Hitting Times for Random Walks with Restarts
2012 (English)In: SIAM Journal on Discrete Mathematics, ISSN 0895-4801, E-ISSN 1095-7146, Vol. 26, no 2, 537-547 p.Article in journal (Refereed) Published
The time it takes a random walker in a lattice to reach the origin from another vertex x has infinite mean. If the walker can restart the walk at x at will, then the minimum expected hitting time gamma(x, 0) (minimized over restarting strategies) is finite; it was called the "grade" of x by Dumitriu, Tetali, and Winkler. They showed that in a more general setting, the grade (a variant of the "Gittins index") plays a crucial role in control problems involving several Markov chains. Here we establish several conjectures of Dumitriu, Tetali, and Winkler on the asymptotics of the grade in Euclidean lattices. In particular, we show that in the planar square lattice, gamma(x, 0) is asymptotic to 2 vertical bar x vertical bar(2) log vertical bar x vertical bar as vertical bar x vertical bar ->infinity. The proof hinges on the local variance of the potential kernel h being almost constant on the level sets of h. We also show how the same method yields precise second order asymptotics for hitting times of a random walk (without restarts) in a lattice disk.
Place, publisher, year, edition, pages
2012. Vol. 26, no 2, 537-547 p.
random walk, hitting time, Gittins index, harmonic function, potential kernel
IdentifiersURN: urn:nbn:se:uu:diva-178157DOI: 10.1137/100796352ISI: 000305962300009OAI: oai:DiVA.org:uu-178157DiVA: diva2:542128