In the paper Existence de processus Markoviens pour des systèmes infinis de particules by Cocozza, C. and Kipnis, C. (Ann. lnst. H. Poincaré, Sect. B, 13, 239-257, 1977), one reads (pg 249)
While $A^{x_1,\ldots,x_k}_{t_1,\ldots,t_k}$ is defined in page 248 by:
The way it is written it is not true that $F_\epsilon \subset \big(A^{x_1,\ldots,x_k}_{t_1,\ldots,t_k}\big)^c$ however, from the computation of $P(t_i - \epsilon_{p,i}<T^{x_i}_p< t_i + \epsilon_{p,i})$ one is lead to believe that there is a typo in the definition of $F_\epsilon$ and that, instead of what is there, one should have $$F_\epsilon = \bigg[\bigcap_{i = 1}^k \bigcap_p (t_i - \epsilon_{p,i} <T^{x_i}_p < t_i + \epsilon_{p,i} ) \bigg] $$
Is this the case?
second typo: instead of $$P_n(N^{x_i}_{t_i + \epsilon_{p,i}}\geq 1) \leq \Bbb{E}_{P_n} (N^{x_i}_{t_i + \epsilon_{p,i}} - N^{x_i}_{t_i - \epsilon_{p,i}}) $$
shouldn't it be: $$P_n(N^{x_i}_{t_i - \epsilon_{p,i},t_i + \epsilon_{p,i}}\geq 1) \leq \Bbb{E}_{P_n} (N^{x_i}_{t_i + \epsilon_{p,i}} - N^{x_i}_{t_i - \epsilon_{p,i}}) $$
third typo: One reads
Here, instead of "il existe une fonction $F_\epsilon$ ($\ldots$)" shouldn't it be "il existe une fonction $f_\epsilon$ ($\ldots$)"?
Yes, those are indeed typos.
The way it's written doesn't make sense, you corrections are attempts at guessing what the author was trying to say. With these corrections it makes sense to go on.