\lambda \phi^4 in de Sitter

 space

Correlation functions in massless \lambda \phi^4 in de Sitter space are IR divergent, indicating that the interacting vacuum of the theory is non-perturbatively far from the free vacuum. This gives rise to worries that de Sitter space might actually be unstable. We introduce two different perturbative schemes for sub-horizon and super-horizon modes. While short modes are solved for with the familiar expansion in powers of the coupling constant \lambda, for the long modes we develop a non-perturbative formalism, that allows us to find a non-perturbative solution and to expand around it in powers of \hbar, gradients and square-root of \lambda. At leading order in all these parameters, the density matrix for the long modes satisfies a Fokker-Planck equation, as envisioned long-time ago by Starobinsky. A generalized equation allows us to obtain the corrections in all the expansion parameters. We compute explicitly the ones in square-root of \lambda, which are the leading ones. We provide several checks, such as the fact that, on the static patch, correlation functions are thermal. Our results provide not only a solution to the problem of \lambda \phi^4 in de Sitter space, but also offer a rigorous proof of the existence of slow-roll eternal inflation.