Partial Differential Equations with Random Noise in Inflationary
  Cosmology
  Random noise arises in many physical problems in which the observer is not
tracking the full system. A case in point is inflationary cosmology, the
current paradigm for describing the very early universe, where one is often
interested only in the time-dependence of a subsystem. In inflationary
cosmology it is assumed that a slowly rolling scalar field leads to an
exponential increase in the size of space. At the end of this phase, the scalar
field begins to oscillate and transfers its energy to regular matter. This
transfer typically involves a parametric resonance instability. This article
reviews work which the author has done in collaboration with Walter Craig
studying the role which random noise can play in the parametric resonance
instability of matter fields in the presence of the oscillatory inflaton field.
We find that the particular idealized form of the noise studied here renders
the instability more effective. As a corollary, we obtain a new proof of
Anderson localization.
