Summary: In a recent paper, E. G. Puckett [Math. Comput. 52, No. 186, 615- 645 (1989; Zbl 0671.60075)] proposed a stochastic particle method for the Kolmogorov-Petrovskij-Piskunov equation in \((0,T] \times\mathbb{R}\) \[ {{\partial u} \over {\partial t}}= Au:= \Delta u+ f(u), \qquad u(0,\cdot)= u_ 0(\cdot), \] where \(1-u_ 0\) is the cumulative function, supposed to be smooth enough, of a probability distribution, and \(f\) is a function describing the reaction. His justification of the method and his analysis of the error were based on a splitting of the operator \(A\). He proved that, if \(h\) is the time discretization step and \(N\) number of particles used in the algorithm, one can obtain an upper bound of the norm of the random error on \(u(T,x)\) in \(L^ 1(\Omega \times\mathbb{R})\) of order \(1/N^{1/4}\), provided \(h- O (1/N^{1/4})\), but conjectured, from numerical experiments, that it should be of order \(O(h)+O(1/ \sqrt{N})\), without any relation between \(h\) and \(N\). We prove that conjecture.
We also construct a similar stochastic particle method for more general nonlinear diffusion-reaction-convection PDEs \({{\partial u}\over {\partial t}}= Lu+ f(u)\), \(u(0,\cdot)= u_ 0(\cdot)\), where \(L\) is a strongly elliptic second-order operator with smooth coefficients, and prove that the preceding rate of convergence still holds when the coefficients of \(L\) are constant, and in the other case \(O (\sqrt{h})+ O(1/ \sqrt{N})\). The construction of the method and the analysis of the error are based on a stochastic representation formula of the exact solution \(u\).