×
McKean, Henry P. jun.

Elementary solutions for certain parabolic partial differential equations. (English) Zbl 0070.32003

Trans. Am. Math. Soc. 82, 519-548 (1956).
Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page
scan of Zbl 0070.32003

Cited in 83 Documents

Keywords:

partial differential equations
Cite Review PDF
Full Text: DOI

References:

[1] William Feller, On second order differential operators, Ann. of Math. (2) 61 (1955), 90 – 105. · Zbl 0064.11301 · doi:10.2307/1969621
[2] William Feller, The general diffusion operator and positivity preserving semi-groups in one dimension, Ann. of Math. (2) 60 (1954), 417 – 436. · Zbl 0057.09805 · doi:10.2307/1969842
[3] Hermann Weyl, Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Ann. 68 (1910), no. 2, 220 – 269 (German). · JFM 41.0343.01 · doi:10.1007/BF01474161
[4] Marshall Harvey Stone, Linear transformations in Hilbert space, American Mathematical Society Colloquium Publications, vol. 15, American Mathematical Society, Providence, RI, 1990. Reprint of the 1932 original. · Zbl 0005.40003
[5] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Oxford, at the Clarendon Press, 1946 (German). · Zbl 0097.27601
[6] Kunihiko Kodaira, The eigenvalue problem for ordinary differential equations of the second order and Heisenberg’s theory of \?-matrices, Amer. J. Math. 71 (1949), 921 – 945. · Zbl 0035.27101 · doi:10.2307/2372377
[7] Joanne Elliott, Eigenfunction expansions associated with singular differential operators, Trans. Amer. Math. Soc. 78 (1955), 406 – 425. · Zbl 0065.10404
[8] Einar Hille, The abstract Cauchy problem and Cauchy’s problem for parabolic differential equations, J. Analyse Math. 3 (1954), 81 – 196. · Zbl 0059.08703 · doi:10.1007/BF02803587
[9] W. Feller, Zur Theorie der stochastischen Prozesse (Existenz und Eindeutigkeitssätze), Math. Ann. vol. 113 (1936) pp. 113-160. · Zbl 0014.22201
[10] Norman Levinson, A simplified proof of the expansion theorem for singular second order linear differential equations, Duke Math. J. 18 (1951), 57 – 71. · Zbl 0044.31302
[11] Norman Levinson, Addendum to ”A simplified proof of the expansion theorem for singular second order linear differential equations.”, Duke Math. J. 18 (1951), 719 – 722. · Zbl 0045.04602
[12] Kôsaku Yosida, On Titchmarsh-Kodaira’s formula concerning Weyl-Stone’s eigenfunction expansion, Nagoya Math. J. 1 (1950), 49 – 58. · Zbl 0038.24802
[13] Kôsaku Yosida, Correction to my paper ”On Titchmarsh-Kodaira’s formula concerning Weyl-Stone’s eigenfunction expansion” in Nagoya Mathematical Journal, vol. 1 (1950), 49 – 58, Nagoya Math. J. 6 (1953), 187 – 188. · Zbl 0038.24802
[14] Samuel Karlin and James McGregor, Representation of a class of stochastic processes, Proc. Nat. Acad. Sci. U. S. A. 41 (1955), 387 – 391. · Zbl 0067.10803
[15] W. Ledermann and G. E. H. Reuter, Spectral theory for the differential equations of simple birth and death processes, Philos. Trans. Roy. Soc. London. Ser. A. 246 (1954), 321 – 369. · Zbl 0059.11704 · doi:10.1098/rsta.1954.0001
[16] William Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (2) 55 (1952), 468 – 519. · Zbl 0047.09303 · doi:10.2307/1969644
[17] Einar Hille, Functional Analysis and Semi-Groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, New York, 1948. · Zbl 0033.06501
[18] R. Courant and D. Hilbert, Methoden der mathematischen Physik, vol. 1, Berlin, Springer, 1931. · Zbl 0001.00501
[19] David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, N. J., 1941. · Zbl 0063.08245
[20] A. Kolmogoroff, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann. 104 (1931), no. 1, 415 – 458 (German). · Zbl 0001.14902 · doi:10.1007/BF01457949
[21] A. Khinchin, Asymptotische Gesetze der Wahrscheinlichkeitsrechnung, Ergebnisse der Mathematik u. ihrer Grenzgebiete, vol. 2 (4), New York, Chelsea, 1948.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.