Finite-dimensional probability distributions in the random Burgers-Riemann problem

Abstract

We fully quantify the uncertainty for the Burgers’ partial differential equation with random Riemann initial states, by obtaining the generalized probability density functions associated to the finite-dimensional distributions of the response. The entropy solution is a mixed random field, decomposed into discrete and continuous parts, therefore the formalism of the Dirac delta function is employed in a novel way to derive closed-form expressions for the generalized densities, both for non-independent and independent initial conditions. This work extends the recent papers that focused on the first and the second finite-dimensional laws.