Abstract: We develop a probabilistic technique to solve first-order conservation partial differential equations that include non-linear terms. For example, u_t = F(u)u_x + u. Our proposed solution is local in time and is presented as an expectation of the total progeny of a sub-critical branching process. The offspring of this process are independent and identically distributed random variables, which absorb the initial conditions and the coefficients of the nonlinear term of the PDEs. Our technique has an interesting probabilistic interpretation that can be understood by building the link to coagulation processes and Erdős–Rényi-like random graphs. Although the technique has a similar to the Feynman–Kac formula flavour, it is applicable to non-linear PDEs and allows for the analysis of their domain of existence, yielding bounds or, in some cases, exact regions. Joint work with Jochem Hoogendijk and Camillo Schenone: https://arxiv.org/pdf/2310.11338 and https://arxiv.org/pdf/2401.12844