  This paper studies the behavior of solitons in the Korteweg-de Vries equation
under the influence of multiplicative noise. We introduce stochastic processes
that track the amplitude and position of solitons based on a rescaled frame
formulation and stability properties of the soliton family. We furthermore
construct tractable approximations to the stochastic soliton amplitude and
position which reveal their leading-order drift. We find that the statistical
properties predicted by our method agree well with numerical evidence.
