Large-order Perturbation Theory and de Sitter/Anti de Sitter Effective   Actions
  We analyze the large-order behavior of the perturbative weak-field expansion of the effective Lagrangian density of a massive scalar in de Sitter and anti de Sitter space, and show that this perturbative information is not sufficient to describe the non-perturbative behavior of these theories, in contrast to the analogous situation for the Euler-Heisenberg effective Lagrangian density for charged scalars in constant electric and magnetic background fields. For example, in even dimensional de Sitter space there is particle production, but the effective Lagrangian density is nevertheless real, even though its weak-field expansion is a divergent non-alternating series whose formal imaginary part corresponds to the correct particle production rate. This apparent puzzle is resolved by considering the full non-perturbative structure of the relevant Feynman propagators, and cannot be resolved solely from the perturbative expansion. 
Ashok Das and Gerald V. Dunne