40

On the Propagation of Chaos 
for Recombination Models

On the Propagation of Chaos  
for Recombination Models

Lu Kevin Ong
Institute of Mathematics
University of the Philippines Diliman

ABSTRACT

We consider binary interactions in an N-particle system. In particular, we use 

probability distributions known as recombination models to describe these 

interactions. Chaos propagates when the stochastic independence of two random 

particles in a particle system persists in time, as the number of particles tends 

to infinity. The concept of propagation of chaos was first introduced by Kac in 

connection with the Boltzmann equation, while modeling binary collisions 

in a gas. We obtain a development of Kac’s program in the framework of 

recombination models. Specifically, our aim is to prove the relevant propagation 

of chaos phenomenon for our particle system. We first show that the solution 

for the master equation of our time-continuous process converges. Then, we use 

this solution together with the concepts of marginal measure and chaos to prove 

our desired result. Our main theorem for this study says that if a sequence of 

measures on our defined particle system is chaotic, then the resulting sequence 

of measures that had undergone the recombination process is also chaotic. This 

implies that the study of one particle after recombination gives information on 

the behavior of a group of particles in our particle system.

Keywords:  particle system, recombination, master equation, propagation of chaos

INTRODUCTION 

In 1956, Kac investigated the probabilistic foundations of kinetic theory and 
introduced the concept of propagation of chaos for the Boltzmann equation. Gottlieb 
(1998) stated in his thesis that Kac invented a class of interacting particle systems 
wherein particles collide at random with each other while the density of particles 
evolves deterministically in the limit of infinite particle number. He also described 
the concept of propagation of chaos in general, and stated that chaos propagates 
when the stochastic independence of two random particles in a particle system 
persists in time, as the number of particles tends to infinity.

Several topics in propagation of chaos play a major role in the field of probability, 
as discussed by Sznitman (1991). In particular, he wrote that one motivation for 

Science Diliman (January-June 2021) 33:1, 40-56



L.K. Ong

41

the subject was to investigate the connection between a detailed and a reduced 
description of the particles’ evolution. He also stated that propagation of chaos deals 
with symmetric evolution of particles. That is, the study of one individual provides 
information on the behavior of the group. Another research related to this topic 
includes the paper of McKean (1967), where he introduced propagation of chaos for 
interacting diffusions and analyzed what are now called McKean-Vlasov equations. 
More recently in 2018, Thai studied a discrete version of the particle approximation 
of the McKean-Vlasov equations, and she proved the propagation of chaos property as 
well.

Many authors are concerned with proving that specific systems propagate chaos. In this 
study, we observe the propagation of chaos property for recombination models. Rabani 
et al. (1995) stated that random matings occur between parental chromosomes via 
a  mechanism known as “crossover”; that is, children inherit pieces of genetic material 
from different parents according to some random rule. Recombination models are 
probability distributions  that represent this randomness.

In 2016, Caputo and Sinclair stated that recombination models based on random   
mating have a number of applications in the natural sciences and play a significant 
role in the analysis of genetic algorithms. Moreover, they used quadratic dynamical 
systems  to study the rate of convergence to equilibrium in terms of relative entropy 
for recombination models.

Let Ω
n
 = {-1,1}n for a fixed natural number n. For each natural number N, we consider 

the state space Ω = Ω
n
N . We think of each state as a sequence of N particles in the 

system, where each particle is represented by a string of bits in Ω
n
. Our main result 

in this paper is on the propagation of chaos phenomenon for recombination models 
that is represented by the interaction in this particle system.

MATERIALS AND METHODS

Recombination Models

Let [n] denote the set {1, ..., n}. Given A ⊂ [n] and σ ∈ Ω
n
, we write σ

A
 for the  

A-component of σ, that is, the subsequence (σ
j
, j ∈ A). For example, suppose n = 5 

and let σ = (−1, −1, 1, −1, 1), A = {2, 3, 5}. Then σ
A
 = (−1, 1, 1).

Definition 1. (Recombination at a Set). Given A ⊂ [n] and (σ, η) ∈ Ω
n
 × Ω

n
, the 

recombination of (σ, η) at A consists in exchanging the A-component of σ with the 
A-component of η. This defines the map

(σ, η) → (η
A
σ

A
c, σ

A
η

A
c ),



On the Propagation of Chaos for Recombination Models

42

On the Propagation of Chaos 
for Recombination Models

where Ac is the complement of set A, and the components of η
A
σ

A
c ∈ Ω

n
 is defined by

(η
A
σ

AC
)
j
=

η
j
 if j ∈A,

σ
j
 if j ∈AC .

⎧
⎨
⎪

⎩⎪

⎫
⎬
⎪

⎭⎪

We have a similar definition for the components of σ
A
η

A
c. For example, suppose n = 

5 and let σ  = (−1, −1, 1, −1, 1), η = (−1, 1, 1, 1, −1), A = {2, 3, 5}. Then

η
A
σ

A
c = (−1, 1, 1, −1, −1) and σ

A
η

A
c = (−1, −1, 1, 1, 1).

The following process is motivated by the discussion of Carlen et al. (2010).

Definition 2. (Recombination Walk). Let S = (σ(1), σ(2), ..., σ(N)) denote a state in Ω, 
where σ(i) ∈ Ω

n
 for each i ∈ [N ]. S represents a set of N particles in our system. 

Recombination walk on Ω is a process that illustrates the evolution of an initial 
state S by applying recombinations on a random pair of particles at each step. We 
describe the process as follows:

1. Randomly pick a pair (k, l) of distinct indices in [N], uniformly chosen from among 
all such pairs.

2.  Choose a subset A ⊂ [n] at random according to some probability distribution 
ν.

3.  Update S by leaving σ(i) unchanged for i ≠ k, l, and updating the particles σ(k) and 
σ(l) via the recombination at A.

 Let R
k,l,A

S denote the new state in Ω. That is, if we assume k < l, the mapping 
R

k,l,A
 on Ω is given by

(σ(1), σ(2), ..., σ(N)) → (σ(1), ..., σ(l)
 
 σ(k)

 
, ..., σ(k)

 
, σ(l)

 
, ..., σ(N)).

Here are some examples of recombination models represented by ν, from Caputo and 
Siclair (2016):

1. Single sire recombination: v(A)
1
n

1(A = {i}
1=1

n

∑ ), where 1 is an indicator function on 
subsets of [n]. That is, 

1(A = {i}) =
1 if A = {i} for some i ∈[n],
0 otherwise.

⎧
⎨
⎪

⎩⎪

⎫
⎬
⎪

⎭⎪

2. One-point crossover: v(A) =
1
n+ 1

1(A = J
i

i=0

n

∑ ),  where J0 = {} and Ji = [i] for  i ≥ 1.

A A
c

A A
c



L.K. Ong

43

3. Uniform crossover: v(A) =
1
2n

,  for all A ⊂ [n].

4. The Bernoulli (q) model: for some q ∈ [0, 1/2],

          v(A) = q
|A|(1−q)n−|A|.

Master Equation

Let S
j
 denote the position after the jth step of the walk and φ ∈ RΩ be a bounded Borel- 

measurable function on Ω. We let T : φ → Q
N
 φ be the Markov transition operator on 

RΩ represented by the transition matrix Q
N
 . Based on the defined process above, the 

function (Q
N
 φ) ∈ RΩ is given by

(Q
N
ϕ)(S) = E[ϕ(S

j+1
) | S

j
= S]

             =  
N
2

⎛

⎝⎜
⎞

⎠⎟

−1

v(A)φ(R
k ,l,A
S).

A
∑

1≤k<l≤N
∑

Let p
i
 be any probability measure on {−1, 1} for i ∈ [n] and let π

n
 be the direct   

product of the p
i
’s, that is, π

n
 = p

1
 ⊗ p

2
 ⊗ ... ⊗ p

n
. It is not difficult to see that 

the product measure π = π
n
N on Ω gives us a reversible process, as defined by Levin 

et al. (2009):

Indeed, for any S = (σ(1), σ(2), ..., σ(N)), we have

 
π(R

k ,l,A
S) = π

n
(σ ( j) )

j≠k ,l
∏
⎡
⎣
⎢

⎤
⎦
⎥πn(σ A

(l)σ
Ac
(k ) )π

n
(σ

A

(k )σ
Ac
(l) ) = π(S).

Since R
k,l,A

(R
k,l,A

S) = S, the transition matrix Q
N
 is symmetric, and so we have

π(S)Q
N
 (S, R

k,l,A
S) = π(R

k,l,A
S)Q

N
 (R

k,l,A
S, S),

where Q
N
 (x, y) = P[S

j+1
 = y|S

j
 = x].

The following definition is adopted from Carlen, Degond and Wennberg (2013).

Definition 3. (Time-continuous Master Equation). The time-continuous master equation 
associated to a discrete Markov process with transition matrix Q is given by

d
dt
F (N )(t,S) = L*F (N ) (t,S)  with  F (N ) (0,S) = F

0

(N )(S),

   



On the Propagation of Chaos for Recombination Models

44

On the Propagation of Chaos 
for Recombination Models

where L* = N [Q* − I] with Q* the adjoint of Q, I is the identity matrix, and F
O

(N) is the 
initial probability density function.

Since the Q
N
 is symmetric, we have Q*

N  =  QN . We now give our time-continuous master 
equation in the following proposition, which is motivated by Carlen et al. (2010).

Proposition 4. If the law µ
0

(N) of the initial state S
0
 has a density F

0
(N) with respect to 

π, then for all t > 0, the law µ
t
(N) 

 
of S

t
 has a density F

t
(N) with respect to π, where 

F
t
(N) is the solution of the following equation

       
d
dt
F

N( ) = L
N
µ(N ) t,S( ) with F (N ) (0,S) =  F0 N( ) S( ),                       (1)

where L
N
 = N (Q

N
 − I) and I is the identity matrix.

Remarks:

1. Note that L
N
 is self-adjoint, since Q

N
 is symmetric.

2.  As defined in the proposition, we have µ
t
(N)(S) = F (N)(t, S)π(S) and thus (1) is 

equivalent to

     
d
dt
µ

N( ) = L
N
µ(N ) t,S( )  with  µ(N ) (0,S) = µ0(N ) S( ).   (2)

3. The solution µ
t
(N) of (2) is given by

µ
t

(N ) = etLN µ
0

(N ) =
t j

j!j=0

∞

∑ LN
j µ

0

(N ).

Propagation of Chaos

We now present some known definitions needed for our theorem. The following definition 
is taken from Carlen et al. (2010).

Definition 5. (Marginal Measure). Let µ(N) be a probability measure on Ω and let 
k be a positive integer with k < N. The marginal measure of µ(N) for the first k 
particles on A⊂ Ωk

n
 is defined as

P
k
(µ(N))[A] = µ(N)[{(σ(1), ..., σ(k)) ∈ A}].

We adopt the following definition from Sznitman (1991).



L.K. Ong

45

Definition 6. (Chaos).  Let  µ  be  a  probability  measure  on  Ω
n
. Let {µ(N)} ∞N =1 be a 

sequence of symmetric probability measures on Ω, i.e., the value is the same 
no matter the order of its arguments. We say that {µ(N)} ∞N =1 is µ-chaotic if for  
{g

k
 : 1 ≤ k < N } ⊂ C

b
(Ω

n
), where C

b
(Ω

n
) is the set of bounded functions from Ω to R, 

we have

         lim
N→∞

g(S)µ(N ) (S)
S∈Ω
∑ = gt (σ )µ

σ ∈Ω
n

∑ (σ ).
⎛

⎝⎜
⎞

⎠⎟t=1

k

∏  ,   (3)

where g = g
1
 ⊗ g

2
 ⊗ ... ⊗ g

k
 ⊗ 1 ⊗ ... ⊗ 1 with N − k copies of the 1.

We adopt the following definition from Gottlieb (1998).

Definition 7. (Propagation of Chaos). A sequence {Q
N
} ∞N =1  whose N -th term is a 

Markov transition function on Ω propagates chaos if, whenever a sequence of 
measures {µ(N)} ∞N =1 ⊂ Ω is µ-chaotic for some measure µ ∈ Ωn, then for any t ≥ 
0, the sequence {µ

t
(N)} ∞N =1 ⊂ Ω which satisfies (2) is µ̄ t-chaotic for some measure 

µ̄
t
 ∈ Ω

n
.

RESULTS AND DISCUSSION

We now present our main result for this study in the following theorem.

Statement of the Main Result

Theorem 8. (Propagation of Chaos for Recombination Models).

Let µ(N) be the probability measure on Ω with probability density F (N) with respect 
to π. Suppose that {µ(N)} ∞N =1  is a µ-chaotic family, where µ is some probability 
measure on Ω

n
. For each natural number N, let F (N)(t, ·) denote the solution of (1) at 

time t, starting from the initial data F (N). Let µ(N)(t, ·) be the measure on Ω with 
probability density F (N)(t, ·), with initial measure µ(N) at t = 0.  Then for any t ≥ 0,  
{µ(N)(t, ·)} ∞N =1  is µ(t, σ)-chaotic, where µ(t, σ) is the solution of the following problem:

 

µ(0,⋅) = µ
d
dt

µ(t,σ ) = v(A)[µ
A
(t,σ

A
)⊗ µ

AC
(t,σ

AC
)− µ(t,σ )],

A
∑

⎧
⎨
⎪

⎩
⎪  (4)

where

µ
A
(t,σ

A
) = µ(t,σ ).

σ AC
∑



On the Propagation of Chaos for Recombination Models

46

On the Propagation of Chaos 
for Recombination Models

Remarks:

1.  For each natural number N, we will use the notation µ(N)(·) instead of µ(N)(0, ·). 

2.  From the discussion of Hauray and Mischler (2014): Suppose µ(N) is symmetric. 
Define the empirical measure of the system as

M
n
(t,∑) =

1
N

δ
σ (1)(t )

t=1

N

∑ (∑),

 where ∑ is a subset of Ω
n
 and δ is the Dirac measure. That is,

δ
σ (1)(t )

(∑) =
1,  if σ (i )(t)∈∑,
0,  otherwise.

⎧
⎨
⎪

⎩⎪

 Then µ(N) is µ-chaotic is equivalent to saying that

lim
N→∞

 M
N
(t,Σ) =  µ(t,Σ).

 Moreover, it is also equivalent to condition (3) with k = 2. We will use this later 
in the proof of our theorem.

3.  The differential equation given in (4) has a unique solution, as shown by Baake 
et al. (2016) in the general case. In our theorem, this solution is given by µ(t, σ), 
which we will choose appropriately in our proof.

We will first prove three lemmas that will be needed for the proof of our theorem.

Three Lemmas

The first lemma that we will prove concerns the convergence of the series that we 
will get from the solution of (2).

Lemma 9. Let g ∈ C
b
(Ω) that depends only on σ(1), ..., σ(k). If T < 1/4, for any natural 

number N, the power series

    etLNg =
t j

j!j=0

∞

∑ LN
j g,    (5)

absolutely converges for t ∈ [0, T].



L.K. Ong

47

Proof: Suppose that g depends only on one particle and, without loss of generality,    
we can set this to be the first particle. That is, for some g : Ωn → R, we have

g(S) = g(σ(1), ..., σ(N)) = g(σ(1)).

Since g ∈ C
b
(Ω), this implies that g  ∈ C

b
(Ω

n
), that is for all σ ∈ Ω

n
, 

|g(σ)| < M  for some M ∈ R.

Let N be a natural number. Since L
N
 = N (Q

N
 − I), by using the definition of Q

N
 , we have 

for any S = (σ(1), σ(2), ..., σ(N)) that

 

L
N
g(S) = N

N
2

⎛

⎝⎜
⎞

⎠⎟

−1

v(A)[g(R
k ,l,A
S

A
∑

1≤k<l≤N
∑ ) − g(s)]

          =
2

N − 1
v(A)[g

A
∑

l=2

N

∑ (σ A
(l)σ

AC
( 1) ) − g(σ ( 1) )]

So that 

  L
N
g(S) <

2
N −1

v(A) (2M) = 4M
A
∑

l=2

N

∑ .   (6)

Define g
2
 by

g
2
(σ ( 1) ,σ (l) ) = v(A)[g

A
∑ (σ A

(l)σ
AC
( 1) ) − g(σ ( 1) )],   for  l ∈{2,...,N}.

Then
L
N
g(S) =

2
N −1

g
2
(

l=2

N

∑ σ (1),σ (l) ).

In general, we will prove by induction that           

   L
N

j g(S) <  j!4 jM,          (7)

for any natural number j.



On the Propagation of Chaos for Recombination Models

48

On the Propagation of Chaos 
for Recombination Models

Indeed j = 1 is true by (6). Suppose L
N

j g(S) <  j!4 jM, for some natural number k. Let

 

g
k+1

= L
N

k g,   where g
k+1

:  Ω
n

k+1 →!.

L
N

k+1g(S) = L
N
g
k+1

(S (k+1) )

            =
2

N − 1
v(A) [g

1≤l
1
<l

2
≤k+1

∑
A
∑ (Rl

1

,
l
2

,AS (k+1) ) − g(S (k+1) )]

           +
2

N − 1
v(A)[g

A
∑

j=k+2

N

∑
l=1

k+1

∑ (σ ( 1) ,...,σ A
( j)σ

AC
(l) ,...,σ (k+1) ) − g(S (k+1) )]

So that

 

L
N

k+1g(S)

<
2

N −1
k +1
2

⎛

⎝⎜
⎞

⎠⎟
2g(S (k+1)) +

2
N −1

(k +1)(N −k −1)2g(S (k+1))

= 4 g(S (k+1)) (k +1)(
2N −K −2
2N −2

)

< 4(k!4kM)(k +1) = (k +1)!4k+1M

Now, for 0 ≤ t < 1/4, the series for etLNg absolutely converges since 

 

t j

j!j=0

∞

∑ LN
J g <

t j

j!j=0

∞

∑ j!4 JM = M (4t)J .
j=0

∞

∑

In general, if we start with g, a functional of p particles where p ≥ 2, (7) is changed 
into

   |Lj
N
g(S)| < (j +  p – 1)!4jM.   (8)

for all natural numbers N, S ∈ Ω. 

Furthermore, for any natural number j,

j + p −1( )!4 jM = j! j + p−1( )!
j!

 4 jM < j! j + p−1( )p−1 4 jM.

So that for 0 ≤ t < 1/4 and any natural number N, we have

t J

j!j=0

∞

∑ LN
J g < (4t)J( j + p−1)p−1M.

j=0

∞

∑
    

   

Then



L.K. Ong

49

Thus, etLNg is absolutely convergent for 0 ≤ t < 1/4. This ends the proof of the 
lemma.

Before we proceed to the second lemma, we first look at the following observation.

For each k ∈ [N − 1], we have

P
k
(µ(N )(t,S (k ) )) = µ(N )(t,S).

σ (k+1) ,...,σ (N )
∑

Let g be a function that depends on σ(1) and define g
2
 as we have defined before.

Then

L
N
g(S)µ(N )

S
∑ (S) = 2

N −1
g
2
(σ (1),σ (l) )µ(N )(S)

l=2

N

∑
σ (3) ,...σ (N )
∑

⎡

⎣
⎢

⎤

⎦
⎥

σ (1) ,σ (2)
∑ .

So that,

lim
N→∞

L
N
g

S∈Ω
∑ (S)µ(N )(S) = 2 µ2(S

(2))g
2
(S (2))

S(2)∈Ω(2)
∑⎡

⎣
⎢

⎤
⎦
⎥,

where

µ
2
(σ (1),σ (2)) = lim

N→∞
P
2
(µ(N )(0,σ (1),σ (2))).

We now prove a more general result in the following lemma.

Lemma 10. Let g be a function of one variable, say σ(1). Then for any natrual number k,

 lim
N→∞

L
N

k

S∈Ω
∑ g(S)µ(N )(S) = 2k µk+1(S

(k+1))g
k+1
(S (k+1))

S(k+1)∈Ω(k+1)
∑⎡

⎣
⎢

⎤
⎦
⎥,   (9)

where µ
k +1
(S (k+1)) = lim

N→∞
P
k +1
(µ(N )(0,S (k+1))),g

1
= g,  and

 g
k+1
(S (k+1)) = v(A) g

k
(σ (1),...,σ (t−1),σ

A

(k+1)σ
Ac
(t ),...,σ (k ) )− g

k
(S (k ) )⎡⎣

⎤
⎦

A
∑

t=1

k

∑ .  (10)

Proof: Note that for k  ≥ 2 we have



On the Propagation of Chaos for Recombination Models

50

On the Propagation of Chaos 
for Recombination Models

 

L
N

k g(S) =
2k

N −1
v(A)

A
∑ [gk−1(

1≤l
1
<l
2
≤k

∑ Rl
1
,l
2
,A
S (k−1))− g

k−1
(S (k−1))]

                  +
2k

N −1
g
k+1
(

J=k+1

N

∑ σ (1),...,σ (k ),σ ( j) )

            = α(S)+ β(S),

where α(S) and β(S) represent the 1st and 2nd set of summations in the right-hand 
side, respectively. Since the summations in α(S) involve a finite number of terms, we 

can write it as α(S) =
C
N −1

 for some constant C. So that,

lim
N→∞

α
S∈Ω
∑ (S)µ(N )(S) = lim

N→∞

C
N −1

µ
S∈Ω
∑

(N )
(S) = lim

N→∞

C
N −1

= 0.

Now, by symmetry we have 

β
S∈Ω
∑ (S)µ(N )(S) = 2

k(N −k)
N −1

g
k+1
(S (k+1))µ(N )(S)

σ (k+2) ,...,σ (N )
∑

⎡

⎣
⎢

⎤

⎦
⎥.

σ (1) ,...,σ (k+1)
∑

So that 

lim
N→∞

β
S∈Ω
∑ (S)µ(N )(S) = 2k µk+1(S

(k+1))g
k+1
(S (k+1))

S(k+1)∈Ω(k+1)
∑⎡

⎣
⎢

⎤
⎦
⎥.

This ends the proof of the lemma.

Finally, we prove the third lemma that will be essential to the proof of our theorem.

Lemma 11. Let g
1
 and h

1
 be functions of a single variable. Let γ

2
 be a function 

of two variables, say σ(1) and σ(2), such that γ
2
(σ(1), σ(2)) = g

1
(σ(1))h

1
(σ(2)). Then for  

0 ≤ t < 1/4,



L.K. Ong

51

  

(2t)l

l!l=0

∞

∑ γ l+2(S
(l+2))

S( l+2)
∑⎡
⎣
⎢

⎤
⎦
⎥ µ1(σ

(t ) )
t=1

l+2

∏

=
(2t) j

j!j=0

∞

∑ gj+1(S
( j+1))

S( j+1)
∑⎡

⎣
⎢

⎤
⎦
⎥ µ1(σ

(t ) )
t=1

j+1

∏

×
(2t)k

k!k=0

∞

∑ hk+1(S
(k+1))

S (k+1)
∑

⎡

⎣
⎢

⎤

⎦
⎥ µ1(σ

(t ) )
t=1

k+1

∏ ,

   (11)

where g
j+1

, h
k+1

, γ
l+2

 are defined inductively as in (10) for j, k, l ≥ 0.

Proof: Let RHS demote the hand side of equation (11). Distributing the terms in the 
RHS gives us.

RHS =
(2t)l

l!l=0

∞

∑
l
m

⎛

⎝⎜
⎞

⎠⎟m=0

l

∑
S( l+2)
∑ gm+1(σ

(1),σ (3),...,σ (m+2))⎡⎣ ⎤⎦

             × h
(l−m)+1

(σ (2),σ (m+3),...,σ (l+2))⎡⎣
⎤
⎦ µ1
t=1

l+2

∏ (σ (t ) ).

The result is proved if we show that for each l ≥ 0,

  

γ
l+2
(S (l+2))G(S (l+2))

S( l+2)
∑

=
l
m

⎛

⎝⎜
⎞

⎠⎟m=0

l

∑
S( l+2)
∑ gm+1(σ

(1),σ (3),...,σ (m+2))⎡⎣ ⎤⎦

            × h
(l−m)+1

(σ (2),σ (m+3),...,σ (l+2))⎡⎣
⎤
⎦G(S

(l+2)),
  (12)

for any symmetric function G.

The case l = 0 is true by the definition of γ
2
, that is

g
1

S(2)
∑ (σ (1))h1(σ

(2))G(S (2)) = γ
2

S(2)
∑ (S (2))G(S (2)).

The proof readily follows by using induction on this initial condition.   



On the Propagation of Chaos for Recombination Models

52

On the Propagation of Chaos 
for Recombination Models

Proof of the Theorem

We now proceed with the proof of our theorem.

Proof Theorem 4: Again, we consider the case when g ∈ C
b
(Ω) that depends only on 

one variable, say σ(1). Since L
N
 is self-adjoint, we have

g(S)µ(N )(t,S)
S∈Ω
∑ = g(S)etLN µ(N )(S)

S∈Ω
∑ = t

k

k!
L
N

k

k=0

∞

∑
S∈Ω
∑ g(S)µ(N )(S).

Since we are assuming that {µ(N)}
N∈!

 is µ- chaotic, we have from (9)

lim
N→∞

Lk
S∈Ω
∑ g(S)µ(N )(S) = 2k gk+1(S

(k+1))
S (k+1)∈Ω(k+1)
∑

⎡

⎣
⎢

⎤

⎦
⎥ µ1(σ

(t ) ),
t=1

k+1

∏

where µ
1
(0) = lim

N→∞
P
1
(µ(N )(0,σ )).

Note also that

g(S)µ(N )(t,S) = g(σ (1) µ(N )(t,S)
σ (2) ,...,σ (N )
∑

⎡

⎣
⎢

⎤

⎦
⎥

σ (1)
∑

S∈Ω
∑

                                = g(σ (1))P1(µ(N )(t,σ (1))).
σ (1)
∑

From (5), it follows that for 0 ≤ t < 1/4,

 g(σ (1))µ
1
(t,σ (1))

σ (1)
∑ = (2t)

k

k!k=0

∞

∑ gk+1(S
(k+1))

s(k+1)∈Ω(k+1)
∑

⎡

⎣
⎢

⎤

⎦
⎥ µ1(σ

(t ) ),
t=1

k+1

⨿  (13)

where µ(t, σ (1)) =  lim
N→∞
P
1
(µ(N )(t,σ (1))).

Starting now from a function γ
2 (σ

(1), σ(2)) = g(σ(1))h(σ(2)) are defining γ
k 
(S(k)) inductively 

as in (10), we obtain again for 0 ≤  t < 1/4, 

g
σ (1) ,σ (2)
∑ (σ (1))h(σ (2))µ2(t,σ

(1),σ (2))

=
(2t)k

k!k=0

∞

∑ γ k+2(S
(k+2))

S(k+2)∈Ω(k+2)
∑⎡

⎣
⎢

⎤
⎦
⎥ µ1(σ

(t ) ),
t=1

k+2

∏



L.K. Ong

53

where µ
2
(t, σ (1),σ (2)) =  lim

N→∞
P
2
(µ(N )(t,σ (1),σ (2))).

From (11) and (13) it follows that 

  

g
σ (1) ,σ (2)
∑ (σ (1))h(σ (2))µ2(t,σ

(1),σ (2))

= g(σ (1))µ
1
(t,σ (1)

σ (1)
∑ )⎡
⎣
⎢

⎤
⎦
⎥ h(σ

(2))µ
1
(t,σ (2)

σ (1)
∑ )⎡
⎣
⎢

⎤
⎦
⎥.   (14)

Since g, h are arbitrarily chosen, we have 0 ≤ t < 1/4,

µ
2
(t,σ (1),σ (2)) = µ

1
(t,σ (1))µ

1
(t,σ (2)).

We only need to show that µ1(t,σ
(1)) is the solution of (4). Similar computations will 

lead to

  

g
2

σ (1) ,σ (2)
∑ (σ (1),σ (2))µ1(t,σ

(1)),µ
2
(t,σ (2))

=
(2t)k

k!k=0

∞

∑ gk+2(S
(k+2))

S(k+2)∈Ω(k+2)
∑⎡

⎣
⎢

⎤
⎦
⎥ µ1(σ

(t )

t=1

k+2

∏ ).   (15)

Now differentiating (13), we get

  

g
σ (1)
∑ (σ (1)) d

d
t

µ
1
(t,σ (1))

=
(2t)k−1

(k −1)!k=1

∞

∑ gk+1(S
(k+1))

S(k+1)∈Ω(k+1)
∑⎡

⎣
⎢

⎤
⎦
⎥ µ1(σ

(t )

t=1

k+1

∏ ).
 



On the Propagation of Chaos for Recombination Models

54

On the Propagation of Chaos 
for Recombination Models

Using a change of variable for the summation in the right hand side,

  

g
σ (1)
∑ (σ (1)) d

d
t

µ
1
(t,σ (1))

=
(2t)k

k!k=0

∞

∑ gk+2(S
(k+2))

S(k+2)∈Ω(k+2)
∑⎡

⎣
⎢

⎤
⎦
⎥ µ1(σ

(t )

t=1

k+2

∏ )

= g
2
(σ (1),σ (2))µ

1
(t,σ (1))µ

1
(t,σ (2))

σ (1) ,σ (2)
∑

= (Q
2
− I)g(σ (1))µ

1
(t,σ (1))µ

1
(t,σ (2))

σ (1) ,σ (2)
∑ .

 

Again, by using the self-adjoint property, we have

g
σ (1)
∑ (σ (1)) d

d
t

µ
1
(t,σ (1))

= g(σ (1))(Q
2
− I)µ

1
(t,σ (1))µ

1
(t,σ (2))

σ (1) ,σ (2)
∑

= g(σ (1))
σ (2)
∑

σ (1)
∑ v(A)[µ1(t,σ A

(2)σ
Ac
(1) − µ

1
(t,σ (1))]

A
∑⎡
⎣⎢

⎤
⎦⎥
µ(t,σ (2))

= g(σ (1))
σ (1)
∑ v(A)

A
∑ µ1(t,σ A

(2)σ
Ac
(1))(µ

1
(t,σ (2))( )− µ(t,σ (1))

σ (2)
∑⎡
⎣
⎢

⎤
⎦
⎥

= g(σ (1))
σ (1)
∑ v(A)

A
∑ [µ1A(t,σ A

(1))⊗ µ
1Ac
(t,σ

Ac
(1))− µ

1
(t,σ (1))]

Finally, we need to remove the restriction on t < 1/4. We can start with some t
0
, 

where 0 ≤  t
0
 < 1/4, and we repeat the argument to extend the proof to time t 

where t
0
 ≤  t < t

0
 + 1/4. We can see that by proceeding in this manner, we can cover 

the whole time range, that is, µ(N)(t, ·) is µ(t, σ)-chaotic for any natural number N and 
t ≥ 0. This ends the proof of the theorem.



L.K. Ong

55

CONCLUSION

In this study, we introduced some definitions and ideas used to prove propagation 
of chaos for a particle system where the binary interactions are represented by 
recombination models. Specifically, we showed that if a sequence of measures {µ(N)}∞

N =1
  

on our defined particle system Ω is chaotic, then for any t ≥ 0, {µ
t
(N)}∞

N =1
 , is also chaotic, 

where each measure µ
t
(N) i s  obtained after applying recombinations on µ(N) for some 

time t.

ACKNOWLEDGMENT

The author thanks the Department of Mathematics and Physics of Roma Tre 
University for its hospitality during his research in Italy. He also thanks the support 
of the Office of the Vice President for Academic Affairs, University of the Philippines, 
during his research in Italy. 

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______

Lu Kevin Ong <lkong@math.upd.edu.ph> is currently a Ph.D. student and an instructor at the 

Institute of Mathematics, University of the Philippines Diliman. His research interests include 

probability theory, mathematical statistics, actuarial mathematics, and dynamical systems. He 

aims to contribute to the development of mathematics in the Philippines through research 

and mentorship.