  We show how to regularize the big bang singularity in the presence of random
perturbations modeled by Brownian motion using stochastic methods. We prove
that the physical variables in a contracting universe dominated by a scalar
field can be continuously and uniquely extended through the big bang as a
function of time to an expanding universe only for a discrete set of values of
the equation of state satisfying special co-prime number conditions. This
result significantly generalizes a previous result \cite{Xue:2014} that did not
model random perturbations. This result implies that the extension from a
contracting to an expanding universe for the discrete set of co-prime equation
of state is robust, which is a surprising result. Implications for a purely
expanding universe are discussed, such as a non-smooth, randomly varying scale
factor near the big bang.
