Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



Analyticity preserving properties of resolvents for degenerate diffusion operators in one dimension

Author: Masaaki Tsuchiya
Journal: Proc. Amer. Math. Soc. 90 (1984), 91-94
MSC: Primary 47D05; Secondary 26E05, 34A25, 35A99, 60J60
MathSciNet review: 722422
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 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
  • 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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47D05, 26E05, 34A25, 35A99, 60J60

Retrieve articles in all journals with MSC: 47D05, 26E05, 34A25, 35A99, 60J60

Additional Information

Article copyright: © Copyright 1984 American Mathematical Society