Maslov tunnel asymptotics and random walks on a discrete-time lattice

In this paper, a method for solving parabolic problems on a lattice will be presented using random walks as an example. Due 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 conditions of nonnegativity and norm conservation. The obtained solution exists in the entire attainability domain of the random walk in the case of a finite initial condition. In this case, 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.