Physical interpretation of Itô--distribution on the basis of local measurement of diffusion

In this paper we provide a physical interpretation of Itô-process resulting in thermal equilibrium distribution of a Brownian particle experiencing coordinate dependent diffusion. Since the local quantities like diffusivity would go through large fluctuations in thermal equilibrium, one needs to take these fluctuation into account. We identify that the definition of local diffusivity is an essential ingredient that effectively modifies Hamiltonian of the system to result in a physically relevant Gibbs measure related to the Itô-distribution.