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

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