Stationarity of inflation and predictions of quantum cosmology

We describe several different regimes which are possible in inflationary cosmology. The simplest one is inflation without self-reproduction of the Universe. In this scenario the Universe is not stationary. The second regime, which exists in a broad class of inflationary models, is eternal inflation with the self-production of inflationary domains. In this regime local properties of domains with a given density and given values of fields do not depend on the time when these domains were produced. The probability distribution to find a domain with given properties in a self-reproducing universe may or may not be stationary, depending on the choice of an inflationary model. We give examples of models where each of these possibilities can be realized, and discuss some implications of our results for quantum cosmology. In particular, we propose a new mechanism which may help solve the cosmological constant problem.