If the effective cosmological constant is nonzero, our observable universe may enter a stage of exponential expansion. In such a case, regions of it may tunnel back to the false vacuum of an inflaton scalar field, and inflation with a high expansion rate may resume in those regions. An ideal eternal observer would then witness an infinite succession of cycles from false vacuum to true, and back. Within each cycle, the entire history of a hot universe would be replayed. If there were several minima of the inflaton potential, our ideal observer would visit each one of these minima with a frequency which depends on the shape of the potential. We generalize the formalism of stochastic inflation to analyze the global structure of the universe when this recycling process is taken into account.