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Variational solutions to nonlinear stochastic differential equations in Hilbert spaces

  • Viorel Barbu
  • Michael Röckner
Article
  • 81 Downloads

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.

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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

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