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Diffusion semigroups on abstract Wiener space


Author: M. Ann Piech
Journal: Trans. Amer. Math. Soc. 166 (1972), 411-430
MSC: Primary 47D05; Secondary 28A40
DOI: https://doi.org/10.1090/S0002-9947-1972-0295141-3
MathSciNet review: 0295141
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Abstract: The existence of a semigroup of solution operators associated with a second order infinite dimensional parabolic equation of the form $ \partial u/\partial t = {L_x}u$ was previously established by the author. The present paper investigates the relationship between $ {L_x}$ and the infinitesimal generator $ \mathcal{U}$ of the semigroup. In particular, it is shown that $ \mathcal{U}$ is the closure of $ {L_x}$ in a natural sense. This in turn implies certain uniqueness results for both the semigroup and for solutions of the parabolic equation.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0295141-3
Keywords: Abstract Wiener space, diffusion semigroups, infinitesimal generator, parabolic equations, Wiener process, uniqueness of semigroup, uniqueness of solutions
Article copyright: © Copyright 1972 American Mathematical Society

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