I would like to ask about slow-roll eternal inflation.
Is it possible that the effects of quantum fluctuation can cause the inflaton value higher in some regions?
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Sure. The fluctuation can send the field into any direction in field space. The key to eternal inflation is that the random fluctuations back up the potential compensate for the classical tendency of the field to roll down the potential. The higher the potential energy, the larger the fluctuation: $\delta \phi \sim V^3/V’^2$, and so models with high energy inflating regions tend to exhibit eternal inflation. Models like Linde’s “chaotic inflation” are a nice prototype for eternal inflation since the random initial field positions across space during inflation ensure that some regions are “way up the potential” with relatively high energy.