Diffusion with resetting in arbitrary spatial dimension

Martin R. Evans 1 Satya N. Majumdar 2

Journal of Physics A: Mathematical and Theoretical, Institute of Physics: Hybrid Open Access, 2014, 47, pp.285001

We consider diffusion in arbitrary spatial dimension d with the addition of a resetting process wherein the diffusive particle stochastically resets to a fixed position at a constant rate $r$. We compute the non-equilibrium stationary state which exhibits non-Gaussian behaviour. We then consider the presence of an absorbing target centred at the origin and compute the survival probability and mean time to absorption of the diffusive particle by the target. The mean absorption time is finite and has a minimum value at an optimal resetting rate $r^*$ which depends on dimension. Finally we consider the problem of a finite density of diffusive particles, each resetting to its own initial position. While the typical survival probability of the target at the origin decays exponentially with time regardless of spatial dimension, the average survival probability decays asymptotically as $\exp -A (\log t)^d$ where $A$ is a constant. We explain these findings using an interpretation as a renewal process and arguments invoking extreme value statistics.

  • 1. SUPA, School of Physics, University of Edinburgh
  • 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques