The notions of the generic Wiener process and the Brownian motion
are sometimes used interchangeably, though there are some fine
differences in their definitions [2, 3]. I shall denote the Wiener process
with W(t) and reserve this term for the standard version (4.1.16), as it
is often done in the literature.
The Brownian motion is the classical topic of statistical physics.
Different approaches for introducing this process are described in the
next section.
4.2 BROWNIAN MOTION
In mathematical statistics, the notion of the Brownian motion is
used for describing the generic stochastic process. Yet, this term
referred originally to Brown’s observation of random motion of
pollen in water. Random particle motion in fluid can be described
using different theoretical approaches. Einstein’s original theory of
the Brownian motion implicitly employs both the Chapman-Kolmo-
gorov equation and the Fokker-Planck equation [1]. However, choos-
ing either one of these theories as the starting point can lead to the
diffusion equation. Langevin offered another simple method for de-
riving the Fokker-Planck equation. He considered one-dimensional
motion of a spherical particle of mass m and radius R that is subjected
to two forces. The first force is the viscous drag force described by the
Stokes formula, F ¼6pZRv, where Z is viscosity and v ¼
dr
dt
is the
particle velocity. Another force, Z, describes collisions of the water
molecules with the particle and therefore has a random nature. The
Langevin equation of the particle motion is
m
dv
dt
¼6pZRv þ Z (4:2:1)
Let us multiply both sides of equation (4.2.1) by r. Since
r
dv
dt
¼
d
dt
(rv) v
2
and rv ¼
1
2
d
dt
(r
2
), then
1
2
m
d
2
dt
2
(r
2
) m
dr
dt

2
¼3pZR
d
dt
(r
2
) þ Zr (4:2:2)
Note that the mean kinetic energy of a spherical particle, E[
1
2
mv
2
],
equals
3
2
kT. Since E[Zr] ¼ 0 due to the random nature of Z, averaging
of equation (4.2.2) yields
32 Stochastic Processes
m
d
2
dt
2
E[r
2
] þ 6pZR
d
dt
E[r
2
] ¼ 6kT (4:2:3)
The solution to equation (4.2.3) is
d
dt
E[r
2
] ¼ kT=(pZR) þ C exp (6pZRt=m) (4:2:4)
where C is an integration constant. The second term in equation
(4.2.4) decays exponentially and can be neglected in the asymptotic
solution. Then
E[r
2
] r
2
0
¼ [kT=(pZR)]t (4:2:5)
where r
0
is the particle position at t ¼ 0. It follows from the compari-
son of equations (4.2.5) and (4.1.15) that D ¼ kT=(pZR).
1
The Brownian motion can be also derived as the continuous limit
for the discrete random walk (see, e.g., [3]). First, let us introduce the
process e(t) that is named the white noise and satisfies the following
conditions
E[e(t)] ¼ 0; E[e
2
(t)] ¼ s
2
;E[e(t) e(s)] ¼ 0, if t s: (4:2:6)
Hence, the white noise has zero mean and constant variance s
2
. The
last condition in (4.2.6) implies that there is no linear correlation
between different observations of the white noise. Such a model repre-
sents an independently and identically distributed process (IID) and is
sometimes denoted IID(0, s
2
). The IID process can still have non-
linear correlations (see Section 5.3). The normal distribution N(0, s
2
)
is the special case of the white noise. First, consider a simple discrete
process
y(k) ¼ y(k 1) þ e(k) (4:2:7)
where the white noise innovations can take only two values
2
e(k) ¼
D, with probability p, p ¼ const < 1
D, with probability (1 p)
(4:2:8)
Now, let us introduce the continuous process y
n
(t) within the time
interval t 2 [0, T], such that
y
n
(t) ¼ y([t=h]) ¼ y([nt=T]), t 2 [0, T] (4:2:9)
Stochastic Processes 33

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