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

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 $\mathbf{\text{SL}}(n)$ 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 $\{F_{\tau }{\}}_{\tau \ge 0}$ on $\mathbf{\text{SL}}(n)$ 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 $\mathbb{E}\mathrm{\mid }{F}_{\tau }{\mathrm{\mid }}^{2p}$ with its anomalous dependence on $p$ (and $n$), and 2) by quantifying a non-tightness of $\mathrm{\mid }{F}_{\tau }{\mathrm{\mid }}^2\mathrm{/}\mathbb{E}\mathrm{\mid }{F}_{\tau }{\mathrm{\mid }}^2$ as $\tau \uparrow \mathrm{\infty }$. It is the second property that transmits to the drift-diffusion process. The arguments rely on stochastic analysis: We write $\{F_{\tau }{\}}_{\tau \ge 0}$ as the solution of $dF=F_{\tau }\circ dB$ with $\{B_{\tau }{\}}_{\tau \ge 0}$ a Brownian motion on the Lie algebra $\mathfrak{s}\mathfrak{l}(n)$. The arguments leverage isotropy: The diffusion projects onto the spectrum of the Gram matrix $G=F^{\ast }F$, as captured by $\mathrm{t}\mathrm{r}{G}^p$.