Stochastic Partial Differential Equations Driven by General Stochastic Measures

  • Vadym Radchenko
Part of the Springer Optimization and Its Applications book series (SOIA, volume 90)


Stochastic integrals of real-valued functions with respect to general stochastic measures are considered in the chapter. For the integrator we assume the σ-additivity in probability only. The chapter contains a review of recent results concerning Besov regularity of stochastic measures, continuity of paths of stochastic integrals, and solutions of stochastic partial differential equations (SPDEs) driven by stochastic measure. Some important properties of stochastic integrals are proved. The Riemann-type integral of random function with respect to the Jordan content is introduced. For the heat equation in \(\mathbb{R}\), we consider the existence, uniqueness, and Hölder regularity of the mild solution. For a general parabolic SPDE in \({\mathbb{R}}^{d}\), we obtain the weak solution. Integrals of random functions with respect to deterministic measures in the equations are understood in Riemann sense.



This research was partially supported by Alexander von Humboldt Foundation, grant no. UKR/1074615.


  1. 1.
    Albeverio, S., Wua, J.-L., Zhang, T.-S.: Parabolic SPDEs driven by Poisson white noise. Stoch. Process. Appl. 74, 21–36 (1998)CrossRefMATHGoogle Scholar
  2. 2.
    Curbera, G.P., Delgado, O.: Optimal domains for L 0-valued operators via stochastic measures. Positivity 11, 399–416 (2007)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Dalang, R.C.: Extending martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e’s, Electron. J. Probab. 4, 1–29 (1999)Google Scholar
  4. 4.
    Dalang, R.C., Sanz-Solé, M.: Regularity of the sample paths of a class of second-order SPDE’s. J. Funct. Anal. 227, 304–337 (2005)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)CrossRefMATHGoogle Scholar
  6. 6.
    Dettweiler, J., Weis, L., van Neerven, J.: Space-time regularity of solutions of the parabolic stochastic Cauchy problem. Stoch. Anal. Appl. 24, 843–869 (2006)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Ilyin, A.M., Kalashnikov, A.S., Oleynik, O.A.: Linear second-order partial differential equations of the parabolic type. J. Math. Sci. (N.Y.) 108, 435–542 (2002)Google Scholar
  8. 8.
    Kamont, A.:A discrete characterization of Besov spaces. Approx. Theory Appl. 13, 63–77 (1997)Google Scholar
  9. 9.
    Kotelenez, P.: Stochastic Ordinary and Stochastic Partial Differential Equations: Transition from Microscopic to Macroscopic Equations. Springer, Berlin (2007)Google Scholar
  10. 10.
    Kwapień, S., Woycziński, W.A.: Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston (1992)CrossRefMATHGoogle Scholar
  11. 11.
    Memin, J., Mishura, Yu., Valkeila, E.: Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion. Stat. Probab. Lett. 27, 197–206 (2001)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Peszat, S., Zabczyk, J.: Stochastic partial differential equations with Lévy noise: an evolution equation approach. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  13. 13.
    Radchenko, V.: Convergence of integrals of unbounded real functions with respect to random measures. Theory Probab. Appl. 42, 310–314 (1998)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Radchenko, V.: Integrals with respect to general stochastic measures. Institute of Mathematics, Kyiv (1999) (in Russian)Google Scholar
  15. 15.
    Radchenko, V.: On the product of a random and of real measure. Theory Probab. Math. Stat. 70, 161–166 (2005)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Radchenko, V.: Besov regularity of stochastic measures. Stat. Probab. Lett. 77, 822–825 (2007)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Radchenko, V.: Parameter-dependent integrals with general random measures. Theory Probab. Math. Stat. 75, 161–165 (2007)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Radchenko, V.: Heat equation and wave equation with general stochastic measures. Ukr. Math. J. 60, 1968–1981 (2008)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Radchenko, V.: Mild solution of the heat equation with a general stochastic measure. Stud. Math. 194, 231–251 (2009)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Radchenko, V.: Sample functions of stochastic measures and Besov spaces. Theory Probab. Appl. 54, 160–168 (2010)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Radchenko, V.: Properties of integrals with respect to a general stochastic measure in a stochastic heat equation. Theory Probab. Math. Stat. 82, 103–114 (2011)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Radchenko, V.: Riemann integral of a random function and the parabolic equation with a general stochastic measure. Teor. Imovir. Mat. Stat. 87, 163–175 (2012)Google Scholar
  23. 23.
    Radchenko, V., Zähle, M.: Heat equation with a general stochastic measure on nested fractals. Stat. Prob. Lett. 82, 699–704 (2012)CrossRefMATHGoogle Scholar
  24. 24.
    Ryll-Nardzewski, C., Woyczyński, W.A.: Bounded multiplier convergence in measure of random vector series. Proc. Am. Math. Soc. 53, 96–98 (1975)CrossRefMATHGoogle Scholar
  25. 25.
    Talagrand, M.: Les mesures vectorielles à valeurs dans L 0 sont bornées. Ann. Sci. École Norm. Sup. (4) 14, 445–452 (1981)Google Scholar
  26. 26.
    Turpin, Ph.: Convexités dans les espaces vectoriels topologiques généraux. Dissertationes Math. 131, pp. 220 (1976)MathSciNetGoogle Scholar
  27. 27.
    Walsh, J.B.: An Introduction to Stochastic Partial Differential Equations. Lecture Notes in Matematics, vol. 1180, pp. 236–434. Springer, Berlin (1984)Google Scholar

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Mathematical AnalysisTaras Shevchenko National University of KyivKyivUkraine

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