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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Analyticity preserving properties of resolvents for degenerate diffusion operators in one dimension
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by Masaaki Tsuchiya PDF
Proc. Amer. Math. Soc. 90 (1984), 91-94 Request permission

Abstract:

Let $L = a(x)({d^2}/d{x^2}) + b(x)(d/dx) + c(x)$ be a diffusion operator on a compact interval $I = [{r_0},{r_1}]$ investigated by S. N. Ethier. Here, assume that $a$, $b$ and $c$ are real analytic functions on $I$, $a(x) > 0$ for $x \in ({r_0},{r_1})$, $a({r_i}) = 0 \leqslant {( - 1)^i}b({r_i})(i = 0,1)$, and both ${r_i}(i = 0,1)$ are simple zeros of $a(x)$. It is shown that the resolvent $\left \{ {{G_\lambda }} \right \}$ for $L$ has the analyticity preserving property for sufficiently large $\lambda$, so that the equation $(L - \lambda )u = f$ is solvable in the space of real analytic functions on $I$. Some examples are given to show that the condition on $L$ is best possible.
References
  • S. N. Ethier, Differentiability-preserving properties of Markov semigroups associated with one-dimensional diffusions, Z. Wahrsch. Verw. Gebiete 45 (1978), no. 3, 225–238. MR 510027, DOI 10.1007/BF00535304
  • M. Hukuhara, Ordinary differential equations, 2nd ed., Iwanami, Tokyo, 1980. (Japanese)
  • Hikosaburo Komatsu, On the regularity of hyperfunction solutions of linear ordinary differential equations with real analytic coefficients, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20 (1973), 107–119. MR 328584
  • L. M. Milne-Thomson, The calculus of finite differences, 2nd. ed., Chelsea, New York, 1981.
  • Oskar Perron, Über Summengleichungen und Poincarésche Differenzengleichungen, Math. Ann. 84 (1921), no. 1-2, 1–15 (German). MR 1512016, DOI 10.1007/BF01458689
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Additional Information
  • © Copyright 1984 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 90 (1984), 91-94
  • MSC: Primary 47D05; Secondary 26E05, 34A25, 35A99, 60J60
  • DOI: https://doi.org/10.1090/S0002-9939-1984-0722422-2
  • MathSciNet review: 722422