A critical drift-diffusion equation: intermittent behavior
Felix Otto, Christian Wagner
TL;DR
The paper addresses the critical 2D drift-diffusion problem with a time-independent, divergence-free Gaussian drift by introducing UV and IR cut-offs and studying harmonic coordinates. The authors develop a scale-by-scale (continuum) homogenization framework and a proxy corrector $\tilde{\phi}$ defined via an Itô SDE in $\tilde{\lambda}^2=1+\varepsilon^2\ln L$ to reveal intermittency in the Jacobian of harmonic coordinates, $\tilde{F}=\mathrm{id}+\nabla\tilde{\phi}$. They prove sharp moment asymptotics for the proxy, notably $\mathbb{E}|\tilde{F}|^2\sim 2\tilde{\lambda}$ and $\mathbb{E}|\tilde{F}|^4\sim \tfrac{8}{3}\tilde{\lambda}^3$, with $\mathbb{E}(\det\tilde{F}-1)^2\ll 1$, while $\mathbb{E}|\tilde{F}|^4\gg(\mathbb{E}|\tilde{F}|^2)^2$ signals strong intermittency akin to Gaussian chaos. By comparison estimates, these proxy properties transfer to the true Jacobian $F$, yielding non-equi-integrability of $|F|^2/\mathbb{E}|F|^2$ and small determinant fluctuations, thereby uncovering a highly intermittent, non-conformal structure of the large-scale Lagrangian coordinates. The results quantify the micro-to-macro mechanism behind intermittency in scale-by-scale homogenization for a critical 2D stochastic drift-diffusion and connect it to Gaussian multiplicative chaos phenomena, with implications for anomalous transport in rough random environments.
Abstract
We consider a drift-diffusion process with a time-independent and divergence-free random drift that is of white-noise character. We are interested in the critical case of two space dimensions, where one has to impose a small-scale cut-off for well-posedness, and is interested in the marginally super-diffusive behavior on large scales. In the presence of an (artificial) large-scale cut-off at scale L, as a consequence of standard stochastic homogenization theory, there exist harmonic coordinates with a stationary gradient $F_L$; the merit of these coordinates being that under their lens, the drift-diffusion process turns into a martingale. It has recently been established that the second moments diverge as $\mathbb{E}|F_L|^2\sim\sqrt{\ln L}$ for $L\uparrow\infty$. We quantitatively show that in this limit, and in the regime of small Péclet number, $|F_L|^2/\mathbb{E}|F_L|^2$ is not equi-integrable, and that $\mathbb{E}|{\rm det}F_L|/\mathbb{E}|F_L|^2 $ is small. Hence the Jacobian matrix of the harmonic coordinates is very peaked and non-conformal. We establish this asymptotic behavior by characterizing a proxy $\tilde F_L$ introduced in previous work as the solution of an Itô SDE w. r. t. the variable $\ln L$, and which implements the concept of a scale-by-scale homogenization based on a variance decomposition and admits an efficient calculus. For this proxy, we establish $\mathbb{E}|\tilde F_L|^4\gg(\mathbb{E}|\tilde F_L|^2)^2$ and $\mathbb{E}({\rm det}\tilde F_L-1)^2\ll 1$. In view of the former property, we assimilate this phenomenon to intermittency. In fact, $\tilde F_L$ behaves like a tensorial stochastic exponential, and as a field can be assimilated to multiplicative Gaussian chaos.