This thesis introduces an effective theory for the long-distance behaviour of scalar fields in de Sitter spacetime, known as the second-order stochastic theory, with the aim of computing scalar correlation functions that are useful in inflationary cosmology. The need for such a theory stems from the challenge standard perturbative quantum field theory calculations face when considering self-interacting scalar fields with a mass $m$. Focussing on quartic self-interactions, parameterised by the coupling $\lambda$, one finds that the perturbative expansion about the free field solution returns correlation functions that are of order $\frac{\lambda H^4}{m^4}$. Thus, the method breaks down beyond the limit $\lambda\ll m^4/H^4$ because the perturbative sum does not converge.

The stochastic theory was introduced as a non-perturbative method for computing these scalar correlation functions. Motivated by the existence of a de Sitter horizon, field modes can be separated into short and long wavelength modes. The expanding spacetime stretches these modes such that the long wavelengths are considered classical. The short wavelength modes contribute to the long-distance behaviour when they cross the horizon, which amounts to a statistical white noise contribution. The result is stochastic equations describing the behaviour of the long-distance modes. One can then apply the formalism of stochastic processes to obtain non-perturbative expressions for stochastic correlation functions. In much of the literature, one introduces a hard cut-off between long and short wavelength modes about the horizon to derive the stochastic equations. The approximation one must make here is $m\ll H$ and $\lambda\ll m^2/H^2$ such that the resulting equations are overdamped.

The second-order stochastic theory is introduced in this thesis to extend the regime of validity of the stochastic approach. Instead of using the cut-off procedure, we write stochastic equations that resemble the field equations of motion, leaving the stochastic parameters - the stochastic mass, quartic coupling and white noise contributions - as free parameters. We employ stochastic techniques to compute correlation functions and then use results from perturbative quantum field theory to fix the parameters such that they return physical quantities. In doing so, we improve upon the overdamped stochastic approach by extending the regime of validity to $m\lesssim H$ and $\lambda^2\ll m^4/H^4$ and by including ultraviolet effects from renormalisation in our stochastic theory. Additionally, we perform non-perturbative computations such that the second-order stochastic theory goes beyond the regime of validity of perturbative quantum field theory.