer Academic, Dordrecht, 1990)Google Scholar
  20. 210.
    W. Stannat, Stochastic partial differential equations: Kolmogorov operators and invariant measures. Jahresber. Dtsch. Math.-Ver. 113(2), 81–109 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 211.
    R.L. Stratonovich, A new representation for stochastic integrals and equations. Vestn. Moskov. Univer., Ser. Mat. Mekhan. 1, 3–12 (1964)Google Scholar
  22. 219.
    J.L. Vázquez, The Porous Medium Equation. Oxford Mathematical Monographs (The Clarendon Press/Oxford University Press, Oxford, 2007)zbMATHGoogle Scholar
  23. 220.
    A.D. Ventzel, On equations of the theory of conditional Markov processes. Theory Probab. Appl. 10(2), 357–361 (1965)CrossRefGoogle Scholar
  24. 223.
    J.B. Walsh, An introduction to stochastic partial differential equations, in ’Ecole d’été de probabilités de Saint-Flour, XIV—1984. Lecture Notes in Mathematics, vol. 1180 (Springer, Berlin, 1986), pp. 265–439Google Scholar
  25. 225.
    G.C. Wick, The evaluation of the collision matrix. Phys. Rev. (2) 80, 268–272 (1950)Google Scholar
  26. 229.
    E. Wong, M. Zakai, Riemann-Stieltjes approximation of stochastic integrals. Z. Wahr. verw. Geb. 120, 87–97 (1969)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Sergey V. Lototsky
    • 1
  • Boris L. Rozovsky
    • 2
  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Division of Applied MathematicsBrown UniversityProvidenceUSA

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