Anomalous diffusion properties of stochastic transport by heavy-tailed jump processes

Abstract

In this work, we investigate the large-scale transport properties of a passive scalar advected by a turbulent fluid, modelled as a superposition of divergence-free vector fields, each weighted by an independent symmetric $α$-stable-like process. Motivated by recent works showing that complex small-scale spatial structures often lead to Brownian dispersion, we study if this principle persists when the driving noise exhibits heavy-tailed jump statistics. Our numerical results show a clear dichotomy linked with the tail behaviour of the noise. When considering standard $α$-stable processes, very large jumps survive the interaction with the spatial complexity and yield anomalous, super-diffusive transport. In contrast, when the $α$-stable noise is either truncated or exponentially tempered, suppressing extremely long jumps, the transport undergoes a transition to a classical diffusive regime.

Figures (7)

• Figure 1: $\mathbb{E}[|Z_t|]$ as a function of time for the $\alpha$-stable process (top-left figure), the tempered process (top-right figure) and the truncated process (bottom figure). For the tempered process we set $A=0.3$; for the truncated process we set $\epsilon=10^{-3}$. Slopes $1/\alpha$ and $1/2$ are represented by the dashed lines.
• Figure 2: Sample paths of the $\alpha$-stable process (solid line), the tempered process (dashed line) and the truncated process (dot-dashed line).
• Figure 3: $\mathbb{E}[|X_t|]$ as a function of time for the $\alpha$-stable process (top-left figure), the tempered process (top-right figure) and the truncated process (bottom figure). The parameters of the driving stochastic processes are the same as in Fig. \ref{['fig:Z_t']}. Slopes $1/\alpha$ and $1/2$ are represented by the dashed lines.
• Figure 4: Sample paths of $X_t$ for driver $Z_t$ being an $\alpha$-stable process (solid line), a tempered process (dashed line) and a truncated process (dot-dashed line).
• Figure 5: Probability distribution function of $X_t$ at the final time $t=1$ for driver $Z_t$ being an $\alpha$-stable process (top-left figure),for driver $Z_t$ being a tempered process (top-right figure) and for driver $Z_t$ being a truncated process (bottom figure).

Theorems & Definitions (2)

• Conjecture 1
• Conjecture 2