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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Asymptotics of parabolic Green’s functions on lattices

Author: P. Gurevich
Original publication: Algebra i Analiz, tom 28 (2016), nomer 5.
Journal: St. Petersburg Math. J. 28 (2017), 569-596
MSC (2010): Primary 35K08
Published electronically: July 25, 2017
MathSciNet review: 3637586
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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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Additional Information

P. Gurevich
Affiliation: Free University of Berlin, Germany; Peoples’ Friendship University, Russia

Keywords: Spatially discrete parabolic equations, asymptotics, discrete Green functions, lattice Green functions, heat kernels of lattices, continuous-time random walks
Received by editor(s): June 22, 2015
Published electronically: July 25, 2017
Article copyright: © Copyright 2017 American Mathematical Society