The cost of stochastic resetting

Abstract

Resetting a stochastic process has been shown to expedite the completion time of some complex tasks, such as finding a target for the first time. Here we consider the cost of resetting by associating to each reset a cost, which is a function of the distance travelled during the reset event. We compute the Laplace transform of the joint probability of first passage time $t_f$, number of resets $N$ and total resetting cost $C$, and use this to study the statistics of the total cost and also the time to completion ${\mathcal T} = C + t_f$. We show that in the limit of zero resetting rate, the mean total cost is finite for a linear cost function, vanishes for a sub-linear cost function and diverges for a super-linear cost function. This result contrasts with the case of no resetting where the cost is always zero. We also find that the resetting rate which optimizes the mean time to completion may be increased or decreased with respect to the case of no resetting cost according to the choice of cost function. For the case of an exponentially increasing cost function, we show that the mean total cost diverges at a finite resetting rate. We explain this by showing that the distribution of the cost has a power-law tail with a continuously varying exponent that depends on the resetting rate.

Fractional advection diffusion asymmetry equation, derivation, solution and application

“…It is unclear to the authors of this manuscript, if equation ( 3) has any connection with the foundational results in [96,97]. (ii) Resetting refers to a category of stochastic processes, where the particle is halted at random or non-random times and then restarted from its initial position or a predetermined location [98][99][100][101][102][103][104]. The standard model for the dynamics is Brownian motion, however due to the wide prevalence of anomalous diffusion, resetting for anomalous process was also studied [99,102,[105][106][107][108].…”

On a diffusion which stochastically restarts from moving random spatial positions: a non-renewal framework

“…Recent references have treated the case of non-instantaneous resetting [54][55][56]66] which introduces, along with the diffusion phase displacement, a return phase to the initial position. It becomes clear that some cost (or penalty) is necessary to be included in the models to turn them more realistic [35,36,71].…”

“…Here we consider one way to measure the cost, which we call quadratic cost, inspired from [35], as follows. Consider the first R resettings performed by the particle.…”

“…temporal relaxation to steadystate [13], aspects of universality regarding the optimal resetting rate [5], autocorrelation [26,27], properties of additive functionals [28,29], optimization of quantiles of FPT w.r.t. resetting rate [30], extreme value statistics [31][32][33][34], cost of resetting [35] and experimental verification of its theoretical predictions [36].…”

“…One essential aspect of the model is that the renewal framework does not apply here, because the noise process Y(t) keeps evolving with time and is not affected by the resetting times T n . When it is possible to be applicable, the renewal approach plays an important role in helping the development of equations [5,27,35,[65][66][67]. On the other hand models with non-renewal characteristics are also relevant.…”