114 Lectures on Dynamics of Stochastic Systems
Differentiating Eq. (4.76) with respect to t, we obtain the integro-differential equation
d
dt
8
a
[t;v(τ )] = iae
−2νt
v(t) − a
2
v(t)
t
Z
0
dt
1
e
−2ν(t−t
1
)
v(t
1
)8
a
[t
1
;v(τ )]. (4.77)
2. Assume now amplitude a is the random quantity with probability density p(a). To obtain
the characteristic functional of process z(t) in this case, we should average Eq. (4.77) with
respect to random amplitude a. In the general case, such averaging cannot be performed
analytically. Analytical averaging of Eq. (4.77) appears to be possible only if probability
density of random amplitude a has the form
p(a) =
1
2
[
δ
(
a −a
0
)
+ δ
(
a +a
0
)
]
, (4.78)
with
h
a
i
= 0 and
a
2
= a
2
0
(in fact, this very case is what is called usually telegrapher’s
process). As a result, we obtain the integro-differential equation
d
dt
8[t;v(τ )] = −a
2
0
v(t)
t
Z
0
dt
1
e
−2ν(t−t
1
)
v(t
1
)8[t
1
;v(τ )]. (4.79)
Now, we dwell on an important limiting theorem concerning telegrapher’s random pro-
cesses.
Consider the random process
ξ
N
(t) = z
1
(t) + ··· + z
N
(t),
where all z
k
(t) are statistically independent telegrapher’s processes with zero-valued means
and correlation functions
h
z(t)z(t + τ )
i
=
σ
2
N
e
−α|τ |
.
In the limit N → ∞, we obtain that process ξ(t) = lim
N→∞
ξ
N
(t) is the Gaussian random
process with the exponential correlation function
h
ξ(t)ξ(t + τ )
i
= σ
2
e
−α|τ |
,
i.e., the Gaussian Markovian process. Thus, process ξ
N
(t) for finite N is the finite-number-
of-states process approximating the Gaussian Markovian process.
Generalized Telegrapher’s Random Process
Consider now generalized telegrapher’s process defined by the formula
z(t) = a
n(0,t)
. (4.80)
Here, n(0, t) is the sequence of integers described above and quantities a
k
are assumed
statistically independent with distribution function p(a). Figure 4.4 shows a possible
realization of such a process.