Geodesic
Asymptotics of parabolic Green's functions on lattices
Algebra i analiz, Tome 28 (2016) no. 5, pp. 21-60.

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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 expansions with respect to powers of time variable $t$ up to an arbitrary order and estimate the remainders 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, allows one to obtain higher-order asymptotics of transition probability functions for continuous-time random walks on $\mathbb Z^d$ and other lattices.
Keywords: spatially discrete parabolic equations, asymptotics, discrete Green functions, lattice Green functions, heat kernels of lattices, continuous-time random walks. 
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P. Gurevich. Asymptotics of parabolic Green's functions on lattices. Algebra i analiz, Tome 28 (2016) no. 5, pp. 21-60. http://geodesic.mathdoc.fr/item/AA_2016_28_5_a1/

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