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 $D\left(x\right)\sim |x-\tilde{x}{|}^{2-2/\alpha }$ 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, Itô, Stratonovich, or Hänggi-Klimontovich, so the existence of an infinite density and the density's shape are both related to the considered interpretation and the structure of $D\left(x\right)$.

• Received 9 August 2018

DOI:https://doi.org/10.1103/PhysRevE.99.042138