It\^o--distribution from Gibbs measure and a comparison with experiment

Langevin dynamics of a confined Brownian particle with coordinate--dependent diffusion involves multiplicative noise. Mathematically, equilibrium of such a stochastic system with multiplicative noise is an It\^o--process. However, in physics literature, the process and resulting It\^o--distribution is not considered to represent equilibrium because it is a modified Boltzmann distribution. It\^o--distribution is derived in this paper from Gibbs measure without involving any convention for stochastic integration, hence, no It\^o vs Stratonovich dilemma results. Then, in the light of an existing experiment reported in 1994 by Faucheux and Libchaber, we compare the Boltzmann distribution with the modified one for thermal equilibrium of Brownian particle near confining walls causing coordinate dependence of diffusion. Distribution corresponding to the It\^o--process (modified Boltzmann) is shown to adequately account for the experimental results where the Boltzmann--distribution fails.