  We show the relation between processes which are modeled by a Langevin
equation with multiplicative noise and infinite ergodic theory. We concentrate
on a spatially dependent diffusion coefficient that behaves as ${D(x)}\sim
|x-\tilde{x}|^{2-2/\alpha}$ in the vicinity of a point $\tilde{x}$, where
$\alpha$ can be either positive or negative. We find that a nonnormalized
state, also called an infinite density, describes statistical properties of the
system. For processes under investigation, the time averages of a wide class of
observables, are obtained using an ensemble average with respect to the
nonnormalized density. A Langevin equation which involves multiplicative noise
may take different interpretation; It\^o, Stratonovich, or
H\"anggi-Klimontovich, so the existence of an infinite density, and the
density's shape, are both related to the considered interpretation and the
structure of $D(x)$.
