The cost of stochastic resetting

John C. Sunil*, Richard A. Blythe, Martin R. Evans, Satya N. Majumdar

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract / Description of output

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 tf , 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 T = C + tf . 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.
Original languageEnglish
Article number395001
Pages (from-to)1-24
Number of pages24
JournalJournal of Physics A: Mathematical and Theoretical
Issue number39
Publication statusPublished - 8 Sept 2023

Keywords / Materials (for Non-textual outputs)

  • cost of resetting
  • diffusion with resetting
  • renewal equation
  • stochastic resetting


Dive into the research topics of 'The cost of stochastic resetting'. Together they form a unique fingerprint.

Cite this