Quantum mechanics and local Lorentz symmetry violation

We show how a Brownian motion on a short scale can originate a relativistic motion on scales larger than the particle's Compton wavelength. Thus, Lorentz symmetry appears to be not a primitive concept, but rather it statistically emerges when a coarse graining average over distances of order, or longer than the Compton wavelength, is taken. We also present the generalizations needed to accommodate in our scheme the doubly special relativistic dynamics. In this way, a previously unsuspected, common stochastic origin of the two frameworks is revealed for the first time. Issues such as generalized commutation relations are also discussed.