On the stochastic Burgers equation with a polynomial nonlinearity in the real line

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.