Stabilization of Starobinsky-Vilenkin stochastic inflation by an environmental noise

We discuss the inflaton $\phi$ in an interaction with an infinite number of fields treated as an environment (noise) with a friction $\gamma^{2}>0$. In a Markovian approximation we obtain a stochastic wave equation (appearing also in the warm inflation models). After the replacement of the environment by the white noise, the stochastic wave equation violates the energy conservation if $\gamma\neq 0$. We introduce a dark energy as a compensation of the inflaton energy-momentum. We add to the classical wave equation the Starobinsky-Vilenkin noise which in the slow-roll approximation describes the quantum fluctuations in an expanding metric. We investigate the resulting consistent Einstein-Klein-Gordon system in the slow-roll regime. We obtain Fokker-Planck equation for the probability distribution of the inflaton assuming that the dependence of the scale factor $a$ and the Hubble variable $ H$ on the field $\phi$ is known. We obtain explicit stationary solutions of the Fokker-Planck equation assuming that $a(\phi)$ and $H(\phi)$ can approximately be determined in a slow-roll regime with the neglect of noise. We extend the results to the multifield D-dimensional configuration space. We show that in the regime $a(\phi)^{3}H(\phi)^{5}\rightarrow \infty$ the quantum noise determines the asymptotic behaviour of the stationary distribution. If $a(\phi)^{3}H(\phi)^{5}$ stays finite then the environmental noise ensures the integrability of the stationary probability. In such a case there is no need to introduce boundary conditions with the purpose to eliminate infinite inflation. The variation of $a(\phi)^{3}H(\phi)^{5}$ could be interpreted as a sign of a transition from cold inflation to warm inflation.