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A pathwise solution for nonlinear parabolic equations with stochastic perturbations. (English) Zbl 1031.35156
Summary: We analyze here a semilinear stochastic partial differential equation of parabolic type where the diffusion vector fields are depending on both the unknown function and its gradient \(\partial_xu\) with respect to the state variable, \(x\in\mathbb{R}^n\). A local solution is constructed by reducing the original equation to a nonlinear parabolic one without stochastic perturbations and it is based on a finite-dimensional Lie algebra generated by the given diffusion vector fields.

35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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