Slow-roll versus stochastic slow-roll inflation

We consider the classical wave equation with a thermal and Starobinsky–Vilenkin noise which in the slow-roll and long wave approximation describes the quantum fluctuations of the gravity-inflaton system in an expanding metric. We investigate the resulting consistent stochastic Einstein-Klein-Gordon system in the slow-roll regime. We show in some models that the slow-roll requirements (of the negligence of $$\partial _{t}^{2}\phi $$) can be satisfied in the probabilistic sense for the stochastic system with quantum and thermal noise for arbitrarily large time and an infinite range of fields. We calculate expectation values of some inflationary variables taking into account quantum and thermal noise. We show that the mean acceleration $$\langle \partial _{t}^{2}a\rangle $$ can be negative or positive (depending on the model) when the random fields take values beyond the classical range of inflation