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

TL;DR

The paper addresses intermittency in a tensorial stochastic exponential, the geometric Brownian motion on $\\textbf{SL}(n)$, arising from a drift-diffusion with a divergence-free, isotropic Gaussian field. It develops a Markovian framework via the Gram matrix $G=F^*F$ and derives closed It\\^o evolutions for moments of $\\mathrm{tr}^p G$ and $\\mathrm{tr} G^p$, establishing exponential growth of $\\mathbb{E}|F_\\tau|^{2p}$ with rate $p+\\frac{2p(p-1)}{n+2}$ and a non-tightness phenomenon for the normalized squared norm. The results extend known $n=2$ intermittency to general $n\\ge 2$ and reveal that the intermittency is governed by the geometry of $\\textbf{SL}(n)$ through spectral projections of $G$, with a Gaussian-like log-behavior for $|F_\\tau|^2$. The work links the stochastic exponential viewpoint to a drift-diffusion process with isotropic scaling and highlights how higher-dimensional geometry controls intermittency and tail behavior.

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 $\{F_τ\}_{τ\ge0}$ 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 $\mathbb{E}|F_τ|^{2p}$ with its anomalous dependence on $p$ (and $n$), and 2) by quantifying a non-tightness of $|F_τ|^2/\mathbb{E}|F_τ|^2$ as $τ\uparrow\infty$. It is the second property that transmits to the drift-diffusion process. The arguments rely on stochastic analysis: We write $\{F_τ\}_{τ\geq 0}$ as the solution of $dF=F_τ\circ dB$ with $\{B_τ\}_{τ\geq 0}$ 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 ${\rm tr}G^p$.

Theorems & Definitions (9)

• Proposition 1: Intermittency
• Proposition 2: Non-tightness
• Remark 3
• Lemma 4: MOW25, Lemma 1
• Lemma 5: MOW25,(74)
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9