abstract: Examples of stochastic resetting are found in a wide variety of situations. A general interesting question is: how does a stochastic resetting affect the temporal evolution of a stochastic system? In this talk I will discuss two qualitatively different resetting mechanism. In the first mechanism, a stochastic process is subjected to a stochastic resetting with a constant rate to its initial condition. The process, under this mechanism, generically reaches a nonequilibrium steady state. I will discuss how the steady state is approached in time through an unusual relaxation mechanism. In the second mechanism, which involve resetting to a dynamically evolving condition, while the process itself does not reach a stationary state, there is a "difference process" which becomes stationary. I will discuss a simple random walk model where in addition to the symmetric walk, at each time step the walker resets to the maximum of the already visited positions with a given probability. In this case, both the average maximum and the average position grow ballistically with a common speed and the fluctuations around their respective averages grow diffusively, again with the same diffusion coefficient. The probability distribution of the difference between the maximum and the location of the walker eventually becomes stationary. The approach to this stationary distribution is accompanied by a dynamical phase transition as in the previous mechanism.
Ref: S. N. Majumdar, S. Sabhapandit, and G. Schehr, Phys. Rev. E 91, 052131 (2015); Phys. Rev. E 92, 052126 (2015)