Infinite Ergodic Theory for Heterogeneous Diffusion Processes

Abstract

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; 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 $D(x)$.

arXiv comment16 pages, 12 figures, 2 tables

References in corpus (18)

• Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes
• Marginally Stable Equilibria in Critical Ecosystems
• Drift without flux: Brownian walker with a space dependent diffusion coefficient
• Population splitting, trapping, and non-ergodicity in heterogeneous diffusion processes
• Non-ergodicity, fluctuations, and criticality in heterogeneous diffusion processes
• Random Walks in Logarithmic and Power-Law Potentials, Nonuniversal Persistence, and Vortex Dynamics in the Two-Dimensional XY Model
• Pesin-Type Identity for Weak Chaos
• Nonergodisity of a time series obeying Lévy statistics
• Infinite densities for Lévy walks
• Solution of the Fokker-Planck equation with a logarithmic potential
• From Non-normalizable Boltzmann-Gibbs statistics to infinite-ergodic theory
• Modeling scaled processes and 1/f^b noise by the nonlinear stochastic differential equations
• First-passage and first-exit times of a Bessel-like stochastic process
• Aging generates regular motions in weakly chaotic systems
• Generalized Arcsine Law and Stable Law in an Infinite Measure Dynamical System
• Isothermal Langevin dynamics in systems with power-law spatially-dependent friction
• Large-deviations for spatial diffusion of cold atoms
• On the connection between dissipative particle dynamics and the Itô-Stratonovich dilemma

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