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Master equation for Ornstein-Uhlenbeck process

  1. May 31, 2012 #1
    Hi everybody...

    I've been working through N.G. Van Kampen's "Stochastic Processes in Physics and Chemistry" and have run into something that has got me sort of stumped. He defines the Ornstein-Uhlenbeck process (a stationary, Gaussian, Markovian random process) in terms of the transition probability between two values [itex]y_{1}[/itex] and [itex]y_{2}[/itex] separated by a time t:

    [itex]T_{t}(y_{2}|y_{1}) = (2π(1-e^{-2t})^{1/2}\exp(\frac{(y_{2}-y_{1}e^{-t})^{2}}{2(1-e^{-2t})})[/itex]

    and the probability distribution

    [itex]P_{1}(y_{1}) = (2π)^{-1/2}e^{-y_{1}^{2}/2}[/itex]

    (This is pg. 83 if you have the book.)

    A little bit later, he defines the master equation by expanding T[itex]_{t}(y_{2}|y_{1})[/itex] in powers of t and defining the coefficient of the linear term as [itex]W(y_{2}|y_{1})[/itex], the transition probability per unit time (pg. 96 if you have the book).

    The thing that's got me stuck here is that, as t→0, T[itex]_{t}(y_{2}|y_{1})[/itex]→δ[itex](y_{2}-y_{1})[/itex], since T[itex]_{t}(y_{2}|y_{1})[/itex] is a Gaussian. I can't seem to come up with a reasonable way to expand this in terms of small t, which makes it difficult to define the master equation.

    So... does anyone know how the Ornstein-Uhlenbeck process is formulated as a master equation? Any ideas would be greatly appreciated.

    --craig
     
  2. jcsd
  3. Jun 1, 2012 #2

    mathman

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    For small t, 1-e-2t ≈ 2t. This gives a Gaussian with variance ≈ 2t, which -> delta function as t -> 0.
     
  4. Jun 1, 2012 #3
    Exactly. But that still doesn't tell me how to get a small-t expansion for [itex]y_{1} ≠ y_{2}[/itex], which is what I need for a meaningful transition rate.
     
  5. Jun 2, 2012 #4

    vanhees71

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    The point is that [itex]y[/itex] is a random variable. I don't know that book, but what he finally should get at, I guess, is that the Fokker-Planck equation for the time evolution of the distribution function is equivalent to a Langevin equation, which is an ordinary stochastic differential equation with Gaussian-distributed (white) noise.

    For a simple derivation (however for the relativistic case) see

    http://fias.uni-frankfurt.de/~hees/publ/hq-qgp4-bibtex.pdf

    p. 41ff.
     
  6. Jun 2, 2012 #5

    mathman

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    e-t≈1-t, so y1-y2e-t≈y1-y2-ty2. The net result for the integrand as t -> 0 is δ( y1-y2)
     
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