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We address the effect of stochastic resetting on diffusion and subdiffusion process. For diffusion we find that mean square displacement relaxes to a constant only when the distribution of reset times possess finite mean and variance. In this case, the leading order contribution to the probability density function (PDF) of a Gaussian propagator under resetting exhibits a cusp independent of the specific details of the reset time distribution. For subdiffusion we derive the PDF in Laplace space for arbitrary resetting protocol. Resetting at constant rate allows evaluation of the PDF in terms of H function. We analyze the steady state and derive the rate function governing the relaxation behavior. For a subdiffusive process the steady state could exist even if the distribution of reset times possesses only finite mean.
We study Brownian motion in a confining potential under a constant-rate resetting to a reset position x(0). The relaxation of this system to the steady-state exhibits a dynamic phase transition, and is achieved in a light cone region which grows linearly with time. When an absorbing boundary is introduced, effecting a symmetry breaking of the system, we find that resetting aids the barrier escape only when the particle starts on the same side as the barrier with respect to the origin. We find that the optimal resetting rate exhibits a continuous phase transition with critical exponent of unity. Exact expressions are derived for the mean escape time, the second moment, and the coefficient of variation (CV).