Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 12, pp. 1–11.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu

DE BRUIJN IDENTITIES IN DIFFERENT MARKOVIAN

CHANNELS

HASSAN EMAMIRAD, ARNAUD ROUGIREL

Communicated by Jerome A. Goldstein

Abstract. De Bruijn’s identity in information theory states that if u is the

solution of the heat equation, then the time derivative of the Shannon entropy
for this solution is equal to the amount of Fisher information at u. In this ar-

ticle, we show how this identity changes if we replace the heat channel by the

Fokker Planck, or passing from Fokker Planck to Ornstein-Uhlenbeck chan-
nels. Through these passages we investigate the different properties of these

solutions. We exclusively dissect different properties of Ornstein-Uhlenbeck

semigroup given by the Mehler formula expression.

1. Introduction

Let the probability triplet be (Ω,F , µ), where Ω is the sample space, F a σ-
algebra and µ is a probability measure. Let ϕ : Rn → R+ with

∫
Rn ϕ(x) dµ(x) = 1

be a density function of a random variable X. We extend the function x ∈ (0,∞) 7→
x lnx by 0 at x = 0 and assume that ϕ lnϕ ∈ L1(Rn, dµ). This function defines so
called Shannon’s entropy or Boltzmann H function

H(ϕ) := −
∫
Rn
ϕ(x) lnϕ(x) dµ(x).

If we place ourselves in a dynamical system at time t, this entropy will be written as
H(ϕ(·, t)). In this context the term of de Bruijn identity was pointed by Stam [5],
which was communicated to him by Prof. de Bruijn and indicate that Shannon’s
entropy decreases in time when u runs through a Gaussian channel with rate equal
to Fisher information.

In mathematical information theory, the Fisher information is a way of measuring
the amount of information that an observable random variable X carries about the
distribution that models ϕ. For example by taking X : Ω×[0,∞) 7→ R a Markovian
process and defining the density function ϕ(·, t) of the random variable valued in
R+ can be considered as a probability distribution depending on t ∈ R+. Formally,
the Fisher information is the variance of the score, which is the gradient of the log-
likelihood function which is logarithm of ϕ(·, t). This is the fundamental concept

2020 Mathematics Subject Classification. 94A17, 94A40.
Key words and phrases. De Bruijn identity; Gaussian; Ornstein-Uhlenbeck channels;

Fokker Planck; relative Fisher information; Kullback-Leibler divergence.
©2023. This work is licensed under a CC BY 4.0 license.

Submitted October 27, 2022. Published February 6, 2023.

1



2 H. EMAMIRAD, A. ROUGIREL EJDE-2023/12

in information theory as it is indicated in the seminal book of Cover and Thomas
[2].

The Fisher information is defined by the following quantity in [0,∞]

I(ϕ) =

∫
Rn

( ∂
∂x

lnϕ(x, t)
)2
ϕ(x, t) dµ(x)

=

∫
Rn

( ∂
∂x
ϕ(x, t)

)2
ϕ(x, t)−1 dµ(x).

The most well-known example in a dynamical system is the unitary heat equation

∂u

∂t
= ∆u, t > 0, x ∈ Rn, (1.1)

with an initial condition u0(·) = u(·, 0) ∈ L1
+(Rn, dµ) where dµ = dx is the standard

Lebesgue measure with ∫
Rn
u0(x)dx = 1. (1.2)

As a straightforward consequence of the explicit expression of the solution in term

of the Green function G(t, x, y) := (4πt)−n/2 exp
(
− |x−y|

2

4t

)
,

u(x, t) =

∫
Rn
u0(y)G(t, x, y) dy, (1.3)

it follows that u(x, t) is positive for any (x, t) ∈ Rn × R+. The mass conservation
implies also that ∫

Rn
u(x, t)dx =

∫
Rn
u0(x)dx = 1. (1.4)

In this case we say that u runs through the Gaussian or heat channel.
The noticeable connection between Fisher information and Shannon’s entropy is

the so-called De Bruijn relation [2] (see also [1, 4]). That is, if u runs through the
Gaussian channel, then

d

dt
H(u) = I(u) (1.5)

(For completeness, the proof is provided in the Appendix).
Before trying to deduce the De Bruijn relation for Fokker-Planck equation

∂

∂t
v(x, t) =

∂2

∂x2
(v(x, t))− ∂

∂x
(xv(x, t)),

in the next section we show how one can derive the Fokker-Planck equation from
the heat equation and in the section 3 we establish the De Bruijn relation for
Fokker-Planck equation.

In section 4 we show how the Ornstein-Uhlenbeck equation ∂w
∂t = ∂2

∂y2w(x, t) −
x ∂
∂xw(x, t) can be deduce from Fokker-Planck equation. For this equation the mass

conservation takes place in L1(Rn, dµ), where dµ is the Gaussian measure. The
section 5 is devoted to Ornstein-Uhlenbeck semigroup in which we prove the hyper-
contractivity of this semigroup which deduces the Chapman-Kolmogorov relation
for its kernel. In the section 6 we recover the De Bruijn relation for this channel.

Finally in section 7 we prove the De Bruijn identity for relative Fisher information
and Kullback-Leibler divergence which is already discussed in [7].



EJDE-2023/12 DE BRUIJN IDENTITIES 3

2. Relationship between Fokker-Planck and heat equation

In general the Fokker-Planck equation in one dimensional space reads as

∂

∂t
v(y, τ) =

∂2

∂y2
(g2(y, t)v(y, τ))− ∂

∂y
(f(y)v(y, τ)), (2.1)

where f(y, t) and g(y, τ) can be arbitrary positive functions define on Ry × R+.
In this section we take g = 1 and f(y) = −y and the following theorem gives an
explicit expression of the solution of (2.1).

Theorem 2.1. If u is the solution of the heat equation (1.1) with initial condition
u0 satisfying (1.2), then v(y, τ) = eτu(eτy, (e2τ − 1)/2), that is

v(y, τ) =
eτ√

2π(e2τ − 1)

∫
R
v0(ξ)e

− (eτ y−ξ)2

2(e2τ−1) dξ, (2.2)

satisfies the Fokker-Planck equation

∂

∂τ
v(y, τ) =

∂2

∂y2
v(y, τ) + v(y, τ) + y

∂

∂y
v(y, τ). (2.3)

Proof. First we remark that for t = (e2r − 1)/2 = 0, we have e2τ − 1 = 0, so τ
should be equal zero. Hence

u0 = v(y, 0) = v0. (2.4)

If we replace u(x, t) by its explicit expression (1.3) and we find (2.2). Now we have
to verify that this function bears out (2.3). Indeed, let us denote

A(τ) =
eτ√

2π(e2τ − 1)
, B := B(τ, y, ξ) =

(eτy − ξ)2

2(e2τ − 1)
,

I(τ, y) =

∫
R
v0(ξ)e−B(τ,y,ξ) dξ,

such that (2.2) can be expressed as

v(y, τ) = A(τ)I(τ, y).

We remark that

∂

∂τ
A(τ) =

eτ√
2π(e2τ − 1)︸ ︷︷ ︸
=A1(τ)

− e3τ√
2π(e2τ − 1)3︸ ︷︷ ︸

=A2(τ)

,

∂

∂τ
B(τ, y, ξ) =

yeτ (eτy − ξ)
e2τ − 1︸ ︷︷ ︸

=B1(τ,y,ξ)

− e2τ (eτy − ξ)2

(e2τ − 1)2︸ ︷︷ ︸
=B2(τ,y,ξ)

,

∂

∂y
B(τ, y, ξ) =

eτ (eτy − ξ)
e2τ − 1︸ ︷︷ ︸

=B3(τ,y,ξ)

,

∂

∂y
B3(τ, y, ξ) =

e2τ

e2τ − 1
.

Hence,
∂

∂y
v(y, τ) = −A(τ)

∫
R
v0(ξ)B3(τ, y, ξ)e−B(τ,y,ξ) dξ



4 H. EMAMIRAD, A. ROUGIREL EJDE-2023/12

and

∂2

∂y2
v(y, τ) = A(τ)

(
−
∫
R
v0(ξ)

e2τ

e2τ − 1
e−B(τ,y,ξ) dξ

+

∫
R
v0(ξ)(B3(τ, y, ξ))2e−B(τ,y,ξ) dξ

)
= −A2(τ)I(τ, y) +A(τ)

∫
R
u0(ξ)(B3(τ, y, ξ))2e−B(τ,y,ξ) dξ

= −A2(τ)I(τ, y) +A(τ)

∫
R
u0(ξ)B2(τ, y, ξ)e−B(τ,y,ξ) dξ.

Consequently,

∂

∂τ
v(y, τ)

= A1(τ)I(τ, y)︸ ︷︷ ︸
=v(y,τ)

−A2(τ)I(τ, y)

−A(τ)
(∫

R
u0(ξ)B1(τ, y, ξ)e−B(τ,y,ξ) dξ︸ ︷︷ ︸

=y
∂v(y,τ)
∂y

−
∫
R
v0(ξ)B2(τ, y, ξ)e−B(τ,y,ξ) dξ

)

= v(y, τ) + y
∂v(y, τ)

∂y
+

∂2

∂y2
v(y, τ).

�

3. De Bruijn identity in Fokker-Planck channels

In this section we will use the above Theorem for obtaining an identity similar
to (1.5).

Theorem 3.1. Assume that t ≥ 0, the Fisher information is defined by the positive
quantity

I(v(·, t)) =

∫
R

( ∂
∂y

ln v(y, t)
)2
v(y, t) dy .

and Shannon’s entropy is

H(v(·, t)) = −
∫
R
v(y, t) ln v(y, t)dy. (3.1)

Then v := v(y, τ) the solution of

d

dτ
v(y, τ) =

∂2

∂y2
v(y, τ) + v(y, τ) + y

∂

∂y
v(y, τ),

v(y, 0) = v0(y), with

∫
R
v0(y) dy = 1

(3.2)

will satisfy the modified De Bruijn identity in Fokker-Planck channels

d

dτ
H(v) = I(v)− 1. (3.3)

Proof. According (2.2),

lim
|y|→∞

|yv| = 0, (3.4)



EJDE-2023/12 DE BRUIJN IDENTITIES 5

and

d

dτ
H(v(·, τ)) = −

∫
R

( ∂2
∂y2

v(y, τ) +
∂

∂y
(yv(y, τ))

)
ln v(y, τ) dy

−
∫
R

( ∂2
∂y2

v(y, τ) +
∂

∂y
(yv(y, τ))

)
dy

=

∫
R

(
(
∂v

∂y
)2/v + y

∂v

∂y
(y, τ)

)
dy

= I(v)−
∫
R
v(y, τ) dy = I(v)− 1,

according to (3.4), Fokker-Planck equation has the mass conservation property and
for t = 0 the mass is equal 1, that is∫

R
v(y, τ) dy = 1 for all τ ∈ R+. (3.5)

�

4. Relationship between Fokker-Planck and Ornstein-Uhlenbeck
equation

Let v∞(y) := 1√
2π
e−y

2/2 be the unique stationary solution of Fokker-Planck

equation (2.3) with
∫
R v∞(y)dy = 1, and denote by dµ = v∞ dx the Gaussian

measure. Now, if we transform the expression (2.2) to the form

v(y, τ) =
1√

2π(1− e−2τ )

∫
R
v0(ξ)e

− (y−ξ/eτ )2

2(1−e−2τ ) dξ, (4.1)

we notice that v(y, τ)→ v∞(y) as τ →∞.

Theorem 4.1. Assume that v is the solution of Fokker-Planck equation (2.3).
Then w = v/v∞ satisfies the Ornstein-Uhlenbeck equation

∂w

∂τ
=

∂2

∂y2
w(y, τ)− y ∂

∂y
w(y, τ), (4.2)

with initial data w0(y) = u0(y)/v∞, u0 ∈ L1
+(Rn, dy) and

∫
Rn u0(y) dy = 1.

Proof. First note that by (2.4), w0 = w(y, 0) = v0(y)/v∞(y) = u0(y)/v∞(y). Con-
sequently, by (3.5) we have

∫
R w(y, τ) dµ = 1.

Now, knowing that v is the solution of (3.1) we can write

∂w

∂τ
=
(∂2v
∂y2

+ v + y
∂v

∂y

)
/v∞. (4.3)

On the other hand, by ∂v∞
∂y = −yv∞, we have

∂w

∂y
= v−1∞ (

∂v

∂y
+ yv),

∂2w

∂y2
= v−1∞

(∂2v
∂y2

+ 2y
∂v

∂y
+ v + y2v

)
.

By insert these expressions in (4.3) we obtain (4.2) . �



6 H. EMAMIRAD, A. ROUGIREL EJDE-2023/12

5. Mehler formula and Ornstein-Uhlenbeck semigroup

On Rn, let µn be the canonical Gaussian measure with density (2π)−n/2e(−|x|
2/2)

with respect to the Lebesgue measure dx. With this measure we consider the

Banach space Lp(Rn, dµn), 0 ≤ p < ∞ with the norm ‖f‖p =
( ∫

Rn |f |
p dµn

)1/p
on which we can define the Ornstein-Uhlenbeck semigroup Pt by mean of Mehler
formula

Ptf(x) :=

∫
Rn
f(e−tx+

√
1− e−2ty) dµn(y), for f ∈ Lp(Rn, dµn). (5.1)

By taking αt = e−t, βt =
√

1− e−2t and making a change of variable z = αtx+βty
in this formula we obtain

Ptf(x) = (2πβ2
t )−n/2

∫
Rn
f(z) exp

(
−|z − αtx|2/2(β2

t )
)
dz . (5.2)

This equality implies that the Gaussian measure dµn is invariant for Pt, that is∫
Rn
Ptf(x) dµn(x) =

∫
Rn
f(x) dµn(x) for all f ∈ Lp(Rn, dµn). (5.3)

To show this we need the following lemma.

Lemma 5.1. For c1, c2 ≥ 0, c1 + c2 6= 0 and a, b ∈ Rn, we have∫
Rn
e−c1|a−z|

2−c2|z−b|2dz =
( π

c1 + c2

)n/2
exp

(
− c1c2
c1 + c2

|a− b|2
)
. (5.4)

Proof. This follows from the unity of the Gaussian measure that for any p ∈ Rn
and α > 0, ∫

Rn
exp

(
−α|x− p|2

)
dx =

(π
α

)n/2
,

which implies ∫
Rn

exp
(
−α|x|2 + 2〈αp, x〉

)
dy = (

π

α
)n/2 exp(α|p|2).

Let α = c1 + c2 and p = (c1a+ c2b)/α, then∫
Rn

exp
(
−(c1 + c2)|x|2 + 2〈(c1a+ c2b), x〉

)
dx

=
( π

c1 + c2

)n/2
exp

( |(c1a+ c2b)|2

c1 + c2

)
.

Now since ∫
Rn

exp
(
−c1|a− x|2 − c2|x− b|2

)
dx

=
( π

c1 + c2

)n/2
exp

( |(c1a+ c2b)|2

c1 + c2
− c1|a|2 − c2|b|2

)
=
( π

c1 + c2

)n/2
exp

(−c1c2|a− b|2
c1 + c2

)
,

we obtain (5.4). �



EJDE-2023/12 DE BRUIJN IDENTITIES 7

For proving (5.3) we write∫
Rn
Ptf(x) dµn(x) = (4π2β2

t )−n/2
∫
Rn

∫
Rn
f(z) exp

(
− |z − αtx|

2 + |βtx|2

2β2
t

)
dx dz.

Taking c1 =
α2
t

2β2
t
, c2 = 1

2 , a = α−1t and b = 0 in Lemma 5.1, since α2
t + β2

t = 1 we

obtain ∫
Rn
Ptf(x) dµn(x) = (2π)−n/2

∫
Rn
f(z) exp

(
− |z|

2

2

)
dz.

which is (5.3).

Theorem 5.2. On Xp := Lp(Rn, dµn) the operator Tt defines a hypercontractive
semigroup; that is,

(i) ‖Tt‖p ≤ ‖f‖q, for all p ≥ q > 1 such that p− 1 ≤ e2t(q − 1);
(ii) limt→0 ‖Ttf − f‖p = 0, for all f ∈ Xp;
(iii) TtTs = Tt+s, for all (t, s) ∈ R2

+.

Proof. (i) Since the constant function 1is in Xp, and the Mehler formula Pt1 = 1,
the equality (5.3) implies that Pt is doubly Markovian in the sense of Nelson (see
[3]). In the same paper (Theorem 2), Nelson shows that an operator which is doubly
Markovian is hypercontractive in the sense of item (i).

(ii) Since α2
t + β2

t = 1, the vectors (αt, βt) and (1, 0) are both on the unit circle
and (αt, βt)→ (1, 0) as t→ 0. For any continuous bounded function f (taking e.g.
f ∈ S ),

f(αtx+ βty)− f(x) = f
(

(αt, βt)

(
x
y

))
− f

(
(1, 0)

(
x
y

))
→ 0

as t→ 0. Furthermore, if M = supx∈R |f(x)|, then |f(αtx+ βty)− f(x)| ≤ 2M in
L1(Rn, dµn). Thus according to Lebesgue’s dominated converence theorem∫

Rn
|f(αtx+ βty)− f(x)| dµn(y)→ 0.

Hence

‖Ttf − f‖pp =

∫
Rn
|Ptf(x)− f(x)|p dµn(x)

=

∫
Rn

∣∣∣ ∫
Rn

[f(αtx+ βty)− f(x)] dµn(y)
∣∣∣p dµn(x)→ 0

Since the Schwartz space S being dense in Xp, this implies item (ii).
(iii) Taking the expression of the Ornstein-Uhlenbeck semigroup Pt (5.2),

PtPsf(x) = (2πβ2
t )−n/2

∫
Rn
Psf(z) exp(−|z − αtx|2/2(β2

t )dz

= (2πβtβs)
−n
∫
Rn
f(y)

∫
Rn

exp
(
− |y − αsz|

2

2β2
s

− |z − αtx|
2

2β2
t

)
dz︸ ︷︷ ︸

=A

dy .

To simplify the expression A we will use the Lemma 5.1. Let

A =

∫
Rn

exp
(
− α2

s

2β2
s

|α−1s y − z|2 − 1

2β2
t

|z − αtx|2
)
dz.



8 H. EMAMIRAD, A. ROUGIREL EJDE-2023/12

Comparing this with (5.4), we obtain

c1 =
α2
s

2β2
s

, c2 =
1

2β2
t

, a = α−1s y, b = αtx.

Hence,

A =
( π
α2
s

2β2
s

+ 1
2β2
t

)n/2
exp

(
−

α2
s

4β2
sβ

2
t

α2
s

2β2
s

+ 1
2β2
t

|α−1s y − αtx|2
)

=
(2π(1− e−2t)(1− e−2s)

1− e−2(t+s)
)n/2

exp
(
− |y − e

−(t+s)x|2

2(1− e−2(t+s))

)
.

(5.5)

Replacing the expression of A in (5.5) we find that

PtPsf(x) = (2π(1− e−2(t+s)))−n2
∫
Rn
f(y) exp

(
− |y − e

−(t+s)x|2

2(1− e−2(t+s))

)
dy = Pt+sf(x).

�

Remark 5.3. From (iii) of the above Theorem one can deduce the Chapman-
Kolmogorov formula∫

Rn
K(x, y, t)K(y, z, s) dy = K(x, z, t+ s) for all x ∈ Rn, t, s > 0, (5.6)

where K(x, y, t) is the heat kernel of Ornstein-Uhlenbeck semigroup, that is

Ptf(x) :=

∫
Rn
K(x, y, t)f(y) dy for all f ∈ Lp(Rn, dµn).

From (5.2) it follows that

K(x, y, t) = (2π(1− t−2t))−n/2 exp
(
− |e

−tx− y|2

2(1− t−2t)

)
.

Hence

PtPsf(x) =

∫
Rn
K(x, y, t)Psf(y) dy

=

∫∫
R2n

K(x, y, t)K(y, z, s)f(z)dzdy

= Pt+sf(x) =

∫
Rn
K(x, z, t+ s)f(z)dz

Since this identity holds for all f ∈ Lp(Rn, dµn), we deduce formula (5.6) for µn-
a.e. z ∈ Rn. The equality holds on Rn by continuity of the left and right hand side
with respect to z.

6. De Bruijn identity in Ornstein-Uhlenbeck channels

In this section we work in L1(R, µ) which is a Lebesgue space with the Gaussian
measure µ := µ1. In this space

∫
R w(·, τ) dµ = 1. If we define the entropy by

Hµ(w(·, τ)) := −
∫
R
w(y, τ) lnw(y, τ) dµ(y) (6.1)

and the Fisher information by

Iµ(w(·, τ)) :=

∫
R
w(y, τ)

( ∂
∂y

lnw(y, τ)
)2
dµ(y), (6.2)



EJDE-2023/12 DE BRUIJN IDENTITIES 9

then the De Bruijn identity in Ornstein-Uhlenbeck channels reads as follows.

Theorem 6.1. Assume that t ≥ 0, then

d

dτ
Hµ(w(·, τ)) = Iµ(w(·, τ)), (6.3)

where w is the solution of Ornstein-Uhlenbeck equation (4.2).

Proof. For the proof we will use

d

dy
µ(y) =

−y√
2π
e−y

2/2 = −yv∞ . (6.4)

The derivative of (6.1) with respect to τ reads

d

dτ
Hµ(w(y, τ))

= −
∫
R

( d
dτ
w(y, τ) lnw(y, τ)

)
dµ

= −
∫
R

( ∂2
∂y2

w(y, τ)− y ∂
∂y
w(y, τ)

)
lnw(y, τ) dµ− d

dτ

∫
R
w(y, τ) dµ︸ ︷︷ ︸

=0 by (3.5)

=

∫
R
w(y, τ)−1

( ∂
∂y
w(y, τ)

)2
dµ(y)−

∫
R

∂

∂y
w(y, τ)(lnw(y, τ))yv∞dy

+

∫
R
y
∂

∂y
w(y, τ)(lnw(y, τ)) dµ = Iµ(w(y, τ)). �

7. De Bruijn identity for relative Fisher information and
Kullback-Leibler divergence

Let ϕ and ψ, be two distribution functions for two random variables X and Y .
The relative Fisher information with respect to a is defined by

Ia(ϕ||ψ) :=

∫
R
ϕ(x)

( ∂
∂x

ln
ϕ(x)

ψ(x)

)2
a(x)dx.

We define the Kullback-Leibler divergence which can be interpreted as the relative
entropy between ϕ and ψ by

DKL(ϕ||ψ) :=

∫
R
ϕ(x) ln

ϕ(x)

ψ(x)
dx.

(see [6]).
The following result establishes that the relative entropy between any two solu-

tions of (2.3) is always decreasing, with a rate given by the relative Fisher infor-
mation:

Theorem 7.1. Assume that ϕ(x, t) and ψ(x, t) two distinct solutions of the Fokker-
Planck equation in its general form:

∂φ

∂t
(x, t) =

∂2

∂x2
(a(x, t)φ(x, t))− ∂

∂x
(b(x, t)φ(x, t)) (7.1)

Then
d

dt
DKL(ϕ||ψ) = −Ib(ϕ||ψ). (7.2)



10 H. EMAMIRAD, A. ROUGIREL EJDE-2023/12

Proof. Let (x, t) ∈ R × R+ and assume ϕ(·, t), ψ(·, t) ∈ H2(R). By differentiating
under the integral sign and using the chain rule, we have (For simplicity we write
ϕ(x) instead of ϕ(x, t))

d

dt
DKL(ϕ||ψ)

=
d

dt

∫
R
ϕ(x) ln

ϕ(x)

ψ(x)
dx

=

∫
R

∂

∂t
ϕ(x) ln

ϕ(x)

ψ(x)
dx+

∫
R
ϕ(x)

∂

∂t
lnϕ(x)dx−

∫
R
ϕ(x)

∂

∂t
lnψ(x)dx

=

∫
R

∂

∂t
ϕ(x) ln

ϕ(x)

ψ(x)
dx+ 0−

∫
R

ϕ(x)

ψ(x)

∂

∂t
ψ(x)dx.

(7.3)

By replacing ∂
∂tϕ(x) in the Fokker-Planck equation (7.1) and using integration by

parts we can write the first integral in (7.3) as∫
R

∂

∂t
ϕ(x) ln

ϕ(x)

ψ(x)
dx

=

∫
R

( ∂2
∂x2

a(x)ϕ(x)− ∂

∂x
b(x)ϕ(x)

)
ln
ϕ(x)

ψ(x)
dx,

=

∫
R

(
a(x)ϕ(x)

∂2

∂x2
ln
ϕ(x)

ψ(x)
+ b(x)ϕ(x)

∂

∂x
ln
ϕ(x)

ψ(x)

)
dx.

(7.4)

Since

∂

∂x
ln
ϕ(x)

ψ(x)
=
(ψ(x)

ϕ(x)

)( ∂
∂x

ϕ(x)

ψ(x)

)
,

and

∂2

∂x2
ln
ϕ(x)

ψ(x)
=
(ψ(x)

ϕ(x)

)( ∂2
∂x2

ϕ(x)

ψ(x)

)
−
( ∂
∂x

ln
ϕ(x)

ψ(x)

)2
,

by replacing these relations in (7.4) and using integration by parts and the Fokker-
Planck equation (7.1) for ψ(x, t) we find that∫

R

∂

∂t
ϕ(x) ln

ϕ(x)

ψ(x)
dx

=

∫
R
a(x)ϕ(x)

(ψ(x)

ϕ(x)

∂2

∂x2
ϕ(x)

ψ(x)
−
(
∂

∂x
ln
ϕ(x)

ψ(x)

)2 )
dx+

∫
R
b(x)ψ(x)

∂

∂x

ϕ(x)

ψ(x)
dx.

Thus,∫
R

∂

∂t
ϕ(x) ln

ϕ(x)

ψ(x)
dx = −Ia(ϕ||ψ) +

∫
R
a(x)ψ(x)

∂2

∂x2
ϕ(x)

ψ(x)
+ b(x)ψ(x)

∂

∂x

ϕ(x)

ψ(x)
dx

Plugging this relation into equation (7.3) and take into account that ψ(x) is also
the solution of the Fokker Planck equation (7.1) we conclude that

d

dt
DKL(ϕ||ψ) = −Ia(ϕ||ψ),

as desired. �



EJDE-2023/12 DE BRUIJN IDENTITIES 11

8. Appendix

Here we give the proof of de Bruijn identity for a function running through a
Gaussian channel.

Let H(u) =
∫
R u(x, ·) lnu(x, ·)dx be Shannon’s entropy, then

d

dt
H(u) =

d

dt

∫
R
u(x, t) lnu(x, t)dx

=

∫
R
(
d

dt
u)(lnu) + u

d

dt
lnu(x, t))dx

=

∫
R

∆u lnudx+

∫
R

d

dt
u(x, t)dx

= −
∫
R

(
(∇u)2/u

)
dx+

d

dt

∫
R
u(x, t)dx.

Since
∫
R u(x, t)dx = 1, we obtain d

dtH(u) = I(u).

References

[1] J.-Ph. Bartier, A. Blanchet, J. Dolbeault, M. Escobedo; Improved intermediate asymptotics
for the heat equation. Appl. Math. Lett., 24 (2011), 76–81.

[2] T. M. Cover, J. A. Thomas. Elements of information theory. Second edition. Wiley-

Interscience , Hoboken, NJ, 2006
[3] E. Nelson; The free Markoff field. J. Funct. Anal., 12 (1973), 211–227.

[4] G. Toscani; The fractional Fisher information and the central limit theorem for stable laws.
Ric. Mat., 65 (2016), 71–91.

[5] A. J. Stam; Some inequalities satisfied by the quantities of information of Fisher and Shannon.

Inf. Contr., 2 (1959), 101–112.
[6] J. Voigt; Stochastic Operators, Information, and Entropy Commun. Math. Phys., 81 (1981)

31–38.

[7] A. Wibisono, V. Jog, P-L. Loh; Information and estimation in Fokker-Plank channels. 2017
IEEE Inter. Symp. on Infor. Theory, arXiv 1702.03656v1.

Hassan Emamirad

Laboratoire de Mathématiques, Université de Poitiers, teleport 2, BP 179, 86960 Chas-

sneuil du Poitou, Cedex, France
Email address: emamirad@math.univ-poitiers.fr

Arnaud Rougirel
Laboratoire de Mathématiques, Université de Poitiers, teleport 2, BP 179, 86960 Chas-

sneuil du Poitou, Cedex, France

Email address: rougirel@math.univ-poitiers.fr


	1. Introduction
	2. Relationship between Fokker-Planck and heat equation
	3. De Bruijn identity in Fokker-Planck channels
	4. Relationship between Fokker-Planck and Ornstein-Uhlenbeck equation
	5. Mehler formula and Ornstein-Uhlenbeck semigroup
	6. De Bruijn identity in Ornstein-Uhlenbeck channels
	7. De Bruijn identity for relative Fisher information and Kullback-Leibler divergence
	8. Appendix
	References