Statistics of the duration time of a random walk given its present position: Dating a random walk

摘要

We consider the distribution of the duration time, the time elapsed since it began, of a diffusion process given itsnpresent position, under the assumption that the process began at the origin. For unbiased diffusion, the distributionndoes not exist (it is identically zero) for one- and two-dimensional systems. We find the explicit expression fornthe distribution for three and higher dimensions and discuss the behavior of the duration time statistics: we findnthat the expected duration time exists only for dimensions five and higher, whereas the variance becomes finitenfor seven dimensions and above.We then turn to the case of biased diffusion. The drift velocity introduces a newntime scale and the resulting statistics arise from the interplay of the diffusive time scale and the drift time scale.nFor these systems all the moments exist and explicit expressions are presented and discussed for the expectednduration time and its variance for all dimensions.