  It was recently proposed that deformations of the relativistic symmetry, as
those considered in Deformed Special Relativity (DSR), can be seen as the
outcome of a measurement theory in the presence of non-negligible (albeit
small) quantum gravitational fluctuations [1,2]. In this paper we explicitly
consider the case of a spacetime described by a flat metric endowed with
stochastic fluctuations and, for a free particle, we show that DSR-like
nonlinear relations between the spaces of the measured and classical momenta,
can result from the average of the stochastic fluctuations over a scale set be
the de Broglie wavelength of the particle. As illustrative examples we consider
explicitly the averaging procedure for some simple stochastic processes and
discuss the physical implications of our results.
