  Classical versions of the Big Bang cosmological models of the universe
contain a singularity at the start of time, hence the time variable in the
field equations should run over a half-line. Nonlocal string field theory
equations with infinite number of derivatives are considered and an important
difference between nonlocal operators on the whole real line and on a half-line
is pointed out.
  We use the heat equation method and show that on the half-line in addition to
the usual initial data a new arbitrary function (external source) occurs that
we call the daemon function. The daemon function governs the evolution of the
universe similar to Maxwell`s demon in thermodynamics. The universe and
multiverse are open systems interacting with the daemon environment. In the
simplest case the nonlocal scalar field reduces to the usual local scalar field
coupled with an external source which is discussed in the stochastic approach
to inflation. The daemon source can help to get the chaotic inflation scenario
with a small scalar field.
