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

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 ${\rm tr}G^p$.