Branching diffusion representation for nonlinear Cauchy problems and Monte Carlo approximation
Résumé
We provide a probabilistic representations of the solution of some semilinear hyperbolic
and high-order PDEs based on branching diffusions. 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 simpli ed scalar version
of the Yang-Mills equation, a fourth-order nonlinear beam equation and the Gross-Pitaevskii PDE
as an example of nonlinear Schrodinger equations.
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