  The phenomenon of stochastic eternal inflation is studied for a chaotic
inflation potential in a Bianchi Type I spacetime background. After deriving
the appropriate stochastic Klein-Gordon equation, we give details on the
conditions for eternal inflation. It is shown that for eternal inflation to
occur, the amount of anisotropy must be small. In fact, it is shown that
eternal inflation will only take place if the shear anisotropy variables take
on values within a small region of the interior of the Kasner circle. We then
calculate the probability of eternal inflation occurring based on techniques
from stochastic calculus.
