Resumen
We study the Cauchy problem for non-linear hyperbolic partial differential equations of first order associated with single conservation laws in one space variable. We consider only initial conditions which are smooth and periodic, both in the deterministic and stochastic sense. Stochastic initial conditions are defined by means of a Gaussian probability measure which we construct on the space of smooth periodic functions. We show solutions of the Cauchy problem exist and are piece-wise smooth with probability one (stochastic case) and for all initial conditions outside of a set? Of first category (deterministic case). Only shock discontinuities (called shock waves) arise for such solutions. For most initial conditions, i.e. with probability one and outside of o only one shock wave exists (modulo 2p) after a certain time and an asymptotic formula is given for it. Stability of the Cauchy problem with respect to perturbations of the initial conditions is then proved (for most initial conditions). In particular the geometry of the shock waves is also stable.