A pathwise solution for nonlinear parabolic equations with stochastic perturbations.

*(English)*Zbl 1031.35156Summary: 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.

##### MSC:

35R60 | PDEs with randomness, stochastic partial differential equations |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

##### Keywords:

semilinear stochastic partial differential equation of parabolic type; diffusion vector fields; finite-dimensional Lie algebra##### References:

[1] | Pierre Lions and Panagiotis E. Souganidis: Uniqueness of weak solutions of fully nonlinear stochastic partial differential equations, C.R. Acad. Sci., Paris, t.331, Série I, 2000, pp. 783-790. · Zbl 0970.60072 |

[2] | Bogdan Iftimie: Qualitative Theory for Diffusion Equations with Applications in Physics, Economy and Techniques, Doctoral Thesis, Institute of Mathematics, Romanian Academy of Sciences, 2001. |

[3] | R. Racke: Lectures on Nonlinear Evolution Equations, Aspects of Mathematics, Vieweg, Berlin, 1992. · Zbl 0811.35002 |

[4] | Constantin Varsan: On Evolution Systems of Differential Equations with Stochastic Perturbations, Preprint No. 4/2001, IMAR, ISSN-02503638. |

[5] | Constantin Varsan: Applications of Lie Algebras to Hyperbolic and Stochastic Differential Equations, Kluwer Academic Publishers, Holland, 1999. · Zbl 0948.35002 |

[6] | Constantin Varsan and Cristina Sburlan: Basics of Equations of Mathematical Physics and Differential Equations, Ex Ponto, Constantza, 2000. |

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