Infinite ergodic theory for heterogeneous diffusion processes

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) similar to vertical bar x-(x) over tilde vertical bar(2-2/alpha )in the vicinity of a point (x) over tilde, 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, Ito, Stratonovich, or Hanggi-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).