Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
MathSciNet review: 0353469
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] R. Courant and D. Hilbert, Methoden der mathematischen Physik. Vol. I, Springer, Berlin, 1931; English transl., Interscience, New York, 1953. MR 16, 426.
  • [2] R. Ellis, Chapman-Enskog-Hilbert expansion for a Markovian model of the Boltzmann equation, Comm. Pure Appl. Math. 26 (1973), 327-359. MR 0469010 (57:8811)
  • [3] -, Limit theorems for random evolutions with explicit error estimates, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 28 (1974), 249-256. MR 0368110 (51:4352)
  • [4] G. Hellwig, Differential operators of mathematical physics. An introduction, Springer-Verlag, Berlin and New York, 1964; English transl., Addison-Wesley, Reading, Mass., 1967. MR 29 #2682; 35 #2174. MR 0211292 (35:2174)
  • [5] H. P. McKean, Chapman-Enskog-Hilbert expansion for a class of solutions of the telegraph equation, J. Mathematical Phys. 8 (1967), 547-552. MR 35 #1981. MR 0211099 (35:1981)
  • [6] E. Nelson, Dynamical theories of Brownian motion, Princeton Univ. Press, Princeton, N.J., 1967. MR 35 #5001. MR 0214150 (35:5001)
  • [7] -, The adjoint Markov process, Duke Math. J. 25 (1958), 671-690. MR 21 #365. MR 0101555 (21:365)
  • [8] S. R. S. Varadhan, Stochastic processes, Notes based on a course given at New York University during the year 1967/68, Courant Inst. of Math. Sci., New York University, New York, 1968. MR 41 #4657. MR 0260028 (41:4657)
  • [9] K. Yosida, Functional analysis, 2nd ed., Die Grundlehren der math. Wissenschaften, Band 123, Academic Press, New York; Springer-Verlag, Berlin, 1968. MR 39 #741. MR 0239384 (39:741)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60J60

Retrieve articles in all journals with MSC: 60J60

Additional Information

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

American Mathematical Society