Stability of the stochastic heat equation in $L^1([0,1])$

We consider the white-noise driven stochastic heat equation on $[0,\infty)\times[0,1]$ with Lipschitz-continuous drift and diffusion coefficients $b$ and $\sigma$. We derive an inequality for the $L^1([0,1])$-norm of the difference between two solutions. Using some martingale arguments, we show that this inequality provides some {\it a priori} estimates on solutions. This allows us to prove the strong existence and (partial) uniqueness of weak solutions when the initial condition belongs only to $L^1([0,1])$, and the stability of the solution with respect to this initial condition. We also obtain, under some conditions, some results concerning the large time behavior of solutions: uniqueness of the possible invariant distribution and asymptotic confluence of solutions.