and a contribution from the fluctuations which in turn can be decomposed in positive and negative contributions $\mu_1^+$ and $\mu_1^-$ respectively. We shall evaluate the latter in this section.

We write \_1=Y\_t-Y\_[t\_0]{} =C\_[|Y]{} (-) +\_1\^[+]{}-\_1\^[-]{}. \[eq:c1\] Using Eq. (\[eq:proba\]) and Eq. (\[eq:shift\]), we get the expression \_1\^[+]{}=\_[t\_0\^]{}\^[t]{}dt\_1 \_0\^[+]{} dC\_1 e\^[-]{} (1+C ) for the positive part of the contribution at $t$ of the fluctuations, and \_1\^[-]{}=\_[t\_0\^]{}\^[t\_0]{}dt\_1 \_0\^[+]{} dC\_1 e\^[-]{} ( [1+C ]{}) subtracts the effect at $t_0$ of the fluctuations occurring at $t_1<t_0$. We have introduced a time $t_0^\prime$ of order 1 as a lower bound in these integrals in order to make these expressions finite. The physical meaning of this cutoff is clear: Before $t_0^\prime$, there is no mean-field regime because the whole system consists in a few particles only.

Let us start with the computation of $\mu_1^+$. It is useful to perform the change of variables = ,  u\_=e\^[-]{}\^[3/2]{}, \[eq:change\] which leads to the following expression of $\mu_1^{+}$: \_1\^[+]{}=\_\^1 \_0\^[u\_0()]{}du\_(1+), \[eq:mu1+\] where $u_0(\lambda)=\frac{t^{3/2}}{C}\lambda^{3/2}\sqrt{1-\lambda}$. $u_0$ is large compared to 1, except when $1-\lambda$ is of order $1/t^3$. But the contribution of the region $[1-1/t^3,1]$ in the $\lambda$-integration is smaller than $\sim 1/t^{3/2}$, and hence negligible. So we may always 