On polynomially bounded harmonic functions on the Zd lattice

• Autorzy: Nayar P.
• Języki publikacji: EN

Abstrakty

EN

We prove that if ƒ : Zd → R is harmonic and there exists a polynomial W : Zd → R such that ƒ + W is nonnegative, then ƒ is a polynomial.

Słowa kluczowe

EN

harmonic functions; integer lattice; polynomially bounded harmonic functions

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2009
• Tom: Vol. 57, no 3-4
• Strony: 231--242
• Opis fizyczny: Bibliogr. 8 poz.

Twórcy

autor

Nayar P.

• Institute of Mathematics, University of Warsaw, 02-097 Warszawa, Poland,

Bibliografia

• [1] D. Blackwell, Extension of the renewal theorem, Pacific J. Math. 3 (1953), 315-320.
• [2] G. Choquet et J. Deny, Sur l’équation de convolution μ = μ * σ, C. R. Acad. Sci. Paris 250 (1960), 799-801.
• [3] G. Darkiewicz, Nonnegative harmonic functions on graphs-probabilistic approach, Master Thesis, Univ. of Warsaw, 2001 (in Polish).
• [4] P. L. Davies and D. N. Shanbhag, A generalization of a theorem of Deny with applications in characterization theory, Quart. J. Math. Oxford Ser. 38 (1987), 13-34.
• [5] J. L. Doob, J. L. Snell and R. E. Williamson, Application of boundary theory to sums of independent random variables, in: Contributions to Probability and Statistics, Stanford Univ. Press, Stanford, CA, 1960, 182-197.
• [6] W. Hebisch and L. Saloff-Coste, Gaussian estimates for Markov chains and random walks on groups, Ann. Probab. 21 (1993), 673-709.
• [7] P. Nayar, The Liouville theorem for harmonic functions on the Zd lattice, Bachelor Thesis, Univ. of Warsaw, 2008 (in Polish).
• [8] W. Woess, Random Walks on Infinite Graphs and Groups, Cambridge Univ. Press, Cambridge, 2000.