Title: Revisiting stochastic inflation with perturbation theory

Abstract: A long-standing problem within the study of cosmic inflation consists in fully reconciling the stochastic approach with perturbation theory. A complete connection between both formalisms has remained elusive even in the simple case of a single scalar field with self interactions determined by an arbitrary potential, in a fixed de Sitter background with a constant expansion rate. Using perturbation theory, we offer an exact calculation of the one-point probability density function for primordial fluctuations, valid to first order in the potential. We examine under which conditions our solution respects the Fokker-Planck equation encountered within the stochastic approach. We identify discrepancies and elucidate their origins, allowing us to shed light on the validity of the stochastic formalism.