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Additional resources for Annales Henri Poincaré - Volume 3

Example text

Stability of instantons, Fermi coordinates. Closeness to M is a consequence of the stability of M under the AC evolution. Under AC, in fact, M attracts exponentially fast all data which are in a small neighborhood, in sup norm, · ∞ , of M: namely there are δ, c and a all positive so that if for some ξ, m − m ¯ ξ ∞ < δ, then there is a ξ for which, for all t, m(t) − m ¯ ξ ∞ ≤ ce−at , m(t) being the solution of the AC equation starting from m. 1) with λ = 0 and γ > 0), but the noise, in a polynomial scale, cannot drive too far away from M because of the exponential attraction of 34 L.

S. 22) which converges, as λ ↓ 0, to 0 uniformly for σ ∈ [δ, τ ], as can be easily checked. 2 is proved. e. such that (λ2 T )−1 |η(i)|n vanishes as λ ↓ 0) does not contribute to the limiting equation for ξn since n ≤ (λ2 T )−1 τ . e. η(1), η(7), and η(10), as we shall see. √ Clearly ηn (2) is not negligible because its typical magnitude is λ T . e. ηn (i), with i from 4 to√9, live on an intermediate stage: they are smaller than the a priori bound λ1−ζ T , yet not small enough to be directly negligible.

5), choosing λ small enough, we have, for n ≤ mλ , √ (1) on the set Bλ,τ Vn,∗ ≤ Cλ1−ζ T (1) which implies mλ = nλ (τ ) on the set Bλ,τ . 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 ∞ .