Maslov Tunnel Asymptotics and Random Walks on a Discrete-Time Lattice

We present a method for solving parabolic problems on a lattice using random walks as an example. Owing to the stochastic properties of random walks, previously obtained interpolation methods for solving hyperbolic problems (Fourier transform; V. A. Kotelnikov’s theorem) cannot be applied on lattices. In this paper, a formal asymptotics of the fundamental solution of the Cauchy problem and boundary value problems for a parabolic random walk on a lattice is constructed based on the representation of the Dirac delta function as a Gaussian exponential and a special partition of unity. This solution satisfies the nonnegativity and norm conservation conditions. The solution exists in the entire attainability domain of the random walk in the case of a finitely supported initial condition. The asymptotics of the solution of the Cauchy problem corresponds to a noncompact Lagrangian manifold such that the projection of its singularity coincides with the boundary of the attainability domain.