Master equation for Ornstein-Uhlenbeck process

[cjolley]

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 $y_{1}$ and $y_{2}$ separated by a time t:

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

and the probability distribution

$P_{1}(y_{1}) = (2π)^{-1/2}e^{-y_{1}^{2}/2}$

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

A little bit later, he defines the master equation by expanding T$_{t}(y_{2}|y_{1})$ in powers of t and defining the coefficient of the linear term as $W(y_{2}|y_{1})$, 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$_{t}(y_{2}|y_{1})$→δ$(y_{2}-y_{1})$, since T$_{t}(y_{2}|y_{1})$ 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

 

[mathman]

For small t, 1-e^-2t ≈ 2t. This gives a Gaussian with variance ≈ 2t, which -> delta function as t -> 0.

 

[cjolley]

Exactly. But that still doesn't tell me how to get a small-t expansion for $y_{1} ≠ y_{2}$, which is what I need for a meaningful transition rate.

 

[vanhees71]

The point is that $y$ 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.

 

[mathman]

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