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Stochastic resets have lately emerged as a mechanism able to generate finiteequilibrium mean square displacement (MSD) when they are applied to diffusivemotion. Furthermore, walkers with an infinite mean first arrival time (MFAT) toa given position $x$, may reach it in a finite time when they reset theirposition. In this work we study these emerging phenomena from a unifiedperspective. On one hand we study the existence of a finite equilibrium MSDwhen resets are applied to random motion with $\langle x^2(t)\rangle _m\simt^p$ for $0<p\leq2$. For exponentially distributed reset times, a compactformula is derived for the equilibrium MSD of the overall process in terms ofthe mean reset time and the motion MSD. On the other hand, we also test therobustness of the finiteness of the MFAT for different motion dynamics whichare subject to stochastic resets. Finally, we study a biased Brownianoscillator with resets with the general formulas derived in this work, findingits equilibrium first moment and MSD, and its MFAT to the minimum of theharmonic potential.

Transport properties and first arrival statistics of random searches with stochastic reset times