Feynman Diagrams for Stochastic Inflation and Quantum Field Theory in de
  Sitter Space
  We consider a massive scalar field with quartic self-interaction
$\lambda/4!\,\phi^4$ in de~Sitter spacetime and present a diagrammatic
expansion that describes the field as driven by stochastic noise. This is
compared with the Feynman diagrams in the Keldysh basis of the Amphichronous
(Closed-Time-Path) Field Theoretical formalism. For all orders in the
expansion, we find that the diagrams agree when evaluated in the leading
infrared approximation, i.e. to leading order in $m^2/H^2$, where $m$ is the
mass of the scalar field and $H$ is the Hubble rate. As a consequence, the
correlation functions computed in both approaches also agree to leading
infrared order. This perturbative correspondence shows that the stochastic
Theory is exactly equivalent to the Field Theory in the infrared. The former
can then offer a non-perturbative resummation of the Field Theoretical Feynman
diagram expansion, including fields with $0\leq m^2\ll\sqrt \lambda H^2$ for
which the perturbation expansion fails at late times.
