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

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