We survey recent results of normal and anomalous diffusion of two types of random motions with long memory in $R^d$ or $Z^d$ . The first class consists of random walks on $Z^d$ 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 diffusion, 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 2012-2017, with some hints to the main ideas of the proofs. No technical details are presented here.
|Title of host publication||Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018.|
|Subtitle of host publication||Invited lectures|
|Editors||Boyan Sirakov, Paulo Ney de Souza, Marcelo Viana|
|Publisher||World Scientific Publisher|
|Publication status||Published - 2018|
- random walk in random environment; self-repelling Brownian polymer; scaling limit; central limit theorem; anomalous diffusion; martingale approximation; resolvent methods