Previously, the author presented a stochastic and variational ap- proach to the Lax-Friedrichs finite difference scheme applied to hyperbolic scalar conservation laws and the corresponding Hamilton-Jacobi equations with convex and superlinear Hamiltonians in the one-dimensional periodic set-ting, showing new results on the stability and convergence of the scheme [Soga, Math. Comp. 84 (2015), 629–651]. In the current paper, we extend these re-sults to the higher dimensional setting. Our framework with a deterministic scheme provides approximation of viscosity solutions of Hamilton-Jacobi equa-tions, their spatial derivatives and the backward characteristic curves at the same time, within an arbitrary time interval. The proof is based on stochastic calculus of variations with random walks, a priori boundedness of minimizers of the variational problems that verifies a CFL type stability condition, and the law of large numbers for random walks under the hyperbolic scaling limit. Convergence of approximation and the rate of convergence are obtained in terms of probability theory. The idea is reminiscent of the stochastic and vari-ational approach to the vanishing viscosity method introduced in [Fleming, J. Differ. Eqs 5 (1969) 515–530].
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