2'4 Brownian Motion: The OrnsteinUhlenbeck Process

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2.4 Brownian Motion The Ornstein-Uhlenbeck
Process If the Wiener process is used to model
Brownian motion, i.e., is chosen as a stochastic
process to represent the position of the Brownian
particle, then the instantaneous velocity is not
defined in this model. It is infinite, since the
sample paths of WI are nowhere differentiable.
This can be avoided by considering the velocity
of the Brownian particle instead as the main
random quantity as done by Uhlenbeck and Ornstein
2.7. This stochastic process is therefore
known as the Ornstein-Uhlenbeck process. The
position is then obtained by integration and not
given any more by the Wiener process. The
starting point for this model of Brownian motion
is the decomposition of the force acting on the
suspended particle into a systematic part, the
friction and into a random part due to the
perpetual kicks of the surrounding fluid
molecules
2.112
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Since the average effect of the fluid molecules
is already taken into account in the friction
term, it is plausible to assume that Ft has mean
value zero, As discussed above, Ft is the sum
of many independent infinitesimally small
contributions, We assume it therefore to be
Gaussian distributed in the light of the central
limit theorem. Furthermore, since Ft is due to
the numerous collisions of the light fluid
molecules with the much heavier Brownian
particles, it is plausible to suppose that Ft
varies on a much faster time scale than v t
Dividing (2.1 t 2) by m, we write
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If we suppose that has a short but
nonvanishing memory, (2.113) can be interpreted
as an ensemble of ordinary differential equations
for the sample paths and thus be solved
realization wise. This point will be commented
upon in much more detail in Chap. 8. In any case
a naive approach to (2.113) is not likely to be
dangerous since it is a linear problem. The
solution of (2.113) is given by
2.114
A linear operator transforms a Gaussian process
into a Gaussian process. Thus the integral over a
Gaussian process is Gaussian 2.1, p. 161 f..
This implies that vt will be a Gaussian process
if the initial condition Vo is a Gaussian random
variable independent of Ft (or a constant).
Remember, Gaussian processes are completely
characterized by the mean value function and the
correlation function. We have
Obviously, in agreement with the physics of the
problem, should be a stationary process, i.e.,
2.117
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Since furthermore varies on a much faster
time scale than vt the function should be sharply
peaked around zero and decrease rapidly to zero
for 'With we obtain for the covariance of the
velocity
If then vt is (at least) a wide sense stationary
Gaussian process with
and
In fact we shall now define the
Ornstein-Uhlenbeck process XI as the
process given by the following hierarchy of
probability densities
2.120
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where
2.121
The so-defined Ornstein-Uhlenbeck process Xt
solves (2.113) if is a Gaussian white noise.
At this point, we do not yet have available all
the necessary mathematical ingredients to
establish this fact. We shall come back to this
problem in Chap. 5. By definition, the
Ornstein-Uhlenbeck process (O-U-process) is like
the Wiener process a Gaussian process. Also it
shares with the Wiener process the property that
it has almost surely continuous sample paths.
Since we shall obtain this result in Chap. 4 as a
byproduct in a more general context, we shall not
go to any lengths now to prove it using
Kolmogorov's criterion.
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The Ornstein- Uhlenbeck process differs from the
Wiener process by two important features
2.124
This expression is in general nonzero. Thus the
changes in the velocity of the Brownian particle
are correlated, and a fortiori stochastically
dependent, over nonoverlapping time intervals.
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Let us now consider the integrated
Ornstein-Uhlenbeck process
2.125
which represents the position of the Brownian
particle, started at time zero at the origin Yo
O. (The integration is understood
realizationwise, i.e.,
a.s
It is well defined since is, with
probability one, a continuous function of s).
Obviously we have
For the correlation function we obtain
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As already mentioned above, the integral over a
Gaussian process is again a Gaussian process.
Thus Yt is completely characterized by its mean
(2.126) and its correlation function (2.127). We
see that the Wiener process as a model
for Brownian motion in position space is
recovered in the limit Such that
( 1 for the standard Wiener processes).
The Wiener process is thus an adequate
description of Brownian motion in the high
friction and high noise intensity limit.
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2.5 The Poisson Process
Wiener and Ornstein-Uhlenbeck processes are two
examples of processes that model physical
quantities, which vary continuously, and have
thus fittingly almost surely continuous
realizations. Let us now turn our attention to
the other extrerpe, i.e., quantities that vary
only by discrete steps. The basic example for I.
this class of processes and in a certain sense
the discrete counterpart to the , Wiener process
is the Poisson process. It measures the number of
times a speciified event occurs during the time
period from 0 to t. Thus each sample function is
a nondecreasing step function, i.e., the Poisson
process does not have almost surely continuous
sample paths. Obviously, by definition, all
sample paths start at zero at time t 0
The Poisson process VI is used to model the
disintegration of radioactive particles, the
number of chromosome breakages under harmful
irradiation, the number of incoming telephone
calls, the arrival of customers for service, etc.
In all these examples, the number of events
occurring in an interval rs, t does not depend,
to a good approximation, on how many events
occurred in a preceding nonoverlapping interval
ru, v. The Poisson process has, as does its
continuous counterpart the Wiener process WI,
independent incremen,ts. To be precise,
its hierarchy of probabilities is given by
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2.130
j. The structure of this hierarchy coincides, as
a simple glance at (2.99,100) reveals completely
with that of the Wiener process, except that the
Gauss distributior has been replaced by the
Poisson distribution. The independent increments
of the Poisson process are also stationary,
whereas in complete analogy to the Wiener
process, the Poisson process itself is not even
stationary in the wide sense, since
As follows from (2.129) and the fact that the
increments are independent, the probability that
during the interval (t, t h) one or more jumps
have taken place is
This obviously tends to zero, if h tends to zero.
In other words, the probability that at a
specified instant of time t the Poisson process
undergoes at least one jump is zero. CF (2.91),
the Poisson process is almost surely continuous.
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(As is easily seen (2.89, 90) are also
fulfilled.) Let us, however, stress again that in
contrast to the Wiener process the Poisson
process does not have almost surely continuous
sample paths. A fortiori, the sample paths of the
Poisson process are not almost surely
differentiable functions.
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