A Small Deviation for Random Walk with Random Environment in Time

We establish a small deviation principle for random walk with random environment in time, where the time-space scaling can be logarithm rate. The classic small deviation principle (for i.i.d. random walk) in Mogul'skiĭ (1974) included all slow growth scaling, while Lv & Hong provided the small deviation principle for random walk with random environment in time under the assumption that the scaling can not slower than power rate. Therefore, in this paper, we fill this gap in some extent.