Title: "New applications of stochastic inflation"

In the stochastic formulation of inflation, the quantum fluctuations arising on small scales are collected in a classical noise term that perturbs the classical evolution on super-Hubble scales. We review the derivation of the Langevin equation for the long-wavelength modes, and analyze the statistical properties of its solution in dependence on the characteristics of the noise. We provide a calculation of the power spectrum of the cosmological perturbations in the case of a colored noise, and we show that a blue tilt on the largest observable scales is found when one properly accounts for the local homogeneity of our patch of the Universe before the Hubble crossing (in agreement with the WMAP data showing a suppression of the low multipoles of the CMB anisotropy). When self-interaction is introduced, the probability distribution of the observable fluctuations preserve some memory of the higly non-Gaussian ultra-large-scale dynamics: as a consequence of the cross talk between scales induced by colored noise the intrinsic non-Gaussianity is substantially enhanced also on observable scales. We finally describe the stochastic evolution of a quintessence field during inflation. In this case quantum fluctuations drive the quintessence field out of the range of values allowed by the observations, unless a constraint on the total amount of inflation is set.