A self-dual variational approach to stochastic partial differential equations |
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Authors: | Shirin Boroushaki Nassif Ghoussoub |
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Institution: | Department of Mathematics, 1984 Mathematics Road, University of British Columbia, BC, V6T 1Z2, Canada |
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Abstract: | Unlike many of their deterministic counterparts, stochastic partial differential equations are not amenable to the methods of calculus of variations à la Euler–Lagrange. In this paper, we show how self-dual variational calculus leads to variational solutions of various stochastic partial differential equations driven by monotone vector fields. We construct solutions as minima of suitable non-negative and self-dual energy functionals on Itô spaces of stochastic processes. We show how a stochastic version of Bolza's duality leads to solutions for equations with additive noise. We then use a Hamiltonian formulation to construct solutions for non-linear equations with non-additive noise such as the stochastic Navier–Stokes equations in dimension two. |
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Keywords: | Stochastic PDE Self-dual variational calculus Bolza duality Maximal monotone vector fields |
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