Generalized Geometric Brownian motion and the Infinite Ergodicity concept

TL;DR

This work generalizes geometric Brownian motion (GBM) by introducing algebraic nonlinearities in drift and diffusion and by varying the stochastic discretization parameter $\alpha$. It derives log-normal solutions in the driftless case and explicit asymptotic invariant densities $P_{\infty}(u)$ for nonlinear drift, highlighting how invariants depend on Itô, Stratonovich, and anti-Itô interpretations. When Stratonovich scaling ($\alpha=\tfrac{1}{2}$) prevents a normalizable invariant density, the authors implement infinite ergodicity to define an invariant density $\mathcal{I}$ and to compute long-time observables, extended to nonlinear diffusion with $m\neq1$. The square-root (CIR-type) diffusion is examined as a physically meaningful nonnegative model for turbulent quantities, linking turbulence phenomenology to stochastic calculus and offering potential applications in physics and finance through the generalized ergodic framework.

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.

Figures (4)

• Figure 1: Left: $P_{\infty}(u)$ on the "Itô-side" $\alpha < 1/2$ (with $n < 1$ and $H_0 < 0$); $H_0 = -3/2$ and $G_0 = 1$; Right: $P_{\infty}(u)$ on the "anti-Itô side", $\alpha > 1/2$ (with $H_0 > 0$ and $n > 1$), with $H_0 =3/2$ and $G_0 = 1$.
• Figure 2: The invariant density $\mathcal{I}$ from eq. (\ref{['inv']}) with different values of $n$. We adopted $H_0=-3/2$ and $G_0=1$.
• Figure 3: Left: $P_{\infty}(u)$ from eq. (\ref{['pdf_nm']}) with eq. (\ref{['cond2']}) satisfied. We adopted $H_0 = - 3/2$, $G_0=1$, $\alpha=1/2$, and $n=3/2$. Right: $P_{\infty}(u)$ from eq. (\ref{['pdf_nm']}) with eq. (\ref{['cond1']}) satisfied. We adopted $H_0 = 3/2$, $G_0=1$, $\alpha=1/2$, and $n=3/2$. The values of $m$ are specified in both panels.
• Figure 4: Plot of the function $P(u,s)$ reported in eq. (\ref{['P-H0-tris']}) with $\overline{u}_0=1$, $G_0=1$. We assumed that $0<u<3$ and $0.02<s<3$. The initial condition corresponds to $P(u,0) = \delta(u - \overline{u}_0)=\delta(u - 1)$. The solution corresponds to reflecting boundary conditions for $u=0$.