## 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.## References

- A. Ashyralyev and P. E. Sobolevskiĭ,
*Well-posedness of parabolic difference equations*, Operator Theory: Advances and Applications, vol. 69, Birkhäuser Verlag, Basel, 1994. Translated from the Russian by A. Iacob. MR**1299329**, DOI 10.1007/978-3-0348-8518-8 - Wolf-Jürgen Beyn,
*Discrete Green’s functions and strong stability properties of the finite difference method*, Applicable Anal.**14**(1982/83), no. 2, 73–98. MR**678496** - J. H. Bramble and V. Thomée,
*Pointwise bounds for discrete Green’s functions*, SIAM J. Numer. Anal.**6**(1969), 583–590. MR**263265**, DOI 10.1137/0706053 - T. Delmotte and J.-D. Deuschel,
*On estimating the derivatives of symmetric diffusions in stationary random environment, with applications to $\nabla \phi$ interface model*, Probab. Theory Related Fields**133**(2005), no. 3, 358–390. MR**2198017**, DOI 10.1007/s00440-005-0430-y - R. J. Duffin,
*Discrete potential theory*, Duke Math. J.**20**(1953), 233–251. MR**70031** - Léonid S. Frank,
*Factorization for difference operators*, J. Math. Anal. Appl.**62**(1978), no. 1, 170–185. MR**488184**, DOI 10.1016/0022-247X(78)90228-7 - F. Alberto Grünbaum and Plamen Iliev,
*Heat kernel expansions on the integers*, Math. Phys. Anal. Geom.**5**(2002), no. 2, 183–200. MR**1918052**, DOI 10.1023/A:1016258207606 - P. Gurevich and S. Tikhomirov,
*Spatially discrete reaction-diffusion equations with discontinuous hysteresis*, http://arxiv.org/abs/1504.02385. - Anthony J. Guttmann,
*Lattice Green’s functions in all dimensions*, J. Phys. A**43**(2010), no. 30, 305205, 26. MR**2659619**, DOI 10.1088/1751-8113/43/30/305205 - Plamen Iliev,
*Heat kernel expansions on the integers and the Toda lattice hierarchy*, Selecta Math. (N.S.)**13**(2007), no. 3, 497–530. MR**2383604**, DOI 10.1007/s00029-007-0046-4 - G. Joyce,
*On the cubic modular transformation and the cubic lattice Green functions*, J. Phys. A: Math. Gen.**31**(1998), 5105–5115. - Shigetoshi Katsura, Tohru Morita, Sakari Inawashiro, Tsuyoshi Horiguchi, and Yoshihiko Abe,
*Lattice Green’s function. Introduction*, J. Mathematical Phys.**12**(1971), 892–895. MR**286421**, DOI 10.1063/1.1665662 - Gregory F. Lawler and Vlada Limic,
*Random walk: a modern introduction*, Cambridge Studies in Advanced Mathematics, vol. 123, Cambridge University Press, Cambridge, 2010. MR**2677157**, DOI 10.1017/CBO9780511750854 - I. K. Lifanov, L. N. Poltavskii, and G. M. Vainikko,
*Hypersingular integral equations and their applications*, Differential and Integral Equations and Their Applications, vol. 4, Chapman & Hall/CRC, Boca Raton, FL, 2004. MR**2053793** - Moshe Mangad,
*Asymptotic expansions of Fourier transforms and discrete polyharmonic Green’s functions*, Pacific J. Math.**20**(1967), 85–98. MR**203368** - Daniel Marahrens and Felix Otto,
*Annealed estimates on the Green function*, Probab. Theory Related Fields**163**(2015), no. 3-4, 527–573. MR**3418749**, DOI 10.1007/s00440-014-0598-0 - Per-Gunnar Martinsson and Gregory J. Rodin,
*Asymptotic expansions of lattice Green’s functions*, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.**458**(2002), no. 2027, 2609–2622. MR**1942800**, DOI 10.1098/rspa.2002.0985 - Minoru Murata,
*Large time asymptotics for fundamental solutions of diffusion equations*, Tohoku Math. J. (2)**37**(1985), no. 2, 151–195. MR**788128**, DOI 10.2748/tmj/1178228678 - J. R. Norris,
*Long-time behaviour of heat flow: global estimates and exact asymptotics*, Arch. Rational Mech. Anal.**140**(1997), no. 2, 161–195. MR**1482931**, DOI 10.1007/s002050050063 - M. M. H. Pang,
*Heat kernels of graphs*, J. London Math. Soc. (2)**47**(1993), no. 1, 50–64. MR**1200977**, DOI 10.1112/jlms/s2-47.1.50 - Yehuda Pinchover,
*Some aspects of large time behavior of the heat kernel: an overview with perspectives*, Mathematical physics, spectral theory and stochastic analysis, Oper. Theory Adv. Appl., vol. 232, Birkhäuser/Springer Basel AG, Basel, 2013, pp. 299–339. MR**3077281**, DOI 10.1007/978-3-0348-0591-9_{6} - Tetsuo Tsuchida,
*Long-time asymptotics of heat kernels for one-dimensional elliptic operators with periodic coefficients*, Proc. Lond. Math. Soc. (3)**97**(2008), no. 2, 450–476. MR**2439669**, DOI 10.1112/plms/pdn014

## Bibliographic Information

**P. Gurevich**- Affiliation: Free University of Berlin, Germany; Peoples’ Friendship University, Russia
- Email: gurevich@math.fu-berlin.de
- Received by editor(s): June 22, 2015
- Published electronically: July 25, 2017
- © Copyright 2017 American Mathematical Society
- Journal: St. Petersburg Math. J.
**28**(2017), 569-596 - MSC (2010): Primary 35K08
- DOI: https://doi.org/10.1090/spmj/1464
- MathSciNet review: 3637586