Klyatskin V.I. Lectures on Dynamics of Stochastic Systems
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136 Lectures on Dynamics of Stochastic Systems
Problem 5.3
Split the correlator
e
ωξ
ξ
and
ξe
ωξ
ξ
assuming that ξ is the Gaussian random
quantity with zero-valued mean.
Solution
e
ωξ
ξ
= exp
ω
2
σ
2
2
,
ξe
ωξ
ξ
= ωσ
2
exp
ω
2
σ
2
2
.
Problem 5.4
Split the correlator
e
ωξ
f (ξ )
ξ
assuming that ξ is the Gaussian random quantity
with zero-valued mean.
Solution
e
ωξ
f (ξ )
ξ
= exp
ω
2
σ
2
2
D
f (z + ωσ
2
)
E
.
Problem 5.5
Split the correlator
h
F[z(τ )]R[z(τ )]
i
.
Solution
h
F[z(τ ) + η
1
(τ )]R[z(τ ) + η
2
(τ )]
i
=
exp
2
1
i
δ
δη
1
(τ )
+
δ
δη
2
(τ )
− 2
1
i
δ
δη
1
(τ )
− 2
1
i
δ
δη
2
(τ )
×
h
F[z(τ ) + η
1
(τ )]
i h
R[z(τ ) + η
2
(τ )]
i
,
where η
1
(τ ) and η
2
(τ ) are arbitrary deterministic functions.
Problem 5.6
Split the correlator
h
F[z(τ )]R[z(τ )]
i
assuming that z(τ ) is the Gaussian
random process.
Solution
h
F[z(τ ) + η
1
(τ )]R[z(τ ) + η
2
(τ )]
i
= exp
∞
Z
−∞
∞
Z
−∞
dτ
1
dτ
2
B(τ
1
, τ
2
)
δ
δη
1
(τ
1
)
δ
δη
2
(τ
2
)
h
F[z(τ ) + η
1
(τ )]
i h
R[z(τ ) + η
2
(τ )]
i
,
Correlation Splitting 137
or
h
F[z(τ )]R[z(τ ) + η(τ )]
i
= F
z(τ ) +
∞
Z
−∞
dτ
1
B(τ, τ
1
)
δ
δη(τ
1
)
h
R[z(τ ) + η(τ)]
i
.
Problem 5.7
Split the correlator
D
e
i
R
∞
−∞
dτ z(τ )v(τ )
R[z(τ )]
E
assuming that z(τ ) is the Gaussian
random process.
Solution
D
e
i
R
∞
−∞
dτ z(τ )v(τ )
R[z(τ )]
E
= 8[v(τ )]
*
R
z(τ ) + i
∞
Z
−∞
dτ
1
B(τ, τ
1
)v(τ
1
)
+
.
Problem 5.8
Split the correlator
D
e
i
R
t
0
dτ z(τ )v(τ )
R[t; z(τ )]
E
assuming that z(τ ) is the Gaussian
random process.
Solution
D
e
i
R
t
0
dτ z(τ )v(τ )
R[t; z(τ )]
E
= 8[t; v(τ )]
*
R
t; z(τ ) + i
t
Z
0
dτ
1
B(τ, τ
1
)v(τ
1
)
+
,
where 8[t; v(τ )] is the characteristic functional of the Gaussian random
process z(t).
Problem 5.9
Calculate the characteristic functional of process z(t) = ξ
2
(t), where ξ(t) is the
Gaussian random process with the parameters
h
ξ(t)
i
= 0,
h
ξ(t
1
)ξ(t
2
)
i
= B(t
1
− t
2
).
Instruction The characteristic functional
8[t; v(τ )] =
h
ϕ[t; ξ(τ )]
i
, where ϕ[t; ξ(τ )] = exp
i
t
Z
0
dτ v(τ )ξ
2
(τ )
138 Lectures on Dynamics of Stochastic Systems
satisfies the stochastic equation
d
dt
8[t; v(τ )] = iv(t)
D
ξ
2
(t)ϕ[t; ξ(τ )]
E
.
One needs to obtain the integral equation for quantity
9(t
1
, t) =
h
ξ(t
1
)ξ(t)ϕ[t; ξ(τ )]
i
,
9(t
1
, t) = B(t
1
− t)8[t; v(τ )] + 2i
t
Z
0
dτ B(t
1
− τ )v(τ )9(τ, t),
and represent function 9(t
1
, t) in the form
9(t
1
, t) = S(t
1
, t)8[t; v(τ )].
Solution The characteristic functional 8[t; v(τ )] has the form
8[t; v(τ )] = exp
i
t
Z
0
dτ v(τ )S(τ, τ )
,
where
S(t, t) =
∞
X
n=0
S
(n)
(t, t),
S
(n)
(t, t) = (2i)
n
t
Z
0
dτ
1
. . .
t
Z
0
dτ
n
v(τ
1
) . . . v(τ
n
)B(t − τ
1
)B(τ
1
− τ
2
) . . . B(τ
n
− t).
Consequently, the n-th cumulant of process z(t) = ξ
2
(t) is expressed in terms of
quantity S
(n−1)
(t, t).
Problem 5.10
Calculate the characteristic functional of process z(t) = ξ
2
(t), where ξ(t) is the
Gaussian Markovian process with the parameters
h
ξ(t)
i
= 0,
h
ξ(t
1
)ξ(t
2
)
i
= B(t
1
− t
2
) = σ
2
exp{−α|t
1
− t
2
|}.
Solution The characteristic functional 8[t; v(τ )] has the form
8[t; v(τ )] = exp
i
t
Z
0
dτ v(τ )S(τ, τ )
,
Correlation Splitting 139
where function S(t, t) can be described in terms of the initial-value problem
d
dt
S(t, t) = −2α
h
S(t, t) − σ
2
i
+ 2iv(t)S
2
(t, t), S(t, t)
|
t=0
= σ
2
.
Problem 5.11
Show that the expression
h
z(t
1
)z(t
2
)R[z(τ )]
i
=
h
z(t
1
)z(t
2
)
i h
R[z(τ )]
i
,
holds for arbitrary functional R[z(τ )] and τ ≤ t
2
≤ t
1
under the condition that
z(t) is telegrapher’s process [48].
Instruction Expand functional R[z(τ )] in the Taylor series in z(τ ) and use
formulas Eq. (4.75).
Problem 5.12
Show that the expression [48]
h
F[z(τ
1
)]z(t
1
)z(t
2
)R[z(τ
2
)]
i
=
h
F[z(τ
1
)]
i h
z(t
1
)z(t
2
)
i h
R[z(τ
2
)]
i
+
h
F[z(τ
1
)]z(t
1
)
i h
z(t
2
)R[z(τ
2
)]
i
,
holds for any τ
1
≤ t
1
≤ t
2
≤ τ
2
and arbitrary functionals F[z(τ
1
)] and R[z(τ
2
)]
under the condition that z(t) is telegrapher’s process with random parameter a
distributed according to the probability density
p(a) =
1
2
[
δ(a − a
0
) + δ(a + a
0
)
]
.
Instruction In terms of the Taylor expansion in z(τ ), functional R[z(τ
2
)] can be
represented in the form
R[z(τ
2
)] =
X
2k
+
X
2k+1
,
where the first sum consists of terms with the even number of co-factors z(τ ) and
the second sum consists of terms with the odd number of such co-factors. The
desired result immediately follows from the fact that
h
R[z(τ
2
)]
i
=
*
X
2k
+
,
h
z(t
2
)R[z(τ
2
)]
i
=
*
z(t
2
)
X
2k+1
+
.
140 Lectures on Dynamics of Stochastic Systems
Problem 5.13
Show that the equality
z(t
0
)R[t; z(τ )]
= e
−2ν(t
0
−t)
h
z(t)R[t; z(τ )]
i
(t
0
> t)
holds for telegrapher’s process z(t).
Instruction Rewrite the correlator
z(t
0
)R[t; z(τ )]
(t
0
≥ t, τ < t) in the form
z(t
0
)R[t; z(τ )]
=
1
a
2
0
z(t
0
)z(t)z(t)R[t; z(τ )]
.
Lecture 6
General Approaches to Analyzing
Stochastic Systems
In this Lecture, we will consider basic methods of determining statistical characteris-
tics of solutions to the stochastic equations.
Consider a linear (differential, integro-differential, or integral) stochastic equation.
Averaging of such an equation over an ensemble of realizations of fluctuating para-
meters will not result generally in a closed equation for the corresponding average
value. To obtain a closed equation, we must deal with an additional extended space
whose dimension appears to be infinite in most cases. This approach makes it possible
to derive the linear equation for average quantity of interest, but this equation will
contain variational derivatives.
Consider some special types of dynamic systems.
6.1 Ordinary Differential Equations
Assume dynamics of vector function x(t) is described by the ordinary differential
equation
d
dt
x(t) = v(x, t) + f (x, t), x(t
0
) = x
0
. (6.1)
Here, function v(x, t) is the deterministic function and f (x, t) is the random function.
The solution to Eq. (6.1) is a functional of f( y, τ ) + v( y, τ ) with τ ∈ (t
0
, t), i.e.,
x(t) = x[t; f( y, τ ) + v( y, τ )].
From this fact follows the equality
δ
δf
j
(y, τ )
F
(
x(t)
)
=
δ
δv
j
( y, τ)
F
(
x(t)
)
=
∂F
(
x(t)
)
∂x
i
δx
i
(t)
δf
j
( y, τ)
valid for arbitrary function F(x). In addition, we have
δ
δf
j
( y,t − 0)
x
i
(t) =
δ
δv
j
(y,t − 0)
x
i
(t) = δ
ij
δ
(
x(t)−y
)
.
Lectures on Dynamics of Stochastic Systems. DOI: 10.1016/B978-0-12-384966-3.00006-1
Copyright
c
2011 Elsevier Inc. All rights reserved.
142 Lectures on Dynamics of Stochastic Systems
The corresponding Liouville equation for the indicator function
ϕ(x, t) = δ(x(t) − x)
follows from Eq. (6.1) and has the form
∂
∂t
ϕ(x, t) = −
∂
∂x
{
[
v(x, t) + f (x, t)
]
ϕ(x, t)
}
, ϕ(x, t
0
) = δ(x − x
0
), (6.2)
from which follows the equality
δ
δf ( y, t − 0)
ϕ(x, t) =
δ
δv( y,t − 0)
ϕ(x, t) = −
∂
∂x
{
δ(x − y)ϕ(x, t)
}
. (6.3)
Using this equality, we can rewrite Eq. (6.2) in the form, which may look at first sight
more complicated
∂
∂t
+
∂
∂x
v(x, t)
ϕ(x, t) =
Z
dyf( y, t)
δ
δv( y, t)
ϕ(x, t). (6.4)
Consider now the one-time probability density for solution x(t) of Eq. (6.1)
P(x, t) =
h
ϕ(x, t)
i
=
h
δ(x(t) − x)
i
.
Here, x(t) is the solution of Eq. (6.1) corresponding to a particular realization of ran-
dom field f(x, t), and angle brackets
h
· · ·
i
denote averaging over an ensemble of reali-
zations of field f(x, t).
Averaging Eq. (6.4) over an ensemble of realizations of field f(x, t), we obtain the
expression
∂
∂t
+
∂
∂x
v(x, t)
P(x, t) =
Z
dy
δ
δv( y, t)
h
f( y, t)ϕ(x, t)
i
. (6.5)
Quantity
h
f ( y, t)ϕ(x, t)
i
in the right-hand side of Eq. (6.5) is the correlator of random
field f (y, t) with function ϕ(x, t), which is a functional of random field f( y, τ ) and is
given either by Eq. (6.2), or by Eq. (6.4).
The characteristic functional
8[t, t
0
; u( y, τ )] =
*
exp
i
t
Z
t
0
dτ
Z
dyf( y, τ )u( y, τ )
+
= exp
{
2[t, t
0
; u( y, τ )]
}
exhaustively describes all statistical characteristics of random field f (y, τ ) for τ ∈
(t
0
, t).
General Approaches to Analyzing Stochastic Systems 143
We split correlator
h
f ( y, t)ϕ(x, t)
i
using the technique of functionals. Introduc-
ing functional shift operator with respect to field v( y, τ), we represent functional
ϕ[x, t; f(y, τ ) + v( y, τ )] in the operator form
ϕ[x, t; f( y, τ ) + v( y, τ )]
= exp
t
Z
t
0
dτ
Z
dyf( y, τ )
δ
δv( y, τ )
ϕ[x, t; v( y, τ )].
With this representation, the term in the right-hand side of Eq. (6.5) assumes the form
Z
dy
δ
δv
j
( y, t)
*
f
j
( y, t) exp
t
Z
t
0
dτ
R
dy
0
f ( y
0
, τ )
δ
δv( y
0
, τ )
+
*
exp
t
Z
t
0
dτ
R
dy
0
f ( y
0
, τ )
δ
δv( y
0
, τ )
+
P(x, t)
=
˙
2
t
t, t
0
;
δ
iδv( y, τ )
P(x, t),
where we introduced the functional
˙
2
t
[t, t
0
; u( y, τ )] =
d
dt
ln 8[t, t
0
; u( y, τ )].
Consequently, we can rewrite Eq. (6.5) in the form
∂
∂t
+
∂
∂x
v(x, t)
P(x, t) =
˙
2
t
t, t
0
;
δ
iδv( y, τ )
P(x, t). (6.6)
Equation (6.6) is the closed equation with variational derivatives in the functional
space of all possible functions {v(y, τ )}. However, for a fixed function v(x, t), we
arrive at the unclosed equation
∂
∂t
+
∂
∂x
v(x, t)
P(x, t) =
˙
2
t
t, t
0
;
δ
iδf ( y, t)
ϕ[x, t; f( y, τ )]
,
P(x, t
0
) = δ(x − x
0
).
(6.7)
Equation (6.7) is the exact consequence of the initial dynamic equation (6.1).
Statistical characteristics of random field f(x, t) appear in this equation only through
functional
˙
2
t
[t, t
0
; u( y, τ )] whose expansion in the functional Taylor series in powers
of u(x, τ ) depends on all space-time cumulant functions of random field f (x, t).
144 Lectures on Dynamics of Stochastic Systems
As we mentioned earlier, Eq. (6.7) is unclosed in the general case with respect to
function P(x, t), because quantity
˙
2
t
t, t
0
;
δ
iδf ( y, τ )
δ(x(t) − x)
appeared in averaging brackets depends on the solution x(t) (which is a functional of
random field f ( y, τ )) for all times t
0
< τ < t. However, in some cases, the variational
derivative in Eq. (6.7) can be expressed in terms of ordinary differential operators. In
such conditions, equations like Eq. (6.7) will be the closed equations in the correspon-
ding probability densities. The corresponding examples will be given below.
Proceeding in a similar way, one can obtain the equation similar to Eq. (6.7) for the
m-time probability density that refers to m different instants t
1
< t
2
< · · · < t
m
P
m
(x
1
, t
1
; . . . ; x
m
, t
m
) =
h
ϕ
m
(x
1
, t
1
; . . . ; x
m
, t
m
)
i
, (6.8)
where the indicator function is defined by the equality
ϕ
m
(x
1
, t
1
; . . . ; x
m
, t
m
) = δ(x(t
1
) − x
1
) . . . δ(x(t
m
) − x
m
).
Differentiating Eq. (6.8) with respect to time t
m
and using then dynamic equa-
tion (6.1), one can obtain the equation
∂
∂t
m
+
∂
∂x
m
v(x
m
,t
m
)
P
m
(x
1
, t
1
; . . . ; x
m
, t
m
)
=
˙
2
t
m
t
m
, t
0
;
δ
iδf ( y, τ )
ϕ
m
(x
1
, t
1
; . . . ; x
m
, t
m
)
.
(6.9)
No summation over index m is performed here. The initial value to Eq. (6.9) can be
derived from Eq. (6.8). Setting t
m
= t
m−1
in Eq. (6.8), we obtain the equality
P
m
(x
1
, t
1
; . . . ; x
m
, t
m−1
) = δ(x
m
− x
m−1
)P
m−1
(x
1
, t
1
; . . . ; x
m−1
, t
m−1
),
which just determines the initial value for Eq. (6.9).
6.2 Completely Solvable Stochastic Dynamic Systems
Consider now several dynamic systems that allow sufficiently adequate statistical ana-
lysis for arbitrary random parameters.
6.2.1 Ordinary Differential Equations
Multiplicative Action
As the first example, we consider the vector stochastic equation with initial value
d
dt
x(t) = z(t)g(t)F(x), x(0) = x
0
, (6.10)
General Approaches to Analyzing Stochastic Systems 145
where g(t) and F
i
(x), i = 1, . . . , N, are the deterministic functions and z(t) is the
random process whose statistical characteristics are described by the characteristic
functional
8[t; v(τ )] =
*
exp
i
t
Z
0
dτ z(τ )v(τ )
+
= e
2[t;v(τ )]
.
Equation (6.10) has a feature that offers a possibility of determining statistical cha-
racteristics of its solutions in the general case of arbitrarily distributed process z(t).
The point is that introduction of new ‘random’ time
T =
t
Z
0
dτ z(τ )g(τ )
reduces Eq. (6.10) to the equation looking formally deterministic
d
dT
x(T) = F(x), x(0) = x
0
,
so that the solution to Eq. (6.10) has the following structure
x(t) = x(T) = x
t
Z
0
dτ z(τ )g(τ )
. (6.11)
Varying Eq. (6.11) with respect to z(τ) and using Eq. (6.10), we obtain the equality
δ
δz(τ )
x(t) = g(τ )
d
dT
x(T) = g(τ )F(x(t)). (6.12)
Thus, the variational derivative of solution x(t) is expressed in terms of the same
solution at the same time. This fact makes it possible to immediately write the closed
equations for statistical characteristics of problem (6.10).
Let us derive the equation for the one-time probability density
P(x, t) =
h
δ(x(t) − x)
i
.
It has the form
∂
∂t
P(x, t) =
˙
2
t
t;
δ
iδz(τ )
δ
(
x(t)−x
)
. (6.13)
Consider now the result of operator δ/δz(τ ) applied to the indicator function
ϕ(x, t) = δ(x(t) − x). Using formula (6.12), we obtain the expression
δ
δz(τ )
δ
(
x(t)−x
)
= −g(τ )
∂
∂x
{
F(x)ϕ(x, t)
}
.