Eternal inflation in a dissipative and radiation environment: Heated
  demise of eternity
  Eternal inflation is studied in the context of warm inflation. We focus on
different tools to analyze the effects of dissipation and the presence of a
thermal radiation bath on the fluctuation-dominated regime, for which the
self-reproduction of Hubble regions can take place. The tools we explore are
the threshold inflaton field and threshold number of e-folds necessary to
establish a self-reproduction regime and the counting of Hubble regions, using
generalized conditions for the occurrence of a fluctuation-dominated regime. We
obtain the functional dependence of these quantities on the dissipation and
temperature. A Sturm-Liouville analysis of the Fokker-Planck equation for the
probability of having eternal inflation and an analysis for the probability of
having eternal points are performed. We have considered the representative
cases of inflation models with monomial potentials of the form of chaotic and
hilltop ones. Our results show that warm inflation tends to initially favor the
onset of a self-reproduction regime for smaller values of the dissipation. As
the dissipation increases, it becomes harder than in cold inflation (i.e., in
the absence of dissipation) to achieve a self-reproduction regime for both
types of models analyzed. The results are interpreted and explicit analytical
expressions are given whenever that is possible.
