The interpretation of the integral representation over the "proper time" τ is that the two point function is the sum over all random walk paths that link position 0 to position "x" over time τ. The density of these paths at time τ at position "x" is Gaussian, but the random walkers disappear at a steady rate proportional to "t" so that the Gaussian at time τ is diminished in height by a factor that decreases steadily exponentially. In the quantum field theory context, these are the paths of relativistically localized quanta in a formalism that follows the paths of individual particles. In the pure statistical context, these paths still appear by the mathematical correspondence with quantum fields, but their interpretation is less directly physical.