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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0632538-9
PII: S 0002-9947(1981)0632538-9
Keywords: Abstract Wiener space, Wiener measure, $ H$-differentiation, number operator, fundamental solution
Article copyright: © Copyright 1981 American Mathematical Society