Modified Special Relativity on a fluctuating spacetime

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 [S. Liberati, S. Sonego, and M. Visser, Phys. Rev. D 71, 045001 (2005).][R. Aloisio, A. Galante, A. Grillo, S. Liberati, E. Luzio, and F. Mendez, Phys. Rev. D 73, 045020 (2006).]. 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 to 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.