We discuss the Cauchy problem for the stochastic
Burgers equation with a nonlinear term of polynomial growth in the whole
real line. We also establish
the existence of an invariant measure when the equation has an additional zero order
dissipation. Many authors have discussed similar issues
for the stochastic Burgers equation in various different contexts. But
our results for the whole real line are new.
Also, our method is different from
those of the previous works on the stochastic Burgers equation. In particular,
our result on the existence of an invariant measure relies on the author's recent work
on a certain class of stochastic evolution equations.