Asymptotic prime ideals of S2-filtrations

Let A be a noetherian ring with total ring of fractions Q(A) and S=⊕n∈ZIntn a noetherian graded ring such that A[t−1]⊆S⊆Q(A)[t,t−1] and S satisfies the (S2) property of Serre. Under mild conditions on the ring A, we study the behavior of the sets of associated prime ideals Ass(A/In∩A) for n≥1. In particular, we consider the case when S is the S2-ification of the extended Rees algebra of an ideal I. As applications, we obtain several results regarding the asymptotic behavior of Ass(A/In) for certain ideals of analytic deviation one. We also prove several consequences about the symbolic powers of a prime ideal.