Levy solutions of a randomly forced Burgers equation

We consider the one dimensional Burgers equation forced by a brownian in space and white noise in time process $\partial_t u + u \partial_x u = f(x,t)$, with $2E(f(x,t)f(y,s)) = (|x|+|y|-|x-y|)\delta(t-s)$ and we show that there are Levy processes solutions, for which we give the evolution equation of the characteristic exponent. In particular we give the explicit solution in the case $u_0(x)=0$.