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 | References | Similar Articles | Additional Information

Abstract: Let be an abstract Wiener space. If is a twice -differentiable function on such that and is of trace class, then we define , where is the Laplacian and denotes the - pairing. The closure of 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) (initial value problem) and (2) . 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.

**[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**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,
-differentiation,
number operator,
fundamental solution

Article copyright:
© Copyright 1981
American Mathematical Society