Abstract
The infinite-dimensional Laplacian (introduced by P. Levy in 1922) has a number of unusual properties. In particular, the supply of harmonic functions of an infinite number of variables connected with this Laplacian is exceptionally large. In this paper, with the help of estimates of the growth of sums of dependent random variables we get (in a certain sense) optimal conditions for functions on a Hilbert space to be harmonic.
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Literature cited
P. Levy, Concrete Problems of Functional Analysis [Russian translation], Nauka, Moscow (1967).
M. N. Feller, “Infinite-dimensional Laplace-Levy differential operators,” Ukr. Mat. Zh.,32, No. 1, 69–79 (1980).
V. V. Petrov, “Order of growth of sums of dependent random variables,” Teor. Veroyatn. Primen.,18, No. 2, 358–360 (1973).
M. N. Feller, “Infinite-dimensional elliptic equations and P. Levy-type operators,” Usp. Mat. Nauk,41, No. 4, 97–140 (1986).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 11, pp. 1576–1579, November, 1990.
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Feller, M.N. Supply of harmonic functions of an infinite number of variables. I. Ukr Math J 42, 1419–1423 (1990). https://doi.org/10.1007/BF01066202
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DOI: https://doi.org/10.1007/BF01066202