Long time scaling behaviour for diffusion with resetting and memory

Denis Boyer 1 Martin R. Evans 2 Satya N. Majumdar 3

Journal of Statistical Mechanics: Theory and Experiment, IOP Science, 2017, 2017 (2), pp.023208

We consider a continuous-space and continuous-time diffusion process under resetting with memory. A particle resets to a position chosen from its trajectory in the past according to a memory kernel. Depending on the form of the memory kernel, we show analytically how different asymptotic behaviours of the variance of the particle position emerge at long times. These range from standard diffusive ($\sigma^2 \sim t$) all the way to anomalous ultraslow growth $\sigma^2 \sim \ln \ln t$.

• 1. UNAM - Universidad Nacional Autónoma de México
• 2. SUPA - School of Physics, University of Edinburgh
• 3. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques