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

By Vincent Rivasseau (Chief Editor)

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31). 7), on the set Bλ,τ . 41) Vol. s. 7), except (n) and Λ3 , and calling R(n,⊥) its orthogonal projection, (n) Γ2 (n,⊥) v (n,⊥) (Tn+1 ) = λz (n,⊥) (Tn+1 ) + Λ3 (Tn+1 ) + R(n,⊥) . 23). 43). 39). 44) k=0 then, for n ≤ nλ (τ ), since m ¯ 1 = 2, |ξn − xn | ≤ nλ (τ ) 2V∗ (τ ) sup n≤nλ (τ ) sup n≤nλ (τ ) m ¯ xn , v (n) (Tn+1 ) + R(v (n) (Tn+1 )) ∞ . 5) sup n≤nλ (τ ) R(v (n) (Tn+1 )) ∞ ≤ CV∗ (τ )3 . 46). 38) holds also for v (n) (Tn ). Indeed, since m we have the following proposition. 37). Then, for each τ, ζ > 0 there is a constant C = C(τ, ζ) such that, for any λ > 0 sup n≤nλ (τ ) Proof.

30). 8) also follows. 1. ¯ xn , is We are going to prove that the component of v (n) (Tn+1 ) orthogonal to m bounded by Cλ1−ζ , thus considerably improving the bound on the full v (n) (Tn+1 ). Let 3 (n,⊥) (n) ¯ gt := gt m ¯ xn | 1 − |m 4 xn the operator whose kernel is (n,⊥) gt 3 (n) ¯ (x)m (x, y) = gt (x, y) − m ¯ xn (y) . t. e. (n) ¯ xn = m ¯ xn . 11) that there are constants α > 0 and C < ∞ gt m so that, for any ϕ, (n,⊥) ϕ ≤ Ce−αt ϕ ∞ . 33) 48 L. Bertini, S. Brassesco, P. Butt` a and E. Presutti Ann.

11) and proved in Section 8. Going back to AC without noise, observe that stability of M does not mean stability of the single instanton: let m be a small deviation from m ¯ ξ , then from what we said above it will relax under AC to some m ¯ ξ , with ξ close but not necessarily equal to ξ. In the space of all profiles m, m ¯ ξ is marginally stable along the direction M while all the other directions are stable. It is then natural to associate to each m (as above) the value ξ of the center of the limit instanton.

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