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The solutions to Burgers equation, in the limit of vanishing viscosity, are investigated when the initial velocity is a Brownian motion (or fractional Brownian motion) function, i.e. a Gaussian process with scaling exponent 0<h<1 (typeA) or the derivative thereof, with scaling exponent −1<h<0 (typeB). Largesize numerical experiments are performed, helped by the fact that the solution is essentially obtained by performing a Legendre transform. The main result is obtained for typeA and concerns the Lagrangian functionx(a) which gives the location at timet=1 of the fluid particle which started at the locationa. It is found to be a complete Devil's staircase. The cumulative probability of Lagrangian shock intervals Δa (also the distribution of shock amplitudes) follows a (Δa)^−h law for small Δa. The remaining (regular) Lagrangian locations form a Cantor set of dimensionh. In Eulerian coordinates, the shock locations are …