Asymptotics of parabolic Green's functions on lattices

For parabolic spatially discrete equations, we considered the Green functions also known as heat kernels on lattices. Their asymptotic expansions with respect to powers of the time variable t are obrained up to an arbitrary order, the remainders are estimated uniformly on the entire lattice. The spatially discrete (difference) operators under consideration are finite-difference approximations of continuous strongly elliptic differential operators (with constant coefficients) of arbitrary even order in ℝd with arbitrary d ∈. This genericity, besides numerical and deterministic lattice-dynamics applications, makes it possible to obtain higher-order asymptotics of transition probability functions for continuous-time random walks on ℤd and other lattices. © 2017 American Mathematical Society.