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We discuss the inflaton $\phi$ in an interaction with an infinite number offields treated as an environment (noise) with a friction $\gamma^{2}>0$. In aMarkovian approximation we obtain a stochastic wave equation (appearing also inthe warm inflation models). After the replacement of the environment by thewhite noise, the stochastic wave equation violates the energy conservation if$\gamma\neq 0$. We introduce a dark energy as a compensation of the inflatonenergy-momentum. We add to the classical wave equation the Starobinsky-Vilenkinnoise which in the slow-roll approximation describes the quantum fluctuationsin an expanding metric. We investigate the resulting consistentEinstein-Klein-Gordon system in the slow-roll regime. We obtain Fokker-Planckequation for the probability distribution of the inflaton assuming that thedependence of the scale factor $a$ and the Hubble variable $ H$ on the field$\phi$ is known. We obtain explicit stationary solutions of the Fokker-Planckequation assuming that $a(\phi)$ and $H(\phi)$ can approximately be determinedin a slow-roll regime with the neglect of noise. We extend the results to themultifield D-dimensional configuration space. We show that in the regime$a(\phi)^{3}H(\phi)^{5}\rightarrow \infty$ the quantum noise determines theasymptotic behaviour of the stationary distribution. If$a(\phi)^{3}H(\phi)^{5}$ stays finite then the environmental noise ensures theintegrability of the stationary probability. In such a case there is no need tointroduce 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 atransition from cold inflation to warm inflation.

Stabilization of Starobinsky-Vilenkin stochastic inflation by an environmental noise