  This study considers the problem of the extreme behavior exhibited by
solutions to Burgers equation subject to stochastic forcing. More specifically,
we are interested in the maximum growth achieved by the "enstrophy" (the
Sobolev $H^1$ seminorm of the solution) as a function of the initial enstrophy
$\mathcal{E}_0$, in particular, whether in the stochastic setting this growth
is different than in the deterministic case considered by Ayala \& Protas
(2011). This problem is motivated by questions about the effect of noise on the
possible singularity formation in hydrodynamic models. The main quantities of
interest in the stochastic problem are the expected value of the enstrophy and
the enstrophy of the expected value of the solution. The stochastic Burgers
equation is solved numerically with a Monte Carlo sampling approach. By
studying solutions obtained for a range of optimal initial data and different
noise magnitudes, we reveal different solution behaviors and it is demonstrated
that the two quantities always bracket the enstrophy of the deterministic
solution. The key finding is that the expected values of the enstrophy exhibit
the same power-law dependence on the initial enstrophy $\mathcal{E}_0$as
reported in the deterministic case. This indicates that the stochastic
excitation does not increase the extreme enstrophy growth beyond what is
already observed in the deterministic case.
