Large prime factors of well-distributed sequences

We study the distribution of large prime factors of a random element u of arithmetic sequences satisfying simple regularity and equidistribution properties. We show that if such an arithmetic sequence has level of distribution $1$ the large prime factors of u tend to a Poisson--Dirichlet process, while if the sequence has any positive level of distribution the correlation functions of large prime factors tend to a Poisson--Dirichlet process against test functions of restricted support. For sequences with positive level of distribution, we also estimate the probability the largest prime factor of u is greater than u1-ε, showing that this probability is O(ε). Examples of sequences described include shifted primes and values of single-variable irreducible polynomials. Our result on the largest prime factor generalizes a very recent result of Ding on shifted primes. The proofs involve (i) a characterization of the Poisson--Dirichlet process due to Arratia-Kochman-Miller and (ii) an upper bound sieve.