Functional renormalization group in stochastic inflation

Abstract: We apply the functional renormalization group to Starobinsky's stochastic equation describing the local dynamics of a light scalar field in de Sitter. After elaborating on the over-damped regime of stochastic dynamics, we introduce an effective average action for the stochastic field, resulting by progressively integrating out frequencies, and study its flow equation in the local potential approximation (LPA). This effective action determines the approach to equilibrium and allows for the computation of unequal time correlators $\left\langle\phi(t)\phi(t+\Delta t)\right\rangle$ for large values of $\Delta t$. The stochastic RG flow in the LPA can be formulated in two ways, one that preserves the stochastic supersymmetry and one that breaks it. We show that both predict a characteristic decay time very close to that determined by the dynamical mass for a massless self-interacting scalar in de Sitter $m^2\sim \sqrt{\lambda}H^2$. Furthermore, the temporal supersymmetric formulation remarkably recovers the flow for the effective potential found using Quantum Field Theory methods and a smoothing over spatial wavelengths. We also discuss how the stochastic framework generically predicts an infrared mass which is a few percent smaller than the dynamical mass obtained in the LPA. Our results further support the notion that stochastic inflation captures the correct IR dynamics of light scalar fields in inflation.