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Applied Mathematics & Optimization

, Volume 78, Issue 2, pp 361–377 | Cite as

Stochastic Nonlinear Parabolic Equations with Stratonovich Gradient Noise

  • Viorel Barbu
  • Zdzisław Brzeźniak
  • Luciano Tubaro
Article
  • 127 Downloads

Abstract

Existence and uniqueness of solutions to stochastic differential equation \(dX-\text {div}\,a(\nabla X)\,dt=\sum _{j=1}^N(b_j\cdot \nabla X)\circ d\beta _j\) in \((0,T)\times \mathcal O\); \(X(0,\xi )=x(\xi )\), \(\xi \in \mathcal O\), \(X=0\) on \((0,T)\times \partial \mathcal O\) is studied. Here \(\mathcal O\) is a bounded and open domain of \(\mathbb R^d\), \(d\ge 1\), \(\{b_j\}\) is a divergence free vector field, \(a:[0,T]\times \mathcal O\times \mathbb R^d\rightarrow \mathbb R^d\) is a continuous and monotone mapping of subgradient type and \(\{\beta _j\}\) are independent Brownian motions in a probability space \((\Omega ,\mathcal F,\mathbb P)\). The weak solution is defined via stochastic optimal control problem.

Keywords

Stochastic variational inequalities Stochastic optimal control Nonlinear singular-degenerate stochastic partial differential equation Multiplicative gradient-type Stratonovich noise 

Notes

Acknowledgements

This work was done while V. Barbu had a visiting position at the Mathematics Department of University of Trento and Z. Brzeźniak was visiting the same Department. The authors are indebted to anonymous reviewer for pertinent observations and suggestions.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Viorel Barbu
    • 1
  • Zdzisław Brzeźniak
    • 2
  • Luciano Tubaro
    • 3
  1. 1.University Al. I. Cuza and Institute of Mathematics Octav MayerIasiRomania
  2. 2.Department of MathematicsUniversity of YorkYorkUK
  3. 3.Department of MathematicsUniversity of TrentoTrentoItaly

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