Switching diffusions and stochastic resetting

We consider a Brownian particle that switches between two different diffusion states (D-0,D-1) according to a two-state Markov chain. We further assume that the particle's position is reset to an initial valueX(r)at a Poisson rater, and that the discrete diffusion state is simultaneously reset according to the stationary distribution rho(n),n= 0, 1, of the Markov chain. We derive an explicit expression for the non-equilibrium steady state (NESS) onR<i, which is given by the sum of two decaying exponentials. In the fast switching limit the NESS reduces to the exponential distribution of pure diffusion with stochastic resetting. The effective diffusivity is given by the meanD over bar =rho 0D0+rho 1D1<i. We then determine the mean first passage time (MFPT) for the particle to be absorbed by a target at the origin, having started at the reset positionX(r)> 0. We proceed by calculating the survival probability in the absence of resetting and then use a last renewal equation to determine the survival probability with resetting. Similar to the NESS, we find that the MFPT depends on the sum of two exponentials, which reduces to a single exponential in the fast switching limit. Finally, we show that the MFPT has a unique minimum as a function of the resetting rate, and explore how the optimal resetting rate depends on other parameters of the system.