  In this paper we provide a physical interpretation of It\^o-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
It\^o-distribution.
