Asymptotics of parabolic Green’s functions on lattices

Gurevich, P.

doi:10.1090/spmj/1464

Asymptotics of parabolic Green’s functions on lattices
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by P. Gurevich

St. Petersburg Math. J. 28 (2017), 569-596

DOI: https://doi.org/10.1090/spmj/1464

Published electronically: July 25, 2017

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Abstract:

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.

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