Stochastic Semigroups and Coagulation Equations
A general class of bilinear systems of discrete or continuous coagulation equations is considered. It is shown that their solutions can be approximated by the solutions of appropriate stochastic systems describing the coagulation process in terms of stochastic semigroups.
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| author | Lachowicz, M. Лачович, М. |
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| description | A general class of bilinear systems of discrete or continuous coagulation equations is considered. It is shown that their solutions can be approximated by the solutions of appropriate stochastic systems describing the coagulation process in terms of stochastic semigroups. |
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UDC 517.21
M. Lachowicz (Inst. Appl. Math. and Mech., Warsaw Univ., Poland)
STOCHASTIC SEMIGROUPS AND COAGULATION EQUATIONS
STOXASTYÇNI NAPIVHRUPY I RIVNQNNQ KOAHULQCI}
A general class of bilinear systems of discrete or continuous coagulation equations is considered. It is shown that
their solutions can be approximated by the solutions of appropriate stochastic systems describing the coagulation
process in terms of stochastic semigroups.
Rozhlqnuto zahal\nyj klas bilinijnyx system dyskretnyx abo neperervnyx rivnqn\ koahulqci]. Poka-
zano, wo ]x rozv’qzkymoΩut\ buty nablyΩeni rozv’qzkamystoxastyçnyx system, qki opysugt\ proces
koahulqci] çerez stoxastyçni napivhrupy.
1. Coagulation equations — mesoscopic description. Coagulation processes are im-
portant in various physical situations. Entities (clusters, particles, droplets, ...) merge
by coalescence to form larger ones. Such a phenomena takes place in polymer science,
atmosphere physics, colloidal chemistry, biology, and immunology — see references in
[1 – 4]. A system of infinite number of equations (discrete model) was introduced by
Smoluchowski [5] to describe the coagulation of colloids moving according to a Brown-
ian motion.
In the discrete model the size of entities (clusters) is characterized by an integer
r ∈ N = {1, 2, . . . } identified with the number of identical elementary entities. In
the continuous model the size of entities is characterized by a real nonnegative number
r ∈ [0,∞[. The discrete Smoluchowski coagulation equation is an infinite system of
bilinear ODEs whereas the continuous Smoluchowski coagulation equation is a bilinear
integro-differential equation. We unify them in the following notation:
∂tf = Q1[f ], t > 0, r ∈ J , (1.1)
where J is either N or [0,∞[ , f = f(t, r) is the density of clusters of size r at time
t ≥ 0,
Q1[f ](r) =
1
2
∫
Jr
α(r − r1, r1)f(r − r1)f(r1) dλ(r1) − f(r)
∫
J
α(r, r1)f(r1) dλ(r1),
Jr =
{
r1 ∈ J : r1 < r
}
, r ∈ J ,
λ is the counting measure in the case J = N or the Lebesgue measure in the case
J = [0,∞[ , α(r, r1) are the coagulation rate.
The Smoluchowski coagulation equation has been the subject of several modifica-
tions and studies. Another coagulation model was proposed by Oort and van de Hulst
(and then by Safronov) to describe the process of aggregation of protoplanetary bodies
in astrophysics (see references in [3]). The Oort – Hulst – Safronov coagulation equation
reads
∂tf = Q0[f ], t > 0, r ∈ [0,∞[ ,
(1.2)
Q0[f ](r) = −∂r
(
f(r)
r∫
0
r1α(r, r1)f(r1) dr1
)
− f(r)
∞∫
r
α(r, r1)f(r1) dr1.
In [3] the following class of generalized coagulation equations was introduced
c© M. LACHOWICZ, 2005
770 ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
STOCHASTIC SEMIGROUPS AND COAGULATION EQUATIONS 771
∂tf = QGC [f ], t > 0, r ∈ [0,∞[ , (1.3)
where
QGC [f ](r) =
1
2
∞∫
0
∞∫
0
A(r; r1, r2)a(r1, r2)f(r1)f(r2) dr1 dr2 −
−f(r)
∫ ∞
0
α(r, r1)f(r1) dr1. (1.4)
A is the weighted probability that the interaction of a cluster of size r1 and another
cluster of size r2 generates a cluster of size r and is a nonnegative function satisfying
A(r; r1, r2) = A(r; r2, r1), r, r1, r2 ∈ [0,∞[, (1.5a)
∞∫
0
r A(r; r1, r2) dr = r1 + r2, r1, r2 ∈ [0,∞[. (1.5b)
The structure of Eq. (1.3) can be related to the large class of bilinear Generalized Kinetic
Models. A general class of bilinear systems of Boltzmann-like integro-differential equa-
tions describing the dynamics of individuals undergoing kinetic (stochastic) interactions
was proposed and analyzed in [6]. These equations can model interactions between pairs
of individuals of various populations at the mesoscopic level. The class of equations in
[6] can be regarded as a generalization of the Jäger and Segel kinetic model [7], as well
as those of Arlotti and Bellomo [8, 9], Arlotti, Bellomo and Lachowicz [10], Lachowicz
and Wrzosek [4], Geigant, Ladizhansky and Mogilner [11]. In the literature these kind of
models are referred to as the GKM — Generalized Kinetic Models. Paper [6] was a first
step in the description of the mathematical properties of a large class of General Kinetic
Models. It provides some existence and uniqueness theorems for the GKM, discusses
its equilibrium solutions, and studies its diffusive limit. In [6] the existence of unstable
equilibrium solutions which are inhomogeneous was proved. The case when only homo-
geneous equilibrium solutions exist was specified. Under suitable scaling it was proved
that the one-dimensional version of the GKM is asymptotically equivalent to the nonlin-
ear porous medium equation also used in mathematical biology as the model for density
dependent population dispersal. In [12] research perspectives for finding possible tran-
sitions from the mesoscopic to the macroscopic level were presented. In [13] the results
on the relationships between the microscopic and the mesoscopic levels and then in [14]
between the microscopic and the macroscopic levels are proved.
Condition (1.5b) ensures that the total volume is preserved during the coagulation
reaction. In fact we have
∞∫
0
QGC [f ]φdr =
=
∞∫
0
r1∫
0
∞∫
0
A(r; r1, r2)φ(r) dr
− φ(r1) − φ(r2)
α(r1, r2)f(r1)f(r2) dr2 dr1
(1.6)
for any test function φ.
In [3] a family of generalized coagulation equations connecting the Smoluchowski
and the OHS equations was introduced. For ε ∈ ]0, 1] and r1, r2 ∈ [0,∞[ it was
defined
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
772 M. LACHOWICZ
Aε(r; r1, r2) = δ
(
r − r1 ∨ r2 − εr1 ∧ r2
)
+ (1 − ε)δ
(
r − r1 ∧ r2
)
(1.7)
where
r1 ∨ r2 = max
{
r1, r2
}
, r1 ∧ r2 = min
{
r1, r2
}
,
δ is the Dirac distribution,
αε(r1, r2) =
α(r1, r2)
ε
. (1.8)
Putting Aε instead of A and aε instead of a in (1.6) we set QGC = Qε. Thus we
have
∂tfε = Qε[f ], t > 0, r ∈ [0,∞[ , (1.9)
where
Qε[f ](r) =
1
ε
r
1+ε∫
0
α(r − εr1, r1)f(r − εr1)f(r1) dr1 +
+
1 − ε
ε
f(r)
∞∫
r
α(r, r1)f(r1) dr1 −
1
ε
f(r)
∞∫
0
α(r, r1)f(r1) dr1 . (1.10)
We have
∞∫
0
Qε[f ]φdr =
=
∞∫
0
r1∫
0
(φ(r1 + εr2) − φ(r1)
ε
− φ(r2)
)
α(r1, r2)f(r1)f(r2) dr1 dr2, (1.11)
for any test function φ.
It is straightforward to see that the choice ε = 1 yields (1.1) in the case J = [0,∞[.
On the other hand in [3] the convergence of the weak solution fε to Eq. (1.9) with the
initial datum f (0) towards a weak solution to Eq. (1.2) with the same initial datum was
proved.
The models presented in this section can be related to the level of statistical description
of test-particle (a “mesoscopic” description). Using the general approach of [13] we are
going to show that the solutions of the mesoscopic models above can be approximated
by the solutions of appropriate systems describing the coagulation process of particles
undergoing stochastic interactions (a “microscopic” description) in terms of stochastic
semigroups.
The mathematical relationships between the particle systems and various Smoluchow-
ski coagulation equations were studied in a number of papers — see [2, 15 – 17], and
references therein. Our approach however is simpler and makes use of a general theory
developed in [13, 18, 19].
2. Particle systems — microscopic description. Here we follow the idea of [20]
and [13]. We show that the solutions of Eq. (1.9) can be approximated by solutions of
(linear) equations describing the dynamics of a suitable system of interacting particles.
For simplicity of notation we consider here only the continuous case J = [0,∞[
but the discrete case J = N can be treated in the same way.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
STOCHASTIC SEMIGROUPS AND COAGULATION EQUATIONS 773
Consider a system composed of N interacting particles. Every particle n ∈
∈ {1, 2, ..., N} is characterized by un = (rn, un), where rn ∈ J characterizes the
size of the n-particle and un ∈ U — its inner state. Here J = [0,∞[ and U = [0, 1].
Actually un plays an auxiliary rôle but it may be related to the measure of “coagulation
intensity” of the particle. The n-particle interacts with the m-particle and the interaction
take place at random times. After the interaction both particles may merge or/and change
their inner state.
Consider the Markov process of N-particles with infinitesimal generator given by
ΛNφ(r1, u1, . . . , rN , uN ) =
=
1
N ε
∑
1≤n,m≤N
n�=m
α(rn, rm)um
(∫
U
(
χ(rm < rn)B1(v, un) ×
×φ
(
r1, u1, . . . , rn−1, un−1, rn + εrm, v, rn+1, un+1, . . . , rN , uN
)
+
+χ(rn < rm)B2(v, un) ×
×φ
(
r1, u1, . . . , rn−1, un−1, rn, v, rn+1, un+1, . . . , rN , uN
))
dv −
−φ
(
r1, u1, . . . , rN , uN
))
, rj ∈ J , uj ∈ U , j = 1, . . . , N, (2.1)
where φ is an appropriate test function, ε ∈ ]0, 1[, B1 and B2 are measurable functions
such that ∫
U
Bi(u, u1) du = 1,
∫
U
uBi(u, u1) du = u1κi,
for a.a. u1 ∈ U , i = 1, 2, κ1 = 1, κ2 = 1 − ε,
(2.2)
χ(true) = 1, χ(false) = 0 .
In the present paper we assume rather restrictive case
0 ≤ α(u,u1) ≤ ca, for a.a. u,u1 ∈ J × U , (2.3)
where ca is a positive constant. However the general case of unbounded α can be treated
by the usual approximation methods (cf. [3]).
Assume that the system is initially distributed according to the probability density
FN ∈ L
(N)
1 , where L
(N)
1 is the space equipped with the norm
‖f‖
L
(N)
1
=
∫
J×U
. . .
∫
J×U
|f(u1, . . . ,uN )| du1 . . . duN .
The time evolution is described by the probability density
fN (t) = exp
(
tΛ∗
N
)
FN . (2.4)
It satisfies
(
in L
(N)
1
)
∂tf
N = Λ∗
NfN , fN
∣∣∣
t=0
= FN , (2.5)
where
Λ∗
Nf(r1, u1, . . . , rN , uN ) =
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
774 M. LACHOWICZ
=
1
N ε
∑
1≤n,m≤N
n�=m
um
(∫
U
(
χ
(
(1 + ε)rm < rn
)
×
×α(rn − εrm, rm)B1(un, v) ×
×f
(
r1, u1, . . . , rn−1, un−1, rn − εrm, v, rn+1, un+1, . . . , rN , uN
)
+
+χ(rn < rm)α(rn, rm)B2(un, v) ×
× f
(
r1, u1, . . . , rn−1, un−1, rn, v, rn+1, un+1, . . . , rN , uN
))
dv −
−α(rn, rm)f
(
r1, u1, . . . , rN , uN
))
, rj ∈ J , uj ∈ U , j = 1, . . . , N, (2.6)
Under the assumptions (2.3) the operator Λ∗
N is a bounded linear operator in the
space L
(N)
1 . Therefore the Cauchy problem (2.5) has a unique solution (2.4) in L
(N)
1 for
all t ≥ 0. Moreover, by standard argument we see that the solution is nonnegative for
nonnegative initial data and the L
(N)
1 -norm is conserved
‖fN (t)‖
L
(N)
1
= ‖FN‖
L
(N)
1
= 1 for t > 0 . (2.7)
We assume that all functions are symmetric
fN (u1, . . . ,uN ) = fN (up1 , . . . ,upN
), (2.8)
for a.a. u1, . . . , uN in J × U and for any permutation {p1, . . . , pN} of the set
{1, . . . , N}. Note that if f is invariant with respect to permutations of variables then
Λ∗
Nf is invariant too.
We introduce the s-individual marginal density ( 1 ≤ s < N )
fN,s(u1, . . . ,us) =
∫
(J×U )N−s
fN (u1, . . . ,uN ) dus+1 . . . duN ,
and fN,N = fN .
The function fN satisfies Eq. (2.5) iff fN,s satisfy the following finite hierarchy of
equations
∂tf
N,s =
s
N
Λ∗
sf
N,s +
N − s
N
Θs+1f
N,s+1, s = 1, . . . , N, (2.9)
where
(Θs+1f)(r1, u1, . . . , rs, us) =
=
1
ε
s∑
n=1
∫
J×U
us+1
(∫
U
(
χ
(
(1 + ε)rs+1 < rn
)
×
×α(rn − εrs+1, rs+1)B1(un, v) ×
×f
(
r1, u1, . . . , rn−1, un−1, rn − εrs+1, v, rn+1, un+1, . . . , rs+1, us+1
)
+
+χ(rn < rs+1)α(rn, rs+1)B2(un, v) ×
×f
(
r1, u1, . . . , rn−1, un−1, rn, v, rn+1, un+1, . . . , rs+1, us+1
))
dv −
−α(rn, rs+1)f
(
r1, u1, . . . , rs+1, us+1
))
drs+1 dus+1. (2.10)
Taking N sufficiently large we may expect that the solution of the finite hierarchy
(2.9) approximates solution of the following infinite hierarchy of equations
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
STOCHASTIC SEMIGROUPS AND COAGULATION EQUATIONS 775
∂tf
s = Θs+1f
s+1, s = 1, 2, . . . . (2.11)
The integral versions of hierarchies (2.9) and (2.11) read
fN,s(t) = FN,s +
s
N
t∫
0
Λsf
N,s(t1) dt1 +
N − s
N
t∫
0
Θs+1f
N,s+1(t1) dt1, (2.12)
s = 1, . . . , N,
and
fs(t) = F s +
t∫
0
Θs+1f
s+1(t1) dt1, s = 1, 2, . . . , (2.13)
respectively.
Definition 2.1. An admissible hierarchy
{
fs
}
s=1,2,3,...
is a sequence of functions
fs satisfying (for s = 1, 2, . . . ):
(i) fs is a probability density on
(
J × U
)s;
(ii) fs(u1, . . . ,us) = fs(ur1 , . . . ,urs
) for a.a. u1, . . . , us in J × U and for
any permutation {r1, . . . , rs} of the set {1, . . . , s};
(iii) fs(u1, . . . ,us) =
∫
J×U
fs+1(u1, . . . ,us+1) dus+1 for a.a. u1, . . . , us in
J × U .
By (2.3) we have
‖Θs+1f‖L
(s)
1
≤ c1s
ε
‖f‖
L
(s+1)
1
, (2.14)
and ∫
(J×U)s
(Θs+1f)(u1, . . . ,us) du1 . . . dus = 0, (2.15)
for all f ∈ L
(s+1)
1 and s = 1, 2, . . . , where c1 is a constant. Moreover,∫
J×U
(Θs+1f)(u1, . . . ,us) dus = (Θsf̂)(u1, . . . ,us−1), (2.16)
for all f ∈ L
(s+1)
1 and s = 1, 2, . . . , where
f̂(u1, . . . ,us) =
∫
J×U
f(u1, . . . ,us−1,us+1,us) dus+1.
We have the following theorem.
Theorem 2.1. Let {F s}s=1,2,... be an admissible hierarchy. Then, for all t > 0,
there exists a unique hierarchy {fs(t)}s=1,2,..., such that fs(t) ∈ L
(s)
1 , s = 1, 2, . . . ,
that is a solution of Eq. (2.13) with initial data fs(0) = F s, s = 1, 2, . . . . Moreover
{fs(t)}s=1,2,..., for all t > 0, is an admissible hierarchy.
The proof is the same as the proof of Theorem 3.1 in [13].
3. The main result — asymptotic relationship. We assume now that the Markov
process starts with chaotic (i.e. factorized) probability density and we consider the hierar-
chy (2.13) with initial data
F s = F ⊗ . . .⊗ F = (F )s⊗, s = 1, 2, . . . , (3.1)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
776 M. LACHOWICZ
i.e. s-fold outer product of a probability density F defined on J × U . We may see that
the propagation of chaos is held and the solution fs(t) to Eq. (2.13) is the s-product of
solution f(t) of the following bilinear equation:
∂tf(r, u) = Γε[f ](r, u), (r, u) ∈ J × U , (3.2)
Γε[f ](r, u) =
1
ε
r
1+ε∫
0
1∫
0
α(r − εr1, r1)B1(u, v)f(r − εr1, v) f̄(r1) dv dr1 +
+
1
ε
∞∫
r
1∫
0
α(r, r1)B2(u, v)f(r, v)f̄(r1) dv dr1 −
1
ε
∞∫
0
α(r, r1)f(r, u)f̄(r1) dr1, (3.3)
where
f̄(r) =
1∫
0
uf(r, u) du . (3.4)
Equation (3.2) can be related to the class of Generalized Kinetic Models. The existence
theory for Eq. (3.2) in the L
(1)
1 setting is standard. From Theorem 2.1 we immediately
have the following corollary.
Corollary 3.1. Let F be a probability density on J × U . Then, for each t0 > 0,
there exists an admissible hierarchy {fs}s=1,2,... such that
(i) it is a unique solution of Eq. (2.13) with chaotic initial data (3.1),
(ii) fs(t) is chaotic
fs(t) =
(
f(t)
)s⊗
,
for all 0 < t ≤ t0 and s = 1, 2, . . . , where f(t) is the unique solution in L
(1)
1 of
Eq. (3.2) with the initial datum F. Moreover, f̄(t), given by (3.4), is a unique solution
of Eq. (1.9) with the initial datum F̄ .
We may now formulate the main result, namely the theorem stating that the solution
of Eq. (1.9) is approximated by the solutions of Eq. (2.5) as N → ∞ (the proof follows
the line of [20], see [13]).
Theorem 3.1. Let F be a probability density on J × U . Then, for each t0 > 0,
there exists N0 such that for N ≥ N0
sup
[0,t0]
‖fN,1 − f‖
L
(1)
1
≤ c2
Nη
,
where the nonnegative functions fN,s ∈ L
(s)
1 , s = 1, . . . , N, form the unique solution
of Eq. (2.12) corresponding to the initial datum
fN,s(0) = (F )s⊗, s = 1, . . . , N ;
f ∈ L
(1)
1 is the unique, nonnegative solution of Eq. (3.2) corresponding to the initial
datum F ; η and c2 are positive constants that depend on t0.
Corollary 3.2. Under the assumptions of Theorem 3.1
sup
t∈[0,t0]
∫
J
∣∣∣f̄N,1(t, r) − f̄(t, r)
∣∣∣ dr ≤ c2
Nη
,
where f̄(t), given by (3.4), is a unique solution of Eq. (1.9) corresponding to the initial
datum F̄ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
STOCHASTIC SEMIGROUPS AND COAGULATION EQUATIONS 777
Corollary 3.2 shows that the solution of the coagulation bilinear integro-differential
equations can be approximated by the solutions of linear equations describing the stochas-
tic system of individuals — provided that the parameters of the stochastic system are
suitably chosen.
The estimates are not optimized. One can hope that some of them can be improved to
make them uniform with respect to t0.
There are some possible generalizations of the above result. With slight modifications
one can consider k interacting clusters (another approach can be find in [21] and [22]).
Acknowledgment. Work supported in part by the EU Programme under contract
MRTN-CT-2004-503661, “Modeling, Mathematical Methods and Computer Simulation
of Tumour Growth and Therapy”. The work bases on Author’s Lecture on the Conference
Recent Trends in Kinetic Theory and its Applications, Kiev, May 11 – 15, 2004.
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Received 21.09.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
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| id | umjimathkievua-article-3643 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:46:19Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2b/8433349b701c2600a8426755c423262b.pdf |
| spelling | umjimathkievua-article-36432020-03-18T20:00:55Z Stochastic Semigroups and Coagulation Equations Стохастичні напівгрупи i рівняння коагуляції Lachowicz, M. Лачович, М. A general class of bilinear systems of discrete or continuous coagulation equations is considered. It is shown that their solutions can be approximated by the solutions of appropriate stochastic systems describing the coagulation process in terms of stochastic semigroups. Розглянуто загальний клас білінійних систем дискретних або неперервних рівнянь коагуляції. Показано, що їх розв'язки можуть бути наближені розв'язками стохастичних систем, які описують процес коагуляції через стохастичні напівгрупи. Institute of Mathematics, NAS of Ukraine 2005-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3643 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 6 (2005); 770–777 Український математичний журнал; Том 57 № 6 (2005); 770–777 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3643/4016 https://umj.imath.kiev.ua/index.php/umj/article/view/3643/4017 Copyright (c) 2005 Lachowicz M. |
| spellingShingle | Lachowicz, M. Лачович, М. Stochastic Semigroups and Coagulation Equations |
| title | Stochastic Semigroups and Coagulation Equations |
| title_alt | Стохастичні напівгрупи i рівняння коагуляції |
| title_full | Stochastic Semigroups and Coagulation Equations |
| title_fullStr | Stochastic Semigroups and Coagulation Equations |
| title_full_unstemmed | Stochastic Semigroups and Coagulation Equations |
| title_short | Stochastic Semigroups and Coagulation Equations |
| title_sort | stochastic semigroups and coagulation equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3643 |
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