Klyatskin V.I. Lectures on Dynamics of Stochastic Systems
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126 Lectures on Dynamics of Stochastic Systems
Replacing differentiation with respect to η(τ ) by differentiation with respect to z(τ )
and setting η(τ ) = 0, we obtain the equality
h
z(t)R[z(τ )]
i
=
∞
Z
−∞
dτ
1
B(t, τ
1
)
δ
δz(τ
1
)
R[z(τ )]
(5.12)
commonly known in physics as the Furutsu–Novikov formula [1, 46].
One can easily obtain the multi-dimensional extension of Eq. (5.12); it can be writ-
ten in the form
z
i
1
,...,i
n
(r)R[z]
=
Z
dr
0
z
i
1
,...,i
n
(r)z
j
1
,...,j
n
(r
0
)
δR[z]
δz
j
1
,...,j
n
(r
0
)
, (5.13)
where r stands for all continuous arguments of random vector field z(r) and i
1
, . . . , i
n
are the discrete (index) arguments. Repeated index arguments in the right-hand side
of Eq. (5.13) assume summation.
Formulas (5.12) and (5.13) extend formulas (4.17), (4.24) to the Gaussian random
processes.
If random process z(τ ) is defined only on time interval [0, t], then functional
2[t, v(τ )] assumes the form
2[t, v(τ )] = −
1
2
t
Z
0
t
Z
0
dτ
1
dτ
2
B(τ
1
, τ
2
)v(τ
1
)v(τ
2
), (5.14)
and functionals [t
0
, t; v(τ )] and [t, t; v(τ )] are the linear functionals
[t
0
, t; v(τ )] =
δ
iδv(t
0
)
2[t, v(τ )] = i
t
Z
0
dτ B(t
0
, τ )v(τ ),
[t, t; v(τ )] =
d
iv(t)dt
2[t, v(τ )] = i
t
Z
0
dτ B(t, τ )v(τ ).
(5.15)
As a consequence, Eqs. (5.6), (5.8) assume the form
z(t
0
)R[t, z(τ )]
=
t
Z
0
dτ B(t
0
, τ )
δR[z(τ )]
δz(τ )
(t
0
6 t) (5.16)
that coincides with Eq. (5.12) if the condition
δR[t; z(τ )]
δz(τ )
= 0 for τ < 0, τ > t (5.17)
holds.
Correlation Splitting 127
5.3 Poisson’s Process
The Poisson process is defined by Eq. (4.66), page 111, and its characteristic functional
is given by Eq. (4.67), page 111. In this case, formulas (5.7) and (5.9) for functionals
[t
0
, t; v(τ )] and [t, t; v(τ )] assume the forms
[t
0
, t; v(τ )] =
δ
iδv(t
0
)
2[t, v(τ )]
= −i
t
0
Z
0
dτ g(t
0
− τ )
˙
W
t
Z
τ
dτ
1
v(τ
1
)g(τ
1
− τ )
,
[t, t; v(τ )] =
d
iv(t)dt
2[t, v(τ )]
= −i
t
Z
0
dτ g(t − τ)
˙
W
t
Z
τ
dτ
1
v(τ
1
)g(τ
1
− τ )
, (5.18)
where
˙
W(v) =
dW(v)
dv
= i
Z
∞
−∞
dξξ p(ξ )e
iξv
.
Changing the integration order, we can rewrite equalities (5.18) in the form
[t
0
, t; v(τ )]
=
∞
Z
−∞
dξξ p(ξ )
t
0
Z
0
dτ g(t
0
− τ ) exp
iξ
t
Z
τ
dτ
1
v(τ
1
)g(τ
1
− τ )
(t
0
6 t). (5.19)
As a result, we obtain that correlations of Poisson’s random process z(t) with func-
tionals of this process are described by the expression
z(t
0
)R[t; z(τ )]
= ν
∞
Z
−∞
dξξ p(ξ )
t
0
Z
0
dτ
0
g(t
0
− τ
0
)
R[t; z(τ ) + ξ g(τ − τ
0
)]
(t
0
6 t). (5.20)
As we mentioned earlier, random process n(0, t) describing the number of jumps
during temporal interval (0, t) is the special case of Poisson’s process. In this case,
p(ξ) = δ(ξ − 1) and g(t) = θ (t), so that Eq. (5.20) assumes the extra-simple form
h
n(0, t)R[t; n(0, τ)]
i
= ν
t
0
Z
0
dτ
R[t; n(0, τ ) + θ(t
0
− τ )]
(t
0
6 t). (5.21)
Equality (5.21) extends formula (4.25) for Poisson’s random quantities to Poisson’s
random processes.
128 Lectures on Dynamics of Stochastic Systems
5.4 Telegrapher’s Random Process
Now, we dwell on telegrapher’s random process defined by formula (4.73), page 112,
−z(t) = a(−1)
n(0,t)
, where a is the deterministic quantity.
As we mentioned earlier, calculation of correlator
h
z(t)R[t; z(τ )]
i
for τ ≤ t assumes
the knowledge of the characteristic functional of process z(t); unfortunately, the cha-
racteristic functional is unavailable in this case. We know only Eqs. (4.77) and (4.79),
page 114, that describe the relationship between the functional
9[t; v(τ )] =
1
iv(t)
d
dt
8[t; v(τ )] =
*
z(t) exp
i
t
Z
0
dτ z(τ )v(τ )
+
and the characteristic functional 8[t; v(τ )] in the form
*
z(t) exp
i
t
Z
0
dτ z(τ )v(τ )
+
= ae
−2νt
+ ia
2
t
Z
0
dt
1
e
−2ν(t−t
1
)
v(t
1
)
*
exp
i
t
1
Z
0
dτ z(τ )v(τ )
+
.
This relationship is sufficient to split correlator
h
z(t)R[t; z(τ )]
i
, i.e., to express it
immediately in terms of the average functional
h
R[t; z(τ )]
i
.
Indeed, the equality
h
z(t)R[t; z(τ ) + η(τ)]
i
=
*
z(t) exp
t
Z
0
dτ z(τ )
δ
δη(τ )
+
R[t; η(τ )]
= aR[t; η(τ )]e
−2νt
+ a
2
t
Z
0
dt
1
e
−2ν(t−t
1
)
δ
δη(t
1
)
h
R[t; z(τ )θ(t
1
− τ ) + η(τ )]
i
,
(5.22)
where η(t) is arbitrary deterministic function, holds for any functional R[t; z(τ )] for
τ ≤ t. The final result is obtained by the limit process η → 0:
h
z(t)R[t; z(τ )]
i
= aR[t; 0]e
−2νt
+ a
2
t
Z
0
dt
1
e
−2ν(t−t
1
)
δ
δz(t
1
)
e
R[t, t
1
; z(τ )]
, (5.23)
Correlation Splitting 129
where functional
e
R[t, t
1
; z(τ )] is defined by the formula
e
R[t, t
1
; z(τ )] = R[t; z(τ )θ(t
1
− τ + 0)]. (5.24)
In the case of random quantity a distributed according to probability density distri-
bution (4.78), page 114, additional averaging (5.23) over a results in the equality
h
z(t)R[t; z(τ )]
i
= a
2
0
t
Z
0
dt
1
e
−2ν(t−t
1
)
δ
δz(t
1
)
e
R[t, t
1
; z(τ )]
. (5.25)
Formula (5.25) is very similar to the formula for splitting the correlator of the
Gaussian process z(t) characterized by the exponential correlation function (i.e., the
Gaussian Markovian process) with functional R[t; z(τ )]. The difference consists in
the fact that the right-hand side of Eq. (5.25) depends on the functional that is cut by
the process z(τ ) rather than on functional R[t; z(τ )] itself.
Let us differentiate Eq. (5.23) with respect to time t. Taking into account the fact
that there is no need in differentiating with respect to the upper limit of the integral in
the right-hand side of Eq. (5.23), we obtain the expression [47]
d
dt
+ 2ν
h
z(t)R[t; z(τ )]
i
=
z(t)
d
dt
R[t; z(τ )]
(5.26)
usually called the differentiation formula.
One essential point should be noticed. Functional R[t; z(τ)] in differentiation for-
mula (5.26) is an arbitrary functional and can simply coincide with process z(t − 0).
In the general case, the realization of telegrapher’s process is the generalized func-
tion. The derivative of this process is also the generalized function (the sequence of
delta-functions), so that
z(t)
d
dt
z(t) 6=
1
2
d
dt
z
2
(t) ≡ 0
in the general case. These generalized functions, as any generalized functions, are
defined only in terms of functionals constructed on them. In the case of our inter-
est, such functionals are the average quantities denoted by angle brackets
h
· · ·
i
, and
the above differentiation formula describes the differential constraint between dif-
ferent functionals related to random process z(t) and its one-sided derivatives for
t → t − 0, such as dz/dt, d
2
z/dt
2
. For example, formula (5.26) allows derivation of
equalities, such as
z(t)
d
dτ
z(τ )
τ =t−0
= 2ν
D
z
2
E
,
z(t)
d
2
dτ
2
z(τ )
τ =t−0
= 4ν
2
D
z
2
E
.
It is clear that these formulas can be obtained immediately by differentiating the corre-
lation function
z(t)z(t
0
)
with respect to t
0
(t
0
< t) followed by limit process t
0
→ t−0.
130 Lectures on Dynamics of Stochastic Systems
Now, we dwell on some extensions of the above formulas for the vector Markovian
process z(t) = {z
1
(t), . . . , z
N
(t)}. For example, all above Markovian processes having
the exponential correlation function
h
z(t)z(t + τ)
i
=
D
z
2
E
e
−α|τ |
(5.27)
satisfy the equality
∂
∂t
+ αk
h
z
1
(t) . . . z
k
(t)R[t; z(τ )]
i
=
z
1
(t) . . . z
k
(t)
∂
∂t
R[t; z(τ )]
. (5.28)
Formula (5.28) defines the rule of factoring the operation of differentiating with res-
pect to time out of angle brackets of averaging; in particular, we have
z
1
(t) . . . z
k
(t)
∂
n
∂t
n
R[t; z(τ )]
=
∂
∂t
+ αk
n
h
z
1
(t) . . . z
k
(t)R[t; z(τ )]
i
, (5.29)
where k = 1, . . . , N.
5.5 Delta-Correlated Random Processes
Random processes z(t) that can be treated as delta-correlated processes are of special
interest in physics. The importance of this approximation follows first of all from
the fact that it is physically obvious in the context of many physical problems and
the corresponding dynamic systems allow obtaining closed equations for probability
densities of the solutions to these systems.
For the Gaussian delta-correlated (in time) process, correlation function has the
form
B(t
1
, t
2
) =
h
z(t
1
)z(t
2
)
i
= B(t
1
)δ(t
1
− t
2
), (
h
z(t)
i
= 0).
In this case, functionals 2[t; v(τ )], [t
0
, t; v(τ )] and [t, t; v(τ )] introduced by
Eqs. (5.14), (5.15) are
2[t; v(τ )] = −
1
2
t
Z
0
dτ B(τ )v
2
(τ ),
[t
0
, t; v(τ )] = iB(t
0
)v(t
0
), [t, t; v(τ )] =
i
2
B(t)v(t),
and Eqs. (5.16) assume significantly simpler forms
z(t
0
)R[t; v(τ )]
= B(t
0
)
δ
δz(t
0
)
R[t; v(τ )]
(0 < t
0
< t),
h
z(t)R[t; v(τ )]
i
=
1
2
B(t)
δ
δz(t)
R[t; v(τ )]
.
(5.30)
Correlation Splitting 131
Formulas (5.30) show that statistical averages of the Gaussian delta-correlated pro-
cess considered here are discontinuous at t
0
= t. This discontinuity is dictated entirely
by the fact that this process is delta-correlated; if the process at hand is not delta-
correlated, no discontinuity occurs (see Eq. (5.16)).
Poisson’s delta-correlated random process corresponds to the limit process
g(t) → δ(t).
In this case, the logarithm of the characteristic functional has the simple form (see
Eq. (4.71)); as a consequence, Eqs. (5.19) for functionals [t
0
, t; v(τ )] and [t, t; v(τ )]
assume the forms
[t
0
, t; v(τ )] = i
∞
Z
−∞
dξξ p(ξ )e
iξv(t
0
)
(t
0
< t),
[t, t; v(τ )] =
ν
iv(t)
∞
Z
−∞
dξξ p(ξ )
h
e
iξv(t)
− 1
i
= ν
∞
Z
−∞
dξp(ξ )
ξ
Z
0
dηe
iηv(t)
,
and we obtain the following expression for the correlation of Poisson random process
z(t) with a functional of this process
z(t
0
)R[t; z(τ )]
= ν
∞
Z
−∞
dξξ p(ξ )
R[t; z(τ ) + ξ δ(τ − τ
0
)]
(t
0
6 t),
h
z(t)R[t; z(τ )]
i
= ν
∞
Z
−∞
dξp(ξ )
ξ
Z
0
dη
h
R[t; z(τ ) + ηδ(t − τ )]
i
.
(5.31)
These expressions also show that statistical averages are discontinuous at t
0
= t. As in
the case of the Gaussian process, this discontinuity is dictated entirely by the fact that
this process is delta-correlated.
In the general case of delta-correlated process z(t), we can expand the logarithm of
characteristic functional in the functional Taylor series
2[t; v(τ )] =
∞
X
n=1
i
n
n!
t
Z
0
dτ K
n
(τ )v
n
(τ ), (5.32)
where cumulant functions assume the form
K
n
(t
1
, . . . , t
n
) = K
n
(t
1
)δ(t
1
− t
2
) . . . δ(t
n−1
− t
n
).
132 Lectures on Dynamics of Stochastic Systems
As can be seen from Eq. (5.32), a characteristic feature of these processes consists in
the validity of the equality
˙
2[t; v(τ )] =
˙
2[t; v(t)]
˙
2[t; v(τ )] =
d
dt
2[t; v(τ )]
, (5.33)
which is of fundamental significance. This equality shows that, in the case of arbitrary
delta-correlated process, quantity 2[t; v(τ )] appears not a functional, but simply a
function of time t. In this case, functionals [t
0
, t; v(τ )] and [t, t; v(τ )] are
[t
0
, t; v(τ )] =
∞
X
n=0
i
n
n!
K
n+1
(t
0
)v
n
(t
0
)
t
0
< t
,
[t, t; v(τ )] =
∞
X
n=0
i
n
(n + 1)!
K
n+1
(t)v
n
(t),
and formulas for correlation splitting assume the forms
z(t
0
)R[t; z(τ )]
=
∞
X
n=0
i
n
n!
K
n+1
(t
0
)
δ
n
R[t; z(τ )]
δz
n
(t
0
)
t
0
< t
,
h
z(t)R[t; z(τ )]
i
=
∞
X
n=0
i
n
(n + 1)!
K
n+1
(t)
δ
n
R[t; z(τ )]
δz
n
(t)
.
(5.34)
These formulas describe the discontinuity of statistical averages at t
0
= t in the general
case of delta-correlated processes.
Note that, for t
0
> t, delta-correlated processes satisfy the obvious equality
z(t
0
)R[t; z(τ )]
=
z(t
0
)
h
R[t; z(τ )]
i
. (5.35)
Now, we dwell on the concept of random fields delta-correlated in time.
We will deal with vector field f (x, t), where x describes the spatial coordinates and
t is the temporal coordinate. In this case, the logarithm of characteristic functional
can be expanded in the Taylor series with coefficients expressed in terms of cumulant
functions of random field f (x, t) (see Secttion 4.2). In the special case of cumulant
functions
K
i
1
,...,i
m
n
(x
1
, t
1
; . . . , x
n
, t
n
)
= K
i
1
,...,i
m
n
(x
1
, . . . , x
n
; t
1
)δ(t
1
− t
2
) . . . δ(t
n−1
− t
n
), (5.36)
Correlation Splitting 133
we will call field f (x, t) the random field delta-correlated in time t. In this case,
functional 2[t; ψ(x
0
, τ )] assumes the form
2[t; ψ(x
0
, τ )] =
∞
X
n=1
i
n
n!
t
Z
0
dτ
Z
. . .
Z
dx
1
. . . dx
n
K
i
1
,...,i
m
n
(x
1
, . . . , x
n
; τ )
× ψ
i
1
(x
1
, τ ) . . . ψ
im
(x
n
, τ ), (5.37)
An important feature of this functional is the fact that it satisfies the equality similar
to Eq. (5.33):
˙
2[t; ψ(x
0
, τ )] =
˙
2[t; ψ(x
0
, t)]. (5.38)
5.5.1 Asymptotic Meaning of Delta-Correlated Processes and Fields
The nature knows no delta-correlated processes. All actual processes and fields are
characterized by a finite temporal correlation radius, and delta-correlated processes
and fields result from asymptotic expansions in terms of their temporal correlation
radii.
We illustrate the appearance of delta-correlated processes using the stationary
Gaussian process with correlation radius τ
0
as an example. In this case, the logarithm
of the characteristic functional is described by the expression
2[t; v(τ )] = −
t
Z
0
dτ
1
v(τ
1
)
τ
1
Z
0
dτ
2
B
τ
1
− τ
2
τ
0
v(τ
2
). (5.39)
Setting τ
1
− τ
2
= ξτ
0
, we transform Eq. (5.39) to the form
2[t; v(τ )] = −τ
0
t
Z
0
dτ
1
v(τ
1
)
τ
1
/τ
0
Z
0
dξB
(
ξ
)
v(τ
1
− ξ τ
0
).
Assume now that τ
0
→ 0. In this case, the leading term of the asymptotic expansion
in parameter τ
0
is given by the formula
2[t; v(τ )] = −τ
0
∞
Z
0
dξB
(
ξ
)
t
Z
0
dτ
1
v
2
(τ
1
)
that can be represented in the form
2[t; v(τ )] = −B
eff
t
Z
0
dτ
1
v
2
(τ
1
), (5.40)
134 Lectures on Dynamics of Stochastic Systems
where
B
eff
=
∞
Z
0
dτ B
τ
τ
0
=
1
2
∞
Z
−∞
dτ B
τ
τ
0
. (5.41)
Certainly, asymptotic expression (5.40) holds only for the functions v(t) that vary
during times about τ
0
only slightly, rather than for arbitrary functions v(t). Indeed, if
we specify this function as v(t) = vδ(t − t
0
), asymptotic expression (5.40) appears
invalid; in this case, we must replace Eq. (5.39) with the expression
2[t; v(τ )] = −
1
2
B(0)v
2
(t > t
0
)
corresponding to the characteristic function of process z(t) at a fixed time t = t
0
.
Consider now correlation
h
z(t)R[t; z(τ )]
i
given, according to the Furutsu–Novikov
formula (5.16), page 126, by the expression
h
z(t)R[t; z(τ )]
i
=
t
Z
0
dt
1
B
t − t
1
τ
0
δ
δz(t
1
)
R[t; z(τ )]
.
The change of integration variable t −t
1
→ ξτ
0
transforms this expression to the form
h
z(t)R[t; z(τ )]
i
= τ
0
t/τ
0
Z
0
dξB
(
ξ
)
δ
δz(t − ξ τ
0
)
R[t; z(τ )]
, (5.42)
which grades for τ
0
→ 0 into the equality obtained earlier for the Gaussian delta-
correlated process
h
z(t)R[t; z(τ )]
i
= B
eff
δ
δz(t)
R[t; z(τ )]
provided that the variational derivative in Eq. (5.42) varies only slightly during times
of about τ
0
.
Thus, the approximation of process z(t) by the delta-correlated one is conditioned
by small variations of functionals of this process during times approximating the pro-
cess temporal correlation radius.
Consider now telegrapher’s and generalized telegrapher’s processes. In the case of
telegrapher’s process, the characteristic functional satisfies Eq. (4.79), page 114. The
correlation radius of this process is τ
0
= 1/2ν, and, for ν → ∞ (τ
0
→ 0), this
equation grades for sufficiently smooth functions v(t) into the equation
d
dt
8[t; v(τ )] = −
a
2
0
2ν
v
2
(t)8[t; v(τ )], (5.43)
Correlation Splitting 135
which corresponds to the Gaussian delta-correlated process. If we additionally assume
that a
2
0
→ ∞ and
lim
ν→∞
a
2
0
/2ν = σ
2
0
,
then Eq. (5.43) appears independent of parameter ν. Of course, this fact does not mean
that telegrapher’s process loses its telegrapher’s properties for ν → ∞. Indeed, for
ν → ∞, the one-time probability distribution of process z(t) will as before correspond
to telegrapher’s process, i.e., to the process with two possible states. As regards the
correlation function and higher-order moment functions, they will possess for ν → ∞
all properties of delta-functions in view of the fact that
lim
ν→∞
n
2νe
−2ν|τ |
o
=
0, if τ 6= 0,
∞, if τ = 0.
Such functions should be considered the generalized functions; their delta-functional
behavior will manifest itself in the integrals of them. As Eq. (5.43) shows, limit pro-
cess ν → ∞ is equivalent for these quantities to the replacement of process z(t)
by the Gaussian delta-correlated process. This situation is completely similar to the
approximation of the Gaussian random process with a finite correlation radius τ
0
by
the delta-correlated process for τ
0
→ 0.
Problems
Problem 5.1
Split the correlator
h
g(ξ)f (ξ )
i
ξ
, where ξ is the random quantity.
Solution
h
g(ξ + η
1
)f (ξ + η
2
)
i
ξ
= exp
2
d
idη
1
+
d
idη
2
− 2
d
idη
1
− 2
d
idη
2
h
g(ξ + η
1
)
i h
f (ξ + η
2
)
i
.
Problem 5.2
Split the correlator
e
ωξ
f (ξ )
ξ
, where parameter ω can assume complex values.
Solution
e
ωξ
f (ξ + η)
ξ
= exp
2
1
i
ω +
d
dη
− 2
d
idη
h
f (ξ + η)
i
ξ
.