  The paper deals with the problem of stability for the flow of the 1D Burgers
equation on a circle. Using some ideas from the theory of positivity preserving
semigroups, we establish the strong contraction in the $L^1$ norm. As a
consequence, it is proved that the equation with a bounded external force
possesses a unique bounded solution on $\mathbb R$, which is exponentially
stable in $H^1$ as $t\to+\infty$. In the case of a random external force, we
show that the difference between two trajectories goes to zero with probability
$1$.
