Functional renormalization group in stochastic inflation
  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.
