We consider the asymptotic behavior in large time of solutions for the KdV equation with multiplicative space-time noise. Under certain assumptions on the noise, we prove that the $L^2$ norm of solutions almost surely decays to zero as time goes to infinity. This phenomena is called stabilization by noise.
"Time decay of solution for the KdV equation with multiplicative space-time noise." Differential Integral Equations 21 (9-10) 959 - 970, 2008. https://doi.org/10.57262/die/1356038594