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Chapman-Enskog-Hilbert expansion for the Ornstein-Uhlenbeck process and the approximation of Brownian motion


Author: Richard S. Ellis
Journal: Trans. Amer. Math. Soc. 199 (1974), 65-74
MSC: Primary 60J60
DOI: https://doi.org/10.1090/S0002-9947-1974-0353469-4
MathSciNet review: 0353469
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Abstract: Let $ (x(t),\upsilon (t))$ denote the joint Ornstein-Uhlenbeck position-velocity process. Special solutions of the backward equation of this process are studied by a technique used in statistical mechanics. This leads to a new proof of the fact that, as $ \varepsilon \downarrow 0,\varepsilon x(t/{\varepsilon ^2})$ tends weakly to Brownian motion. The same problem is then considered for $ \upsilon (t)$ belonging to a large class of diffusion processes.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0353469-4
Keywords: Chapman-Enskog-Hilbert expansion, Ornstein-Uhlenbeck process, Brownian motion, essentially selfadjoint operator, weak solution of a partial differential equation, diffusion process
Article copyright: © Copyright 1974 American Mathematical Society

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