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A pathwise solution for nonlinear parabolic equations with stochastic perturbations

Bogdan Iftimie;  Constantin Varsan — 2003

Open Mathematics

We analyse 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 ∂ xu with respect to the state variable, ∈ ℝn. 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.

Homogenization of a singular random one-dimensional PDE

Bogdan Iftimie;  Étienne Pardoux;  Andrey Piatnitski — 2008

Annales de l'I.H.P. Probabilités et statistiques

This paper deals with the homogenization problem for a one-dimensional parabolic PDE with random stationary mixing coefficients in the presence of a large zero order term. We show that under a proper choice of the scaling factor for the said zero order terms, the family of solutions of the studied problem converges in law, and describe the limit process. It should be noted that the limit dynamics remain random.

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