​​We provide a probabilistic representations of the solution of some semi linear hyperbolic and high-order PDEs based on branching diffusion. This is a direct extension of our previous work in the context of semi linear parabolic PDEs based on the classical Mc Kean representation for KPP equations. These representations pave the way for a Monte-Carlo approximation of the solution, thus bypassing the curse of dimensionality. We illustrate the numerical implications in the context of some popular PDEs in physics such as nonlinear Klein-Gordon equation, a simplified scalar version of the Yang-Mills equation, a fourth-order nonlinear beam equation and the Gross- Pitaevskii PDE as an example of nonlinear Schr ̈odinger equations.

Sponsored by the Van Eenam Lecture Series