Branching diffusion representation for nonlinear Cauchy problems and Monte Carlo approximation

We provide probabilistic representations of the solution some semilinear hyperbolic and high-order PDEs based on branching diffusions. These pave way for an approximation by standard Monte Carlo method, whose error estimate is controlled central limit theorem, thus partly bypassing curse dimensionality. illustrate numerical implications in context popular physics such as nonlinear Klein–Gordon equation, a simplified scalar version Yang–Mills fourth-order beam equation Gross–Pitaevskii PDE example Schrödinger equations.