Stabilization by noise for a class of stochastic reaction-diffusion equations

We prove uniqueness, ergodicity and strongly mixing property of the invariant measure for a class of stochastic reaction-diffusion equations with multiplicative noise, in which the diffusion term in front of the noise may vanish and the deterministic part of the equation is not necessary asymptotically stable. To this purpose, we show that the L-1-norm of the difference of two solutions starting from any two different initial data converges P-a.s. to zero, as time goes to infinity.