AbstractWe discuss the relevance of the classical and quantum rolling tachyons inflation in the frame of the standard, p‐adic and adelic minisuperspace quantum cosmology. The field theory of tachyon matter proposed by Sen in a zero‐dimensional version suggested by Kar leads to a model of a particle moving in a constant external field with quadratic damping. We calculate the exact quantum propagator of the model, as well as, the vacuum states and conditions necessary to construct an adelic generalization.