Long Time Behavior of Stochastic Thin Film Equation

We consider the stochastic thin-film equation with linear deterministic and stochastic Itô perturbations. The existence of nonnegative weak martingale solutions on the semi-axis is established, and their asymptotic behavior as $t\to \mathrm{\infty }$ is investigated. It is shown that in square mean the $L^{\mathrm{\infty }}$ norm of the solution converges to the spatial mean value of the initial condition, multiplied by a random factor similar to a geometric Wiener process.