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.