Probabilistic representation and Monte Carlo method for nonlinear Dirac equations

Source abstract

Dirac equations describe the relativistic evolution of quantum particles. Using the point-process stochastic calculus of piecewise-deterministic Markov processes (PDMPs) on general state spaces, which naturally allows additional Feynman-Kac potentials, we develop a probabilistic representation for the nonlinear Dirac equations in $1+1$ spacetime dimension with cubic nonlinearity and $L_\infty$-type problem data using a multi-type branching telegraph process together with both continuous and jump-type multiplicative functionals. Our main theorem leads to a classification result which, when applied to the specific Soler model, gives a sharp finite-time admissibility of the probabilistic representation. By extending the one-step representation to a multi-step, time-marching representation using an analytically-derived a priori $L_\infty$ estimate, we observe a ``local probabilistic representation, but global PDE well-posedness" phenomenon. We proceed to propose a time-marching Monte Carlo algorithm as a global numerical solver for the Soler system with a rigorous convergence proof and establish an additional stability result on the PDE side. Finally, while the underlying process parameters do not affect the validity of the probabilistic representation, we observe their potential for variance reduction in the Monte Carlo methods.