(Non-)perturbative techniques in de Sitter space

In this thesis, we develop computational techniques for calculating correlation functions of a spectator scalar field in de Sitter space within the long-wavelength approximation. Our main objectives are threefold: to obtain perturbative quantum field theory results, to compare them with the Starobinsky-Yokoyama stochastic approach, and to extract non-perturbative information solely from perturbative data. We introduce a method for incorporating mass into the truncated Yang-Feldman equation. Unlike the standard theory of a massive scalar field based on the de Sitter-invariant vacuum, we develop a framework that does not require de Sitter invariance. Our framework yields a smooth massless limit for the infrared part of correlation functions, thereby bridging the gap between massless and massive perturbative results in the literature. By iterating the massive Yang-Feldman equation, we compute the two-point function to three-loop order and the four-point correlation function to one-loop order. We also establish the correspondence between integral structures arising from the massive Yang-Feldman-type equation and the Schwinger-Keldysh ("in-in") formalism. Within the stochastic framework, we derive non-perturbative results for arbitrary 2n-point correlation functions using recurrence relations. We further propose an alternative method for computing perturbative series for equal- and multi-time correlation functions. We derive a simple first-order differential equation from the Fokker-Planck or forward Kolmogorov equation. Its formal solution connects various correlation functions at different orders in the self-interaction coupling constant λ. Our method unifies the Starobinsky-Yokoyama stochastic approach with quantum field theory results, extending this consistency beyond the equal-time and stationary cases. Finally, we construct a renormalization group-inspired autonomous equation for the long-wavelength part of ⟨ϕ²(t, x⃗)⟩. Integrating its approximate form, we obtain an expression that is non-analytic in λ. In the late-time limit, it almost coincides with the result obtained within the Starobinsky-Yokoyama stochastic approach across the entire interval 0 ≤ π²m⁴/(3λH⁴) < ∞.