Equilibrium with coordinate dependent diffusion: Comparison of different stochastic processes

We show that, simultaneous local scaling of coordinate and time keeping the velocity unaltered is a symmetry of an It\^o-process. Using this symmetry, any It\^o-process can be mapped to a universal additive Gaussian-noise form. We use this mapping to separate the canonical and micro-canonical part of stochastic dynamics of a Brownian particle undergoing coordinate dependent diffusion. We identify the equilibrium distribution of the system and associated entropy induced by coordinate dependence of diffusion. Equilibrium physics of such a Brownian particle in a heat-bath of constant temperature is that of an It\^o-process.