Elementary solutions for certain parabolic partial differential equations
Author:
Henry P. McKean
Journal:
Trans. Amer. Math. Soc. 82 (1956), 519-548
MSC:
Primary 35.0X
DOI:
https://doi.org/10.1090/S0002-9947-1956-0087012-3
MathSciNet review:
0087012
Full-text PDF Free Access
References | Similar Articles | Additional Information
- [1] William Feller, On second order differential operators, Ann. of Math. (2) 61 (1955), 90–105. MR 68082, https://doi.org/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. MR 65809, https://doi.org/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). MR 1511560, https://doi.org/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. MR 1451877
- [5] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Oxford, at the Clarendon Press, 1946 (German). MR 0019765
- [6] Kunihiko Kodaira, The eigenvalue problem for ordinary differential equations of the second order and Heisenberg’s theory of 𝑆-matrices, Amer. J. Math. 71 (1949), 921–945. MR 33421, https://doi.org/10.2307/2372377
- [7] Joanne Elliott, Eigenfunction expansions associated with singular differential operators, Trans. Amer. Math. Soc. 78 (1955), 406–425. MR 68701, https://doi.org/10.1090/S0002-9947-1955-0068701-2
- [8] Einar Hille, The abstract Cauchy problem and Cauchy’s problem for parabolic differential equations, J. Analyse Math. 3 (1954), 81–196. MR 62934, https://doi.org/10.1007/BF02803587
- [9] W. Feller, Zur Theorie der stochastischen Prozesse (Existenz und Eindeutigkeitssätze), Math. Ann. vol. 113 (1936) pp. 113-160.
- [10] Norman Levinson, A simplified proof of the expansion theorem for singular second order linear differential equations, Duke Math. J. 18 (1951), 57–71. MR 41313
- [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. MR 44705
- [12] Kôsaku Yosida, On Titchmarsh-Kodaira’s formula concerning Weyl-Stone’s eigenfunction expansion, Nagoya Math. J. 1 (1950), 49–58. MR 42016
- [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. MR 58074
- [14] Samuel Karlin and James McGregor, Representation of a class of stochastic processes, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 387–391. MR 71665, https://doi.org/10.1073/pnas.41.6.387
- [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. MR 60103, https://doi.org/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. MR 47886, https://doi.org/10.2307/1969644
- [17] Einar Hille, Functional Analysis and Semi-Groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, New York, 1948. MR 0025077
- [18] R. Courant and D. Hilbert, Methoden der mathematischen Physik, vol. 1, Berlin, Springer, 1931.
- [19] David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923
- [20] A. Kolmogoroff, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann. 104 (1931), no. 1, 415–458 (German). MR 1512678, https://doi.org/10.1007/BF01457949
- [21] A. Khinchin, Asymptotische Gesetze der Wahrscheinlichkeitsrechnung, Ergebnisse der Mathematik u. ihrer Grenzgebiete, vol. 2 (4), New York, Chelsea, 1948.
Retrieve articles in Transactions of the American Mathematical Society with MSC: 35.0X
Retrieve articles in all journals with MSC: 35.0X
Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1956-0087012-3
Article copyright:
© Copyright 1956
American Mathematical Society