Variational solutions to nonlinear stochastic differential equations in Hilbert spaces

Article

Abstract

One introduces a new variational concept of solution for the stochastic differential equation \(dX+A(t)X\,dt+{\lambda }X\,dt=X\,dW, t\in (0,T)\); \(X(0)=x\) in a real Hilbert space where \(A(t)={\partial }{\varphi }(t), t\in (0,T)\), is a maximal monotone subpotential operator in H while W is a Wiener process in H on a probability space \(\{{\Omega },{\mathcal {F}},\mathbb {P}\}\). In this new context, the solution \(X=X(t,x)\) exists for each \(x\in H\), is unique, and depends continuously on x. This functional scheme applies to a general class of stochastic PDE so far not covered by the classical variational existence theory (Krylov and Rozovskii in J Sov Math 16:1233–1277, 1981; Liu and Röckner in Stochastic partial differential equations: an introduction, Springer, Berlin, 2015; Pardoux in Equations aux dérivées partielles stochastiques nonlinéaires monotones, Thèse, Orsay, 1972) and, in particular, to stochastic variational inequalities and parabolic stochastic equations with general monotone nonlinearities with low or superfast growth to \(+\infty \).

Keywords

Brownian motion Maximal monotone operator Subdifferential Random differential equation Minimization problem 

Mathematics Subject Classification

Primary 60H15 Secondary 47H05 47J05 

Notes

Acknowledgements

This work was supported by the DFG through CRC 1283. V. Barbu was also partially supported by CNCS-UEFISCDI (Romania) through the Project PN-III-P4-ID-PCE-2016-0011.

References

  1. 1.
    Barbu, V.: Nonlinear Differential Equations of Monotone Type in Banach Spaces. Springer Monographs in Mathematics. Springer, New York (2010)CrossRefMATHGoogle Scholar
  2. 2.
    Barbu, V.: A variational approach to stochastic nonlinear problems. J. Math. Anal. Appl. 384, 2–15 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Barbu, V.: Optimal control approach to nonlinear diffusion equations driven by Wiener noise. J. Optim. Theory Appl. 153, 1–26 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Barbu, V.: A variational approach to nonlinear stochastic differential equations with linear multiplicative noise (submitted)Google Scholar
  5. 5.
    Barbu: Existence for nonlinear finite dimensional stochastic differential equations of subgradient type. Math. Control Relat. Fields (to appear)Google Scholar
  6. 6.
    Barbu, V., Brzezniak, Z., Hausenblas, E., Tubaro, L.: Existence and convergence results for infinite dimensional nonlinear stochastic equations with multiplicative noise. Stoch. Process. Appl. 123, 934–951 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Barbu, V., Da Prato, G., Röckner, M.: Existence of strong solutions for stochastic porous media equations under general monotonicity conditions. Ann. Probab. 37(2), 428–452 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Barbu, V., Da Prato, G., Röckner, M.: Stochastic Porous Media Equations. Lecture Notes in Mathematics, vol. 2163. Springer, Berlin (2016)MATHGoogle Scholar
  9. 9.
    Barbu, V., Röckner, M.: Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise. Arch. Ration. Mech. Anal. 209, 797–834 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Barbu, V., Röckner, M.: An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise. J. Eur. Math. Soc. 17, 1789–1815 (2015)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Brezis, H., Ekeland, I.: Un principe variationnel associé à certains équations paraboliques, le cas indépendent du temps. C.R. Acad. Sci. Paris 282, 971–974 (1976)MathSciNetMATHGoogle Scholar
  12. 12.
    Brooks, J.K., Dinculeanu, N.: Weak compactness in spaces of Bochner integrable functions and applications. Adv. Math. 24, 172–188 (1977)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, 1992, 2nd edn. Cambridge University Press, Cambridge (2008)MATHGoogle Scholar
  14. 14.
    Gess, B., Röckner, M.: Stochastic variational inequalities and regularity for degenerate stochastic partial differential equations. Trans. Am. Math. Soc. 369, 3017–3045 (2017)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Krylov, N.V., Rozovskii, B.L.: Stochastic evolution equations. J. Sov. Math. 16, 1233–1277 (1981)CrossRefMATHGoogle Scholar
  16. 16.
    Liu, W., Röckner, M.: Stochastic Partial Differential Equations: An Introduction. Springer, Berlin (2015)CrossRefMATHGoogle Scholar
  17. 17.
    Pardoux, E.: Equations aux Dérivées Partielles Stochastiques Nonlinéaires Monotones. Thèse, Orsay (1972)MATHGoogle Scholar
  18. 18.
    Prevot, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics, 1905, Springer, Berlin (2007)Google Scholar
  19. 19.
    Rockafellar, R.T.: Integrals which are convex functionals. Pac. J. Math. 24, 525–539 (1968)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Visintin, A.: Extension of the Brezis-Ekeland-Nayroles principle to monotone operators. Adv. Math. Sci. Appl. 18, 633–680 (2008)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Octav Mayer Institute of Mathematics of Romanian AcademyIaşiRomania
  2. 2.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  3. 3.Academy of Mathematics and System Sciences, CASBeijingChina

Personalised recommendations