Asymptotics of parabolic Green's functions on lattices

2015/04/10 by Gurevich, Pavel
#05C81 #34A33 #35K08 #35K25 #65M80 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1504.02673

Abstract

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 whole 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∈\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.

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