We study transient, one-dimensional random walks in a random environment (RWRE). A well known result of Kesten, Kozlov, and Spitzer gives the limiting distribution of such RWRE under the averaged probability measure (averaging over all environments). However, it was shown recently that for certain distributions on environments there may not be any quenched limiting distributions. That is, for a fixed environment (with probability 1) the random walk does not have a limiting distribution. In this talk I will describe recent work with Gennady Samorodnitsky that explains why (strong) quenched limiting distributions fail to exist. Viewing the quenched distribution of the hitting times as random probability measures, we show that they converge in distribution on the space of probability measures to a random probability measure with interesting stability properties.

Note: these results have also been obtained independently by Dolgopyat and Goldsheid and also Enriquez, Sabot, Tournier and Zindy.