IR finite correlation functions in de Sitter space, a smooth massless limit, and an autonomous equation

We explore two-point and four-point correlation functions of a massive scalar field on the flat de Sitter background in the long-wavelength approximation. By employing the Yang-Feldman-type equation, we compute the two-point correlation function up to $\lambda^3$ order and the four-point correlation function up to $\lambda^2$ order. In contrast to the standard theory of a massive scalar field based on the de Sitter-invariant vacuum, we develop a vacuum-independent reasoning that may not possess de Sitter invariance but results in a smooth massless limit of the correlation function's infrared part. Our elaboration allows us to calculate correlation functions of a free massive scalar field and to proceed with quantum corrections, relying only on the known infrared part of the two-point correlation function of a free massless one. Remarkably, the two-point correlation function of a free massive scalar field coincides with that of the Ornstein-Uhlenbeck stochastic process and has a clear physical interpretation. We compare our results with those obtained using the Schwinger-Keldysh diagrammatic technique, Starobinsky's stochastic approach, and the Hartree-Fock approximation. At last, we construct a renormalization group-inspired autonomous equation for the two-point correlation function. Integrating its approximate version, one obtains the non-analytic expression with respect to a self-interaction coupling constant $\lambda$. That solution reproduces the correct perturbative series up to the two-loop level. In the late-time limit, it almost coincides with the result of Starobinsky's stochastic approach over the whole interval of a new dimensionless parameter $0 \leq \tfrac{\pi^2 m^4}{3\lambda H^4} < \infty$.