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