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Fundamental solutions for differential equations associated with the number operator


Author: Yuh Jia Lee
Journal: Trans. Amer. Math. Soc. 268 (1981), 467-476
MSC: Primary 35R15; Secondary 28C20, 46G99, 58D20, 60J99
DOI: https://doi.org/10.1090/S0002-9947-1981-0632538-9
Correction: Trans. Amer. Math. Soc. 276 (1983), 621-624.
MathSciNet review: 632538
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Abstract: Let $ (H,\,B)$ be an abstract Wiener space. If $ u$ is a twice $ H$-differentiable function on $ B$ such that $ Du(x) \in {B^{\ast}}$ and $ {D^2}u(x)$ is of trace class, then we define $ \mathfrak{N}u(x) = - \Delta u(x) + (x,\,Du(x))$, where $ \Delta u(x) = {\operatorname{trace} _H}\,{D^2}u(x)$ is the Laplacian and $ ( \cdot ,\, \cdot )$ denotes the $ B$- $ {B^{\ast}}$ pairing. The closure $ \overline{\mathfrak{N}} $ of $ \mathfrak{N}$ is known as the number operator. In this paper, we investigate the existence, uniqueness and regularity of solutions for the following two types of equations: (1) $ {u_t} = - \mathfrak{N}u$ (initial value problem) and (2) $ {\mathfrak{N}^k}u = f(k \geqslant 1)$. We show that the fundamental solutions of (1) and (2) exist in the sense of measures and we represent their solutions by integrals with respect to these measures.


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  • [1] L. Gross, Abstract Wiener spaces, Proc. 5th Berkeley Sympos. Math. Statist. Prob., vol. 2, Univ. California Press, Berkeley, 1965, pp. 31-42. MR 0212152 (35:3027)
  • [2] -, Potential theory on Hilbert space, J. Funct. Anal. 1 (1967), 123-181. MR 0227747 (37:3331)
  • [3] H.-H. Kuo, Integration by parts for abstract Wiener measures, Duke Math. J. 41 (1974), 373-379. MR 0341085 (49:5835)
  • [4] -, Gaussian measure in Banach spaces, Lecture Notes in Math., vol. 463, Springer-Verlag, Berlin and New York, 1975. MR 0461643 (57:1628)
  • [5] -, Distribution theory on Banach spaces, Probability in Banach Spaces, Lecture Notes in Math., vol. 576, Springer-Verlag, Berlin and New York, 1976. MR 0453961 (56:12213)
  • [6] -, Potential theory associated with Uhlenbeck-Ornstein process, J. Funct. Anal. 21 (1976), 63-75. MR 0391285 (52:12106)
  • [7] Y.-J. Lee, Applications of the Fourier-Wiener transform to differential equations on infinite dimensional spaces. I, Trans. Amer. Math. Soc. 262 (1980), 259-283. MR 583855 (82k:35098)
  • [8] -, Parabolic equations on infinite dimensional spaces, Bull. Inst. Math. Acad. Sinica (to appear). MR 625721 (84d:35143)
  • [9] E. Nelson, Probability theory and Euclidean field theory, Lecture Notes in Physics, vol. 25, Springer-Verlag, Berlin and New York, 1973. MR 0395513 (52:16310)
  • [10] M. A. Piech, A fundamental solution of the parabolic equation on Hilbert space, J. Funct. Anal. 3 (1969), 85-114. MR 0251588 (40:4815)
  • [11] -, Parabolic equations associated with the number operator, Trans. Amer. Math. Soc. 194 (1974), 213-222. MR 0350231 (50:2724)
  • [12] -, The Ornstein-Uhlenbeck semigroup in an infinite dimensional $ {L_2}$ setting, J. Funct. Anal. 18 (1975), 271-285. MR 0381014 (52:1911)
  • [13] I. E. Segal, Tensor algebras over Hilbert spaces, Trans. Amer. Math. Soc. 81 (1956), 106-134. MR 0076317 (17:880d)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0632538-9
Keywords: Abstract Wiener space, Wiener measure, $ H$-differentiation, number operator, fundamental solution
Article copyright: © Copyright 1981 American Mathematical Society

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