A related and perhaps easier question might be the following: What is the distribution of the diameter of the smallest sphere entirely containing the random walk. This question was discussed by Weiss and Rubin here in 1983. Is it possible that the question is still open? I tried combing the literature, but could not find anything. Perhaps I have missed some very important works.

In 1941 Daniels was the first to investigate the extent or span of a 1D random walk. He actually determined the probability density of the span of a random walk in 1D. G.H. Weiss and R.J. Rubin generalized the notion of span to multi-dimensional random walks. Spans are the sides of the smallest rectangular box with sides parallel to the coordinate axes that entirely contains the random walk. In 2D, they were able to determine the probability densities for the smallest and largest spans. This is as close as I could get to the answer.