By Vincent Rivasseau (Chief Editor)

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Main diﬃculties and outline of proof. The heuristic arguments outlined above are essentially based on a linear approximation, their validity therefore rests on a rigorous proof that the non linear eﬀects are negligible. Since the strength of √ the noise is γ, we cannot hope to improve the a priori bounds beyond m(t) − √ m ¯ ξ(t) ∞ ≈ γ. Then the non linear terms which are, to lowest order, quadratic have order γ; since they act for a time λ−2 τ (that is γ −1 τ ), a naive estimate gives a non–negligible contribution.

Bertini, S. Brassesco, P. Butt` a and E. Presutti Ann. 6), and get v (n) (t) = (n) (n) (n) (n) (n) (n) Γ1 (t) + Γ2 (t) + Γ4 (t) + Γ5 (t) + Γ9 (t) + Γ10 (t) t +λ Tn (n) ds gt−s ps−Tn h(n−1) (Tn ) + v (n) (Tn ) . 7) we thus need to show t λ Tn (n) (n) (n) ds gt−s ps−Tn h(n−1) (Tn ) + v (n) (Tn ) = Λ3 (t) + Γ6 (t) . s. 16). e. in the deﬁnition of Λ3 (t)). 6) to write h(n−1) (Tn ) + v (n−1) (Tn ) = pTn −Tn−1 h(n−1) (Tn−1 ) + v (n−1) (Tn−1 ) + Tn ds Tn−1 ∂pTn −s (n−1) v (s ). s. e. the deﬁnition of Γ6 (t).

10). We have not worked out the details, but we can at least present some educated guess. 12). Let us then Vol. 17) 2π(t − s) 0 both when θ vanishes and when it diverges. Set y(t) = θa x(θc t), then y(t) = θa θc/2 b(t) − 3θθc/2 t ds 0 y(s) . 12). 12). Concluding remarks and perspectives. 12). This can be characterized in terms of a Brownian motion with absorption at the origin, which in turn is reduced to the well studied one dimensional Schr¨ odinger equation with Dirac’s delta potentials. 12), is then available, in √ particular it shows that the displacements ξ(τ ) of the front have typical size log τ for τ large; “aging phenomena” are also present.

### Annales Henri Poincaré - Volume 3 by Vincent Rivasseau (Chief Editor)

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