Abstract

The subject of Large Deviations measures the probabilities of atypically large fluctuations of random process from their means. On the other hand, small ball probabilities measure the atypically small fluctuations from the mean. While there is an extensive literature on large deviations, small ball probabilities have not been well understood. There has been quite some work on Gaussian processes but far less for general processes. We are interested in understanding the small ball problem in the setting of the stochastic heat equation (SHE); this has not been explored before. Our results would be a first step in understanding behaviour of the paths of the SHE at a fixed time t under the influence of a random field. Examples of random fields are obstacles or a penalty for paths (at fixed time) to self intersect.