Traveled lengths statistic is a key quantity for characterizing stochastic
    processes in bounded domains. For straight lines and diffusive random walks,
    the average length of the trajectories through the domain is independent of the
    random walk characteristics and depends only on the ratio of the volume domain
    over its surface, a behavior that has been recently observed experimentally for
    exponential jump processes. In this article, relying solely on geometrical
    considerations, we extend this remarkable property to all d-dimensional random
    curves of arbitrary lengths (finite or infinite), thus including all kind of
    random walks as well as fibers processes. Integral geometry will be central to
    establishing this universal property of random trajectories in bounded domains.

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