Diffusive-limit conjecture for stochastic transport by heavy-tailed jump processes

Let ${\mathbf{x}}\in {\mathbb{R}}^2$. For the stochastic transport model with regimes $(J_1)$, $(J_2)$, and $(J_3)$, let $({\mathbf{X}}^\eta_t({\mathbf{x}}))_{t\ge 0}$ denote the tracer process and let $\eta$ be the small-scale parameter.

Diffusive-limit conjecture. As $\eta$ goes to zero, the processes $({\mathbf{X}}^\eta_t({\mathbf{x}}))_{t\ge 0}$ converge in law to the $2$-dimensional process $({\mathbf{x}+\overline{{\mathbf{X}}}_t})_{t\ge 0}$, where $(\overline{{\mathbf{X}}}_t)_{t\ge 0}$ is an isotropic $\alpha$-stable process in case $(J_1)$, and a centred Brownian motion in cases $(J_2)$ and $(J_3)$.

The conjecture predicts anomalous super-diffusive transport for stable noise and classical diffusive transport for truncated or tempered noise. The stated validity remains an open question; the paper's numerical results provide evidence for the proposed limiting behaviours.