6. Random evolution and first-order hyperbolic PDE systems

In this final section, we show how random evolutions constructed from Markov jump processes provide a probabilistic representation of the solution of first-order hyperbolic PDE systems. We restrict ourselves to the simple case of jump processes with values in a finite set, and illustrate our results with an example related to the telegraph equation. At the cost of additional technical difficulties, it is possible to generalize these results to the case of jump processes with values in a non-countable set such as ${ℝ}^d$ , which makes it possible to represent the solution of transport equations of the linearized Boltzmann equation type. These probabilistic representations are particularly well suited to Monte Carlo simulation of the solutions...