Fluctuation of solutions to linear elliptic equations with noisy diffusion coefficients

Antoine Gloria (1)

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We consider a linear elliptic equation in divergence form on a bounded domain (or on $\R^d$) in dimension $d\geq 2$, whose coefficients are perturbed by a stationary noise of correlation length $\e>0$. We give estimates on the fluctuation of the solution in function of the correlation length $\e$ of the noise, both in terms of strong $L^2$ and weak $L^1$ norms. This result can be seen as a quantification of the propagation of uncertainties in linear elliptic partial differential equations.

Mots clés: quantification of uncertainty; fluctuation; stochastic perturbation; variance estimate; stochastic homogenization.; stochastic homogenization

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Antoine Gloria. Fluctuation of solutions to linear elliptic equations with noisy diffusion coefficients. Communications in Partial Differential Equations, 2013, 38 (2), pp.35. ⟨10.1080/03605302.2012.715319⟩. ⟨hal-00674445v2⟩