Long-time average of field measured by an ornstein-uhlenbeck wanderer

For a given field V(x) on \\mathbbRd, we shall find (i) the exponent ν such that XT[ω]=T-ν∫0TV(ω(t))dt converges as T->∞ to have a non-trivial probability distribution, and (ii) the distribution itself, where ω(t) denotes the d-dimensional Ornstein-Uhlenbeck process starting from ω(0)=x at t=0. T...

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Bibliographic Details
Published in:Journal of the Physical Society of Japan 2006-09, Vol.75 (9), p.94001
Main Authors: EZAWA, Hiroshi, NAKAMURA, Tom, WATANABE, Keiji
Format: Article
Language:English
Subjects:
Exact sciences and technology
Markov processes
Mathematical methods in physics
Physics
Probability theory, stochastic processes, and statistics
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Summary:For a given field V(x) on \\mathbbRd, we shall find (i) the exponent ν such that XT[ω]=T-ν∫0TV(ω(t))dt converges as T->∞ to have a non-trivial probability distribution, and (ii) the distribution itself, where ω(t) denotes the d-dimensional Ornstein-Uhlenbeck process starting from ω(0)=x at t=0. The exponent and the probability distribution are different depending on whether or not a weighted average ∫\\mathbbRdV(x)exp[-βx2/2D]ddx of V vanishes, where D and β are constants of the Ornstein-Uhlenbeck process. In the non-vanishing case, they are ν=1 and a Dirac delta distribution, while in the vanishing case, ν=1/2 and the Gaussian distribution. The results have applications to the physics of chemoreception in immune systems.
ISSN:0031-9015
1347-4073
DOI:10.1143/jpsj.75.094001