Elementary solutions for certain parabolic partial differential equations

McKean, Henry P.

doi:10.1090/S0002-9947-1956-0087012-3

Elementary solutions for certain parabolic partial differential equations
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by Henry P. McKean

Trans. Amer. Math. Soc. 82 (1956), 519-548

DOI: https://doi.org/10.1090/S0002-9947-1956-0087012-3

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References

William Feller, On second order differential operators, Ann. of Math. (2) 61 (1955), 90–105. MR 68082, DOI 10.2307/1969621
William Feller, The general diffusion operator and positivity preserving semi-groups in one dimension, Ann. of Math. (2) 60 (1954), 417–436. MR 65809, DOI 10.2307/1969842
Hermann Weyl, Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Ann. 68 (1910), no. 2, 220–269 (German). MR 1511560, DOI 10.1007/BF01474161
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, DOI 10.1090/coll/015
E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Oxford, at the Clarendon Press, 1946 (German). MR 0019765
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, DOI 10.2307/2372377
Joanne Elliott, Eigenfunction expansions associated with singular differential operators, Trans. Amer. Math. Soc. 78 (1955), 406–425. MR 68701, DOI 10.1090/S0002-9947-1955-0068701-2
Einar Hille, The abstract Cauchy problem and Cauchy’s problem for parabolic differential equations, J. Analyse Math. 3 (1954), 81–196. MR 62934, DOI 10.1007/BF02803587

Zur Theorie der stochastischen Prozesse

Existenz und Eindeutigkeitssätze

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
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
Kôsaku Yosida, On Titchmarsh-Kodaira’s formula concerning Weyl-Stone’s eigenfunction expansion, Nagoya Math. J. 1 (1950), 49–58. MR 42016
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
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, DOI 10.1073/pnas.41.6.387
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, DOI 10.1098/rsta.1954.0001
William Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (2) 55 (1952), 468–519. MR 47886, DOI 10.2307/1969644
Einar Hille, Functional Analysis and Semi-Groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, New York, 1948. MR 0025077

Methoden der mathematischen Physik

David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, vol. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923
A. Kolmogoroff, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann. 104 (1931), no. 1, 415–458 (German). MR 1512678, DOI 10.1007/BF01457949

Asymptotische Gesetze der Wahrscheinlichkeitsrechnung