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 ¼6pZRv, 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

¼6pZRv 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

¼3pZR

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 6¼ 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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