# Diffusive and Super-Diffusive Limits for Random Walks and Diffusions with Long Memory

## TL;DR

This survey reviews recent findings on both normal and anomalous diffusion behaviors of long-memory random motions in various dimensions, highlighting differences between turbulent flow models and self-repelling diffusions.

## Contribution

It summarizes recent results on diffusive limits for random walks and diffusions with long memory, including new insights into their behavior across different dimensions.

## Key findings

- Normal diffusion in dimensions three and higher.
- Anomalously fast diffusion in one and two dimensions.
- Results are based on studies from 2012-2018.

## Abstract

We survey recent results of normal and anomalous diffusion of two types of random motions with long memory in ${\Bbb R}^d$ or ${\Bbb Z}^d$. The first class consists of random walks on ${\Bbb Z}^d$ in divergence-free random drift field, modelling the motion of a particle suspended in time-stationary incompressible turbulent flow. The second class consists of self-repelling random diffusions, where the diffusing particle is pushed by the negative gradient of its own occupation time measure towards regions less visited in the past. We establish normal diffusion (with square-root-of-time scaling and Gaussian limiting distribution) in three and more dimensions and typically anomalously fast diffusion in low dimensions (typically, one and two). Results are quoted from various papers published between 2012-2018, with some hints to the main ideas of the proofs. No technical details are presented here.