Nested Stochastic Resetting: Nonequilibrium Steady States and Exact Correlations

Stochastic resetting breaks detailed balance and drives the formation of nonequilibrium steady states. Here, we consider a chain of diffusive processes x_{i}(t) that interact unilaterally: at random time intervals, the process x_{n} stochastically resets to the instantaneous value of x_{n-1}. We derive analytically the steady-state statistics of these nested stochastic resetting processes including the stationary distribution for each process as well as its moments. We are also able to calculate exactly the steady-state two-point correlations ⟨x_{n}x_{n+j}⟩ between processes by mapping the problem to one of the ordering statistics of random counting processes. Understanding statistics and correlations in many-particle nonequilibrium systems remains a formidable challenge and our results provide an example of such tractable correlations. We expect this framework will both help build a model-independent framework for random processes with unilateral interactions and find immediate applications, e.g., in the modeling of lossy information propagation.