Intermittency of geometric Brownian motion on $ \textbf{SL}(n) $

TL;DR

This paper investigates the intermittency properties of geometric Brownian motion on SL(n), revealing how moments grow anomalously and how the process exhibits non-tightness, with implications for drift-diffusion processes.

Contribution

It quantifies intermittency of geometric Brownian motion on SL(n) in higher dimensions using stochastic analysis and spectral projections, extending previous understanding.

Findings

Exponential growth rate of moments identified.

Non-tightness of normalized process established.

Intermittency characterized for dimensions n>2.

Abstract

This short note is motivated by a recently discovered connection between a drift-diffusion process in $n$-dimensional Euclidean space with a divergence-free drift sampled from a stationary and isotropic Gaussian ensemble of critical scaling on the one hand, and a geometric Brownian motion on $\textbf{SL}(n)$ on the other hand. This can be seen as a tensorial form of a stochastic exponential; it thus is naturally intermittent, which transfers to the pair distance of the drift-diffusion process. In this note, we quantify the intermittency of the geometric Brownian motion $\{F_\tau\}_{\tau\ge0}$ on $\textbf{SL}(n)$ also in dimensions $n>2$. We do so in two (related) ways: 1) by identifying the exponential growth rate for the $2p$-th stochastic moment $\mathbb{E}|F_\tau|^{2p}$ with its anomalous dependence on $p$ (and $n$), and 2) by quantifying a non-tightness of $|F_\tau|^2/\mathbb{E}|F_\tau|^2$ as $\tau\uparrow\infty$. It is the second property that transmits to the drift-diffusion process. The arguments rely on stochastic analysis: We write $\{F_\tau\}_{\tau\geq 0}$ as the solution of $dF=F_\tau\circ dB$ with $\{B_\tau\}_{\tau\geq 0}$ a Brownian motion on the Lie algebra $\mathfrak{sl}(n)$. The arguments leverage isotropy: The diffusion projects onto the spectrum of the Gram matrix $G=F^*F$, as captured by ${\rm tr}G^p$.