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A fundamental solution of the parabolic equation on Hilbert space. II. The semigroup property


Author: M. Ann Piech
Journal: Trans. Amer. Math. Soc. 150 (1970), 257-286
MSC: Primary 47.50; Secondary 35.00
DOI: https://doi.org/10.1090/S0002-9947-1970-0278116-8
MathSciNet review: 0278116
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Abstract: The existence of a family of solution operators $ \{ {q_t}:t > 0\} $ corresponding to a fundamental solution of a second order infinite-dimensional differential equation of the form $ \partial u/\partial t = Lu$ was previously established by the author. In the present paper, it is established that these operators are nonnegative, and satisfy the condition $ {q_s}{q_t} = {q_{s + t}}$.


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  • [1] Z. Ciesielski, On Haar functions and on the Schauder basis of the space $ {C_{ < 0,1 > }}$, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 7 (1959), 227-232. MR 24 #A1599. MR 0131751 (24:A1599)
  • [2] F. G. Dressel, The fundamental solution of the parabolic equation, Duke Math. J. 7 (1940), 186-203. MR 2, 204. MR 0003340 (2:204b)
  • [3] -, The fundamental solution of the parabolic equation. II, Duke Math. J. 13 (1946), 61-70. MR 7, 450. MR 0015637 (7:450c)
  • [4] E. B. Dynkin, Markov processes, Fizmatgiz, Moscow, 1963; English transl., Vols. I, II, Academic Press, New York and Springer-Verlag, Berlin and New York, 1965. MR 33 #1886; 1887. MR 0193670 (33:1886)
  • [5] A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Englewood Cliffs, N. J. 1964. MR 31 #6062. MR 0181836 (31:6062)
  • [6] L. Gross, Integration and nonlinear transformations in Hilbert space, Trans. Amer. Math. Soc. 94 (1960), 404-440. MR 22 #2883. MR 0112025 (22:2883)
  • [7] -, Measurable functions on Hilbert space, Trans. Amer. Math. Soc. 105 (1962), 372-390. MR 26 #5121. MR 0147606 (26:5121)
  • [8] -, Abstract Wiener spaces, Proc. Fifth Berkeley Sympos. Math. Statist. and Prob. (Berkey, Calif., 1965/66), vol. II, part I, Univ. of California Press, Berkeley, 1967, pp. 31-42. MR 35 #3027. MR 0212152 (35:3027)
  • [9] -, Potential theory on Hilbert space, J. Functional Analysis 1 (1967), 123-181. MR 37 #3331. MR 0227747 (37:3331)
  • [10] A. M. Il'in, A. S. Kalašnikov and O. A. Oleĭnik, Second-order linear equations of parabolic type, Uspehi Mat. Nauk 17 (1962), no. 3 (105), 3-146 = Russian Math. Surveys 17 (1962), no. 3, 1-143. MR 25 #2328. MR 0138888 (25:2328)
  • [11] S. Itô, The fundamental solution of the parabolic equation in a differentiable manifold, Osaka Math. J. 5 (1953), 75-92. MR 15, 36. MR 0056178 (15:36c)
  • [12] M. A. Piech, A fundamental solution of the parabolic equation on Hilbert space, J. Functional Analysis 3 (1969), 85-114. MR 0251588 (40:4815)
  • [13] -, Regularity properties for families of measures on a metric space, Proc. Amer. Math. Soc. 24 (1970), 307-311. MR 0260962 (41:5582)
  • [14] R. Schatten, A theory of cross-spaces, Ann. of Math. Studies, no. 26, Princeton Univ. Press, Princeton, N. J., 1950. MR 12, 186. MR 0036935 (12:186e)
  • [15] B. Sz.-Nagy, Introduction to real functions and orthogonal expansions, Oxford Univ. Press, New York, 1965. MR 31 #5938. MR 0181711 (31:5938)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1970-0278116-8
Keywords: Hilbert space, abstract Wiener space, Wiener measure, parabolic equation, fundamental solution, semigroups of operators
Article copyright: © Copyright 1970 American Mathematical Society

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