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For parabolic spatially discrete equations, we consider Green's functions,also known as heat kernels on lattices. We obtain their asymptotic expansionswith respect to powers of time variable $t$ up to an arbitrary order andestimate the remainders uniformly on the whole lattice. The spatially discrete(difference) operators under consideration are finite-difference approximationsof continuous strongly elliptic differential operators (with constantcoefficients) of arbitrary even order in $\mathbb R^d$ with arbitrary$d\in\mathbb N$. This genericity, besides numerical and deterministiclattice-dynamics applications, allows one to obtain higher-order asymptotics oftransition probability functions for continuous-time random walks on $\mathbbZ^d$ and other lattices.

Asymptotics of parabolic Green's functions on lattices