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]**Leonard Gross,*Abstract Wiener spaces*, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 31–42. MR**0212152****[2]**Leonard Gross,*Potential theory on Hilbert space*, J. Functional Analysis**1**(1967), 123–181. MR**0227747****[3]**Hui Hsiung Kuo,*Integration by parts for abstract Wiener measures*, Duke Math. J.**41**(1974), 373–379. MR**0341085****[4]**Hui Hsiung Kuo,*Gaussian measures in Banach spaces*, Lecture Notes in Mathematics, Vol. 463, Springer-Verlag, Berlin-New York, 1975. MR**0461643****[5]**Hui Hsiung Kuo,*Distribution theory on Banach space*, Probability in Banach spaces (Proc. First Internat. Conf., Oberwolfach, 1975), Springer, Berlin, 1976, pp. 143–156. Lecture Notes in Math., Vol. 526. MR**0453961****[6]**Hui Hsiung Kuo,*Potential theory associated with Uhlenbeck-Ornstein process*, J. Functional Analysis**21**(1976), no. 1, 63–75. MR**0391285****[7]**Yuh Jia Lee,*Applications of the Fourier-Wiener transform to differential equations on infinite-dimensional spaces. I*, Trans. Amer. Math. Soc.**262**(1980), no. 1, 259–283. MR**583855**, https://doi.org/10.1090/S0002-9947-1980-0583855-1**[8]**Yuh Jia Lee,*Parabolic equations on infinite-dimensional spaces*, Bull. Inst. Math. Acad. Sinica**9**(1981), no. 2, 279–292. MR**625721****[9]**G. Velo and A. Wightman (eds.),*Constructive quantum field theory*, Springer-Verlag, Berlin-New York, 1973. The 1973 “Ettore Majorana” International School of Mathematical Physics, Erice (Sicily), 26 July–5 August 1973; Lecture Notes in Physics, Vol. 25. MR**0395513****[10]**M. Ann Piech,*A fundamental solution of the parabolic equation on Hilbert space*, J. Functional Analysis**3**(1969), 85–114. MR**0251588****[11]**M. Ann Piech,*Parabolic equations associated with the number operator*, Trans. Amer. Math. Soc.**194**(1974), 213–222. MR**0350231**, https://doi.org/10.1090/S0002-9947-1974-0350231-3**[12]**M. Ann Piech,*The Ornstein-Uhlenbeck semigroup in an infinite dimensional 𝐿² setting*, J. Functional Analysis**18**(1975), 271–285. MR**0381014****[13]**I. E. Segal,*Tensor algebras over Hilbert spaces. I*, Trans. Amer. Math. Soc.**81**(1956), 106–134. MR**0076317**, https://doi.org/10.1090/S0002-9947-1956-0076317-8

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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