Optimizing search processes in systems with state toggling: exact condition delimiting the efficacy of stochastic resetting strategy

Will the strategy of resetting} help a stochastic process to reach its target efficiently, with its environment continually toggling between a strongly favourable and an unfavourable (or weakly favourable) state? A diffusive run-and-tumble motion, transport of molecular motors on or off a filament, and motion under flashing optical traps are special examples of such state toggling. For any general process with toggling under Poisson reset, we derive a mathematical condition for continuous transitions where the advantage rendered by resetting vanishes. For the case of diffusive motion with linear potentials of unequal strength, we present exact solutions which reveal that there is quite generically a re-entrance of the advantage of resetting as a function of the strength of the strongly favourable potential. This result is shown to be valid for quadratic potential traps by using the general condition of transition.