  In this paper, we consider the problem of existence and uniqueness of
solutions to the Einstein field equations for a spatially flat FLRW universe in
the context of stochastic eternal inflation where the stochastic mechanism is
modelled by adding a stochastic forcing term representing Gaussian white noise
to the Klein-Gordon equation. We show that under these considerations, the
Klein-Gordon equation actually becomes a stochastic differential equation.
Therefore, the existence and uniqueness of solutions to Einstein's equations
depend on whether the coefficients of this stochastic differential equation
obey Lipschitz continuity conditions. We show that for any choice of $V(\phi)$,
the Einstein field equations are not globally well-posed, hence, any solution
found to these equations is not guaranteed to be unique. Instead, the
coefficients are at best locally Lipschitz continuous in the physical state
space of the dynamical variables, which only exist up to a finite explosion
time. We further perform Feller's explosion test for an arbitrary power-law
inflaton potential and prove that all solutions to the EFE explode in a finite
time with probability one. This implies that the mechanism of stochastic
inflation thus considered cannot be described to be eternal, since the very
concept of eternal inflation implies that the process continues indefinitely.
We therefore argue that stochastic inflation based on a stochastic forcing term
would not produce an infinite number of universes in some multiverse ensemble.
In general, since the Einstein field equations in both situations are not
well-posed, we further conclude that the existence of a multiverse via the
stochastic eternal inflation mechanism considered in this paper is still very
much an open question that will require much deeper investigation.
