Generalized Geometric Brownian motion and the Infinite Ergodicity concept

TL;DR

This paper explores generalized geometric Brownian motion processes, focusing on cases lacking a standard invariant measure, and introduces the concept of infinite ergodicity to understand their long-term behavior.

Contribution

It extends the theory of geometric Brownian motion by analyzing conditions for invariant measure existence and applying infinite ergodicity to non-standard stochastic processes.

Findings

Invariant measure existence depends on drift, diffusion, and discretization.

Infinite ergodicity provides a framework for non-standard stochastic processes.

Application to turbulence models illustrates theoretical insights.

Abstract

We investigate stochastic processes that generalize geometric Brownian motion, focusing on cases where the standard invariant measure, i.e. the solution of the stationary Fokker-Planck equation does not necessarily exist. We demonstrate that the existence of such a measure depends sensitively on the structure of the drift and diffusion terms, as well as on the chosen discretization scheme of the underlying stochastic dynamics. To ground our discussion, we draw motivation from phenomenological models in statistical theories of turbulence, where geometric Brownian motion serves as a classical example. To address situations where the standard invariant measure fails to exist, we heuristically explore the concept of infinite ergodicity, a notion recently introduced in the context of statistical physics for drift-diffusion stochastic processes.