  The author presented a stochastic and variational approach 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 setting, showing
new results on the stability and convergence of the scheme [Soga, Math. Comp.
(2015)]. In the current paper, we extend these results to the higher
dimensional setting. Our framework with a deterministic scheme provides
approximation of viscosity solutions of Hamilton-Jacobi equations, 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; 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 variational
approach to the vanishing viscosity method introduced in [Fleming, J. Differ.
Eqs (1969)].
