We consider a Burgers equation perturbed by a multiplicative white noise: \[ d_t u(t, x)= \Biggl({\partial^2 u(t, x)\over \partial x^2}+ {1\over 2} {\partial\over \partial x} u^2(t, x)\Biggr) dt+ g(u(t, x)) dW_t,\;t> 0,\;x\in [0, 1], \]
\[ u(t, 0)= u(t, 1)= 0,\quad t> 0,\quad u(0, x)= u_0(x),\quad x\in [0, 1], \] where \(W\) is defined by \(W(t)= \sum^\infty_{h= 1} \beta_he_h\), where \(\{e_h\}\) is an orthonormal basis of \(L^2(0, 1)\) and \(\{\beta_h\}\) is a sequence of mutually independent real Brownian motions in a fixed probability space \((\Omega, {\mathcal F}, {\mathbf P})\) adapted to a filtration \(\{{\mathcal F}_t\}_{t\geq 0}\). Moreover, \(g\) is a real valued function that is supposed to be Lipschitz continuous and bounded.
We prove the existence and uniqueness of the global solution as well as the strong Feller property and irreducibility for the corresponding transition semigroup. We close the paper by proving existence and uniqueness of an invariant measure.