Asymptotics of parabolic Green’s functions on lattices
Pavel Gurevich · St Petersburg Mathematical Journal · 2017
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 ${\mathbb R}^d$ with arbitrary $d\in {\mathbb N}$. 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 ${\mathbb Z}^d$ and other lattices.