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Existence for nonlinear finite dimensional stochastic differential equations of subgradient type

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  • One proves via variational techniques the existence and uniqueness of a strong solution to the stochastic differential equation $dX+{\partial} {\varphi} (t,X)dt\ni \sum\limits^N_{i = 1}σ_i(X)d{β}_i,\ X(0) = x,$ where ${\partial}{\varphi} :{\mathbb{R}}^d\to2^{{\mathbb{R}}^d}$ is the subdifferential of a convex function ${\varphi}:{\mathbb{R}}^d\to{\mathbb{R}}$ and $σ_i∈ L({\mathbb{R}}^d,{\mathbb{R}}^d)$, $1≤ d<{∞}$.

    Mathematics Subject Classification: Primary: 60H15, 47H05, 49K15; Secondary: 47J05, 47N10.


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  • [1] V. Barbu, Nonlinear Differential Equations Of Monotone Type In Banach Spaces, Springer, 2010. doi: 10.1007/978-1-4419-5542-5.
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    [3] V. Barbu, A variational approach to stochastic nonlinear parabolic problems, J. Math. Annal. Appl., 384 (2011), 2-15.  doi: 10.1016/j.jmaa.2010.07.016.
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    [5] V. Barbu, A variational approach to nonlinear stochastic differential equations with linear multiplicative noise, to appear.
    [6] V. BarbuG. Da Prato and M. Röckner, Existence of strong solutions for stochastic porous media equations under general monotonicity conditions, Ann. Probab., 37 (2009), 428-452.  doi: 10.1214/08-AOP408.
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    [8] V. Barbu and M. Röckner, An operatorial approach to stochastic partial differential equations driven by linear multiplicataive noise, J. Eur. Math.Soc., 17 (2015), 1789-1815.  doi: 10.4171/JEMS/545.
    [9] R. BuckdahnL. MaticiucE. Pardoux and A. Rǎşcanu, Stochastic variational inequalities on nonconvex domains, J. Diff. Equations, 259 (2015), 7332-7374.  doi: 10.1016/j.jde.2015.08.023.
    [10] C. Prevot and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Math., 1905, Springer, 2007.
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