doi: 10.17951/a.2016.70.1.47
ANNALES
UNIVERSITATIS MARIAE CURIE-SKŁODOWSKA
LUBLIN – POLONIA
VOL. LXX, NO. 1, 2016
SECTIO A
47–74
KRZYSZTOF PILORZ
A kinetic equation for repulsive coalescing
random jumps in continuum
Abstract. A continuum individual-based model of hopping and coalescing
particles is introduced and studied. Its microscopic dynamics are described
by a hierarchy of evolution equations obtained in the paper. Then the passage
from the micro- to mesoscopic dynamics is performed by means of a Vlasovtype scaling. The existence and uniqueness of solutions of the corresponding
kinetic equation are proved.
1. Introduction. In this paper, we introduce and study the dynamics of
an infinite system of particles located in Rd , which jump and merge (coalesce). Both jumping and coalescing are repulsive. The proposed model is
individual based, which means that the description of its Markov dynamics
is performed in terms of random changes of states of individual particles.
In the proposed model, such changes include: (a) the particle located at a
given x ∈ Rd changes its position to y ∈ Rd (jumping); (b) two particles,
located at x ∈ Rd and y ∈ Rd , merge into a single particle located at z ∈ Rd
(coalescing). The rates of these events depend on the configuration of all
particles. Similar single type population models are used to describe predation in ecology, see, e.g., [6]. The model proposed can be viewed as an
extension of the Kawasaki model with repulsion studied in [5] where only
random jumps with repulsion are taken into account. To the best of our
2010 Mathematics Subject Classification. 60K35, 35Q83, 82C22.
Key words and phrases. Coalescence, coagulation, hopping particles, individual-based
model, configuration spaces, infinite particle system, microscopic dynamics, Vlasov scaling, kinetic equation.
48
K. Pilorz
knowledge, the microscopic modeling of this kind of merging is performed
here for the first time. Previous theories, see, e.g., [1, 2, 3, 4, 11, 13, 14],
deal with the particle densities and hence do not take into account the corpuscular structure of the system or, see, e.g., [15, 16], while introducing
individual-based models, describe only finite systems with coalescence between two particles dependent on characteristics of the particles involved
and the mass of entire population. Interactions between particles are not
taken into account. The most important aspects of the model proposed
in this paper are: (a) the coalescence rate depends on the neighbourhood
of the coalescing particles; (b) infinite systems of particles are described.
However, the particle mass is not taken into account, which is supposed to
be done in an extension of the present model.
The state space of the system considered is the configuration space, that
is, the set of all locally finite subsets of Rd
n
o
Γ = Γ(Rd ) = γ ⊂ Rd : γ ∩ Λ is finite for every compact Λ ⊂ Rd .
It can be given a measurability structure. Take
F = σ(Γn,Λ : n ∈ N0 , Λ ⊂ Rd - compact),
that is the smallest σ-field generated by the cylinders
Γn,Λ = {γ ∈ Γ : |γ ∩ Λ| = n}.
This allows one to consider probability measures on standard Borel space
(Γ, F) as states of the system. To characterize them one uses observables,
which are appropriate functions F : Γ → R. For an observable F and a
state µ, the number
Z
F dµ
Γ
is the µ-expected value of F . Then the evolution of states µ0 7→ µt can
be described via the dual evolution F0 7→ Ft based on the following duality
relation
Z
Z
F0 dµt =
Ft dµ0 ,
t > 0.
Γ
Γ
The evolution of observables is obtained in turn from the Kolmogorov equation
d
Ft = LFt ,
dt
Ft |t=0 = F0 ,
A kinetic equation for repulsive coalescing random jumps...
49
in which the “operator” L characterizes the model, see [7, 8, 9, 10] for more
details. In the proposed model, it has the following form
X Z
c̃1 (x, y; z; γ) F γ\{x, y} ∪ z − F (γ) dz
LF (γ) =
(1.1)
{x,y}⊂γ Rd
+
X Z
x∈γ
c̃2 (x; y; γ) F γ\x ∪ y − F (γ) dy.
Rd
The first term of L describes the coalescence occurring with intensity
c̃1 (x, y; z; γ). The particles located at x and y merge into a new particle
located at a point z. The second term describes the jump of the particle
located at x to a point y with intensity c̃2 (x; y; γ). The model with L consisting of the second term only is the Kawasaki model studied in [5]. The
kernels c̃1 and c̃2 take into account also the influence of the whole configuration, which is supposed to be repulsive, see (3.2) below. Note that L is
linear and independent of time and hence we deal with Markov dynamics.
In the present research, we follow the statistical approach, see, e.g., [5,
7, 8, 9, 10], in which the dynamics of the model are described by means of
that of the corresponding correlation functions obtained from the following
Cauchy problem
d
kt = L∆ kt , kt=0 = k0 ,
dt
in which the “operator” L∆ is related to L in a certain way. In Section 3,
we calculate L∆ following the scheme developed in [10]. Usually, equations
for kt are studied in scales of the corresponding Banach spaces. However, as
the structure of L∆ obtained below is too complicated, in this work we do
not study this equation, which is supposed to be done in a separate work.
Instead, in Section 4 we consider a simplified version obtained by means of a
Vlasov-type scaling procedure developed in, e.g., [8], which is equivalent to
passing to the so-called mesoscopic description. In particular, we informally
obtain the kinetic equation and study its local solutions in an appropriate
Banach space, showing their existence and uniqueness. In the next section,
we introduce necessary notions and technical tools.
2. Basic notions and tools. In this section we introduce basic notions
and tools used for proving the results in the following sections. We give
only a short description with references to the corresponding sources.
Note that each element γ ∈ Γ is at most countable without finite limiting
points. Γ is endowed with the vague topology, which is the weakest topology
that makes continuous the mappings
X
γ→
f (x)
x∈γ
50
K. Pilorz
for all continuous compactly supported functions f : Rd → R. For the
general discussion on configuration spaces we recommend [7, 9].
One can also consider the space of all finite configurations
n
o
Γ0 = Γ0 (Rd ) = η ∈ Γ : η is finite ,
By Bbs (Γ0 ) we denote the set of all bounded measurable functions G :
Γ0 → R such that for some N ∈ N0 and compact Λ ⊂ Rd , we have
G(n) (x1 , . . . , xn ) = 0, whenever n ≥ N or one of the arguments xi ∈
/ Λ. Then
the K-transform is defined as follows. For any G ∈ Bbs (Γ0 ), KG : Γ → R is
X
G(η),
(2.1)
(KG)(γ) =
ηbγ
where η b γ means that η is a finite sub-configuration of γ. Obviously, the
K-transform is linear. It acts to the set of all measurable cylinder functions
F : Γ → R, that is satisfying F (γ) = F (γΛ ) for some compact Λ ⊂ Rd ,
where γΛ = γ ∩ Λ. See [12] for more details. It is also invertible with the
inverse given by
X
(K −1 F )(η) =
(−1)|η\ξ| F (ξ).
ξ⊂η
For G1 , G2 ∈ Bbs (Γ0 ), it is known that
(KG1 ) · (KG2 ) = K(G1 ? G2 ),
(2.2)
where G1 ? G2 is the “convolution” given by the formula
X
X
(2.3)
(G1 ? G2 )(η) =
G1 (ξ)
G2 (η\ξ ∪ ζ) ∈ Bbs (Γ0 ).
ξ⊂η
ζ⊂ξ
Denote
e(f, γ) =
(2.4)
Y
f (x).
x∈γ
For a measurable compactly supported function f : Rd → R, the following
holds
(2.5)
K e(f, ·) (γ) = e(1 + f, γ).
A probability measure µ on (Γ, B(Γ)) is said to have finite local moments
of all orders if for any n ∈ N and a bounded Borel Λ ⊂ Rd ,
Z
|γΛ |n µ(dγ) < ∞,
Γ0
where |η| stands for the cardinality of η ∈ Γ0 . For such a measure µ, one
can define a correlation measure, ρµ , on (Γ0 , B(Γ0 )) by
Z
Z
G(η)ρµ (dη) = (KG)(γ)µ(dγ), G ∈ Bbs (Γ0 ).
Γ0
Γ
A kinetic equation for repulsive coalescing random jumps...
51
By λ we denote the Lebesgue–Poisson measure on (Γ0 , B(Γ0 )), that is,
the correlation measure for the homogeneous Poisson measure with unit intensity. The Lebesgue–Poisson measure is uniquely defined by the following
formula
Z
Z
∞
X
1
(0)
G(η)λ(dη) = G +
(2.6)
G(n) (x1 , x2 , . . . , xn )dx1 dx2 · · · dxn ,
n!
n=1
Γ0
(Rd )n
which has to hold for all G ∈ Bbs (Γ0 ). The Minlos lemma (see, e.g., [7, eq.
(2.2)]), states that
Z
Z
. . . G(η1 ∪ η2 ∪ · · · ∪ ηn )H(η1 , η2 , . . . , ηn )λ(dη1 )λ(dη2 ) . . . λ(dηn )
(2.7)
Γ0
Γ0
Z
=
G(η)
X
H(η1 , η2 , . . . , ηn )λ(dη),
Γ0
where n is a positive integer, G : Γ0 → R, H : (Γ0 )n → R are positive and
measurable and the sum is taken over all n-part partitions (η1 , . . . , ηn ) of η,
where parts being empty configurations are also considered. For n = 2, we
can rewrite (2.7) in the following form
Z Z
Z
X
(2.8)
G(η ∪ ξ)H(η, ξ)λ(dη)λ(dξ) = G(η)
H(ξ, η\ξ)λ(dη).
Γ0 Γ0
Γ0
ξ⊂η
By taking
(
h(x, η2 ), η1 = {x},
H(η1 , η2 ) =
0,
|η1 | =
6 1
and using (2.6) we obtain the following special case of the Minlos lemma
Z Z
Z X
(2.9)
G(η ∪ x)h(x, η)dxλ(dη) =
G(η)h(x, η\x)λ(dη).
Γ0 Rd
Γ0
x∈η
Analogously, for
(
h(x, y, η3 ),
H(η1 , η2 , η3 ) =
0,
η1 = {x}, η2 = {y},
|η1 | =
6 1 or |η2 | =
6 1,
we have
1
2
(2.10)
Z Z Z
G(η ∪ {x, y})h(x, y, η)dxdyλ(dη)
Γ0 Rd Rd
Z
=
Γ0
X
{x,y}⊂η
G(η)h(x, y, η\{x, y})λ(dη).
52
K. Pilorz
3. Dynamics of the correlation functions. In this section, we follow
the approach described in [10], obtaining operator L∆ acting on correlation
functions, corresponding to (1.1). First, by using the K-transform (2.1) we
obtain the operator L̂ (see (3.4) below) and then L∆ related to L̂ by
Z
Z
(L̂G)(η)k(η)λ(dη) = G(η)(L∆ k)(η)λ(dη).
(3.1)
Γ0
Γ0
Recall, that
L = L1 + L2 ,
where
L1 F (γ) =
X
Z
c̃1 (x, y; z; γ) F γ\{x, y} ∪ z − F (γ) dz
{x,y}⊂γ Rd
and
L2 F (γ) =
X Z
x∈γ
c̃2 (x; y; γ) F γ\x ∪ y − F (γ) dy.
Rd
We assume that (see notion (2.4) for definition of e(f, γ))
(3.2)
c̃1 (x, y; z; γ) = c1 (x, y; z)e(t(1)
z , γ\{x, y}),
c̃2 (x; y; γ) = c2 (x; y)e(t(2)
y , γ\x),
with
−φ1 (z−u)
−φ2 (y−u)
t(1)
, t(2)
.
z (u) = e
y (u) = e
and c1 , c2 , φ1 , φ2 being positive real functions.
The above assumptions are important part of the model, as they guarantee that both jumping and coalescing are repulsive. Indeed, if γ\{x, y}
is nonempty and at least one of its elements, say u, is such that z − u lies
in the support of φ1 , then the coalescence rate c̃1 (x, y; z; γ) is smaller than
c1 (x, y; z). If there are no such elements in γ\{x, y}, then c̃1 (x, y; z; γ) =
c1 (x, y; z). It means that the particles which are not involved in the coalescence, may influence it and this influence is repulsive. The analogous
reasoning is valid for the jump rate c̃2 (x; y; γ).
We add some technical assumptions listed below. Namely, let
(3.3)
1
c̃1 (x, y; z; γ) = (KCx,y;z
)(γ\{x, y}),
2
c̃2 (x; y; γ) = (KCx;y
)(γ\x)
A kinetic equation for repulsive coalescing random jumps...
53
1
2 . We discuss the form of these functions later in
for some Cx,y;z
and Cx;y
this section. Additionally,
c1 (x, y; z) = c1 (y, x; z),
Z
c1 (x1 , x2 ; x3 )dxi dxj = hc1 i < ∞, i, j = 1, 2, 3, i 6= j,
(Rd )2
Z
Z
c2 (x; y)dy = hc2 i < ∞,
c2 (x; y)dx =
Rd
Rd
Z
Z
φ2 (x)dx = hφ2 i < ∞.
φ1 (x)dx = hφ1 i < ∞,
Rd
Rd
Suppose that F = KG, where G : Γ0 → R. Then, by writing K L̂G = LF ,
we define
L̂ = K −1 LK.
(3.4)
By the properties of the K-transform we derive an explicit formula for L̂.
Proposition 3.1. L̂ defined as above has the following form
Z X h
i
1
1
L̂G(η) =
Cx,y;z
? Hx,y;z
(η\{x, y})dz
Rd {x,y}⊂η
+
Z Xh
Rd
i
2
2
Cx;y
? Hx;y
(η\x)dy,
x∈η
where
(3.5)
1
Hx,y;z
(η) = G(η ∪ z) − G(η ∪ x) − G(η ∪ y) − G(η ∪ {x, y}),
2
Hx;y
(η) = G(η ∪ y) − G(η ∪ x).
The next step is to pass with the action of the operator L̂ to the correlation functions, i.e. to obtain L∆ . The latter is defined by the pairing
hhL̂G, kii = hhG, L∆ kii, see (3.1). Let us consider the integral on the left
hand side of equation (3.1). Using the Minlos lemma (2.7), we can transform
it so that we can obtain L∆ .
Proposition 3.2. L∆ defined as above is of the form
∆
L∆ = L∆
1 + L2 ,
54
K. Pilorz
where
L∆
1 k(η)
1
=
2
Z
Z X
(Rd )2 Γ0
c1 (x, y; z)k(η\z ∪ ξ ∪ {x, y})
z∈η
(1)
× e(t(1)
z − 1, ξ)e(tz , η\z)λ(dξ)dxdy
Z Z X
1
c1 (x, y; z)k(η ∪ ξ ∪ y)
−
2
x∈η
(Rd )2 Γ0
(1)
× e(t(1)
z − 1, ξ)e(tz , η\x)λ(dξ)dydz
Z Z X
1
−
c1 (x, y; z)k(η ∪ ξ ∪ x)
2
y∈η
(Rd )2 Γ0
(1)
× e(t(1)
z − 1, ξ)e(tz , η\y)λ(dξ)dxdz
Z Z X
−
c1 (x, y; z)k(η ∪ ξ)
Rd Γ0 {x,y}⊂η
(1)
× e(t(1)
z − 1, ξ)e(tz , η\{x, y})λ(dξ)dz
and
L∆
2 k(η) =
Z Z X
R d Γ0
k(η\y ∪ ξ ∪ x)c2 (x; y)
y∈η
(2)
× e(t(2)
y − 1, ξ)e(ty − 1, η\y)λ(dξ)dx
Z Z
X
−
k(η ∪ ξ)
c2 (x; y)
x∈η
R d Γ0
×
Y
(2)
e(t(2)
y − 1, ξ)e(ty − 1, η\x)λ(dξ)dy.
u∈ξ
Proof of Proposition 3.1. First let us rewrite the operator L in a more
convenient form. Using (3.3) and recalling that any configuration treated
as a subset of Rd is Lebesgue measure-zero as it is countable, we have
Z h
X
1
L1 F (γ) =
KCx,y;z
(·) KG · ∪z
{x,y}⊂γ Rd \γ
− KG · ∪{x, y}
i
γ\{x, y} dz.
Observe that for any ξ ∈ Γ, x, y, z ∈
/ ξ we have
i
h
i
X
Xh
KG(ξ ∪ z) =
G(η) =
G(η) + G(η ∪ z) = K G(·) + G(· ∪ z) (ξ)
ηbξ∪z
ηbξ
A kinetic equation for repulsive coalescing random jumps...
55
and analogously
h
i
KG(ξ ∪ {x, y}) = K G(·) + G(· ∪ x) + G(· ∪ y) + G(· ∪ {x, y}) (ξ).
Using linearity of the K-transform and above observations, we obtain
X
L1 F (γ) =
Z h
1
KCx,y;z
(·)K G(· ∪ z) − G(· ∪ x)
{x,y}⊂γ Rd
i − G(· ∪ y) − G(· ∪ {x, y}) (·) (γ\{x, y})dz.
Considering the second part of the operator, we have
L2 F (γ) =
X Z
x∈γ
h
i
2
KCx;y
(γ\x) KG(γ\x ∪ y) − KG(γ) dy
Rd
h
i X Z 2
=
KCx;y
(·)K G(· ∪ y) − G(· ∪ x) (·) (γ\x)dy.
x∈γ
Rd
Using notion (3.5) and property (2.2) of the product of K-transforms, we
derive
h
i
X Z
1
1
L1 F (γ) =
K Cx,y;z
? Hx,y;z
(γ\{x, y})dz,
{x,y}⊂γ Rd
L2 F (γ) =
X Z
x∈γ
h
i
2
2
K Cx;y
? Hx;y
(γ\x)dy.
Rd
Therefore
LF (γ) =
X
Z
h
i
1
1
K Cx,y;z
? Hx,y;z
(γ\{x, y})dz
{x,y}⊂γ Rd
+
X Z
x∈γ
h
i
2
2
(γ\x)dy.
K Cx;y
? Hx;y
Rd
Recalling the definition (3.4) of the operator L̂ and denoting
L̂1 G(η) = K −1 L1 F (η),
L̂2 G(η) = K −1 L2 F (η),
56
K. Pilorz
we obtain
L̂1 G(η) =
X
=
h
i
1
1
K Cx,y;z
? Hx,y;z
(ξ\{x, y})dz
{x,y}⊂ξ Rd
ξ⊂η
Z
Z
X
(−1)|η\ξ|
X
h
i
1
1
(−1)|η\{x,y}\ξ| K Cx,y;z
? Hx,y;z
(ξ)dz
X
Rd {x,y}⊂η ξ⊂η\{x,y}
Z
=
h
i
1
1
K −1 K Cx,y;z
? Hx,y;z
(η\{x, y})dz
X
Rd {x,y}⊂η
Z
=
i
X h
1
1
Cx,y;z
? Hx,y;z
(η\{x, y})dz
Rd {x,y}⊂η
and analogously
L̂2 G(η) =
=
h
i
X
X Z
2
2
(−1)|η\ξ|
K Cx;y
? Hx;y
(ξ\x)dy
ξ⊂η
x∈ξ
Z X X
h
i
2
2
(−1)|η\x\ξ| K Cx;y
? Hx;y
(ξ)dy
Rd
=
x∈η ξ⊂η\x
Z X
Rd
Rd
h
i
2
2
K −1 K Cx;y
? Hx;y
(η\x)dy
x∈η
Z Xh
i
2
2
Cx;y
? Hx;y
(η\x)dy.
=
Rd
x∈η
Therefore
Z
i
X h
1
1
Cx,y;z
? Hx,y;z
(η\{x, y})dz
L̂G(η) =
Rd {x,y}⊂η
+
Z Xh
Rd
2
Cx;y
?
2
Hx;y
i
(η\x)dy.
x∈η
Proof of Proposition 3.2. Using the special case (2.10) of the Minlos
lemma and proposition (3.1), we have
Z
(L̂1 G)(η)k(η)λ(dη)
Γ0
Z Z
=
X h
Γ0 Rd {x,y}⊂η
i
1
1
Cx,y;z
? Hx,y;z
(η\{x, y})k(η)dzλ(dη)
A kinetic equation for repulsive coalescing random jumps...
Z
1
=
2
Z h
57
i
1
1
Cx,y;z
? Hx,y;z
(η)k(η ∪ {x, y})λ(dη)dxdydz.
(Rd )3 Γ0
Recalling the definition (2.3) of the convolution ? and using the Minlos
lemma in the form (2.8) twice, we obtain
Z
Z Z X
X
1
1
1
(L̂1 G)(η)k(η)λ(dη) =
Cx,y;z
(ξ)
Hx,y;z
(η\ξ ∪ ζ)
2
ξ⊂η
(Rd )3 Γ0
Γ0
ζ⊂ξ
× k(η ∪ {x, y})λ(dη)dxdydz
Z Z Z
X
1
1
1
Cx,y;z
(ξ)
Hx,y;z
(η ∪ ζ)
=
2
ζ⊂ξ
(Rd )3 Γ0 Γ0
× k(η ∪ ξ ∪ {x, y})λ(dη)λ(dξ)dxdydz
Z Z Z Z
1
1
1
=
Cx,y;z
(ξ ∪ ζ)Hx,y;z
(η ∪ ζ)
2
(Rd )3 Γ0 Γ0 Γ0
× k(η ∪ ξ ∪ ζ ∪ {x, y})λ(dη)λ(dξ)λ(dζ)dxdydz.
Using again the Minlos lemma (2.8), but in the opposite direction, we have
Z Z Z X
Z
1
1
1
Cx,y;z
(ξ ∪ ζ)Hx,y;z
(η)
(L̂1 G)(η)k(η)λ(dη) =
2
ζ⊂η
(Rd )3 Γ0 Γ0
Γ0
× k(η ∪ ξ ∪ {x, y})λ(dη)λ(dξ)dxdydz
Z Z
hZ
1
1
=
Hx,y;z (η)
k(η ∪ ξ ∪ {x, y})
2
(Rd )3 Γ0
×
X
Γ0
i
1
Cx,y;z
(ξ ∪ ζ)λ(dξ) λ(dη)dxdydz.
ζ⊂η
1
Let us rewrite above using the definition (3.5) of Hx,y;z
(η).
Z
(L̂1 G)(η)k(η)λ(dη)
Γ0
Z
1
=
2
Z h
i
G(η ∪ z) − G(η ∪ x) − G(η ∪ y) − G(η ∪ {x, y})
(Rd )3 Γ0
×
hZ
Γ0
k(η ∪ ξ ∪ {x, y})
X
ζ⊂η
i
1
Cx,y;z
(ξ ∪ ζ)λ(dξ) λ(dη)dxdydz.
58
K. Pilorz
Using the special cases (2.9) and (2.10) of the Minlos lemma, we obtain
Z
(L̂1 G)(η)k(η)λ(dη)
Γ0
Z
=
G(η)
h1 Z
2
Γ0
−
1
2
Z X
Z
1
2
(Rd )2 Γ0
X
k(η ∪ ξ ∪ y)
1
Cx,y;z
(ξ ∪ ζ)λ(dξ)dydz
ζ⊂η\x
X
k(η ∪ ξ ∪ x)
y∈η
X
−
1
Cx,y;z
(ξ ∪ ζ)λ(dξ)dxdy
ζ⊂η\z
x∈η
Z X
Z Z
X
k(η\z ∪ ξ ∪ {x, y})
z∈η
(Rd )2 Γ0
Z
(Rd )2 Γ0
−
Z X
1
Cx,y;z
(ξ ∪ ζ)λ(dξ)dxdz
ζ⊂η\y
Rd Γ0 {x,y}⊂η
i
1
Cx,y;z
(ξ ∪ ζ)λ(dξ)dz λ(dη).
X
k(η ∪ ξ)
ζ⊂η\{x,y}
Employing the same technique to the second part of the operator L̂, we
derive
Z
(L̂2 G)(η)k(η)λ(dη)
Γ0
Z
Z Z X
=
2
2
Cx;y
(ξ ∪ ζ)Hx;y
(η)k(η ∪ ξ ∪ x)λ(dη)λ(dξ)dxdy
ζ⊂η
(Rd )2 Γ0 Γ0
Z
=
G(η)
Z Z X
Γ0
R d Γ0
y∈η
Z Z
−
k(η\y ∪ ξ ∪ x)
k(η ∪ ξ)
X
2
Cx;y
(ξ ∪ ζ)λ(dξ)dx
ζ⊂η\y
X X
2
Cx;y
(ξ ∪ ζ)λ(dξ)dy λ(dη).
x∈η ζ⊂η\x
R d Γ0
Therefore, we obtain
L∆ k(η) =
(3.6)
=
Z
1
2
Z X
(Rd )2 Γ0
(3.7)
−
1
2
Z
z∈η
Z X
(Rd )2 Γ0
(3.8)
−
1
2
Z
y∈η
1
Cx,y;z
(ξ ∪ ζ)λ(dξ)dxdy
ζ⊂η\z
k(η ∪ ξ ∪ y)
x∈η
Z X
(Rd )2 Γ0
X
k(η\z ∪ ξ ∪ {x, y})
X
1
Cx,y;z
(ξ ∪ ζ)λ(dξ)dydz
ζ⊂η\x
k(η ∪ ξ ∪ x)
X
ζ⊂η\y
1
Cx,y;z
(ξ ∪ ζ)λ(dξ)dxdz
A kinetic equation for repulsive coalescing random jumps...
Z Z
X
−
(3.9)
Rd Γ0 {x,y}⊂η
+
(3.10)
Z Z X
Rd Γ0
−
(3.11)
1
Cx,y;z
(ξ ∪ ζ)λ(dξ)dz
ζ⊂η\{x,y}
X
k(η\y ∪ ξ ∪ x)
y∈η
Z Z X
Rd Γ0
X
k(η ∪ ξ)
59
2
Cx;y
(ξ ∪ ζ)λ(dξ)dx
ζ⊂η\y
k(η ∪ ξ)
x∈η
X
2
Cx;y
(ξ ∪ ζ)λ(dξ)dy.
ζ⊂η\x
Note that so far we have not used any assumption about coefficients c̃1 and
c̃2 apart from that they can be written as results of action of the K-transform
1
2 . Let us calculate explicit forms
on corresponding functions Cx,y;z
and Cx;y
of these functions. Recall that
c̃1 (x, y; z; γ) = c1 (x, y; z)
Y
e−φ1 (z−u) ,
u∈γ\{x,y}
Y
c̃2 (x; y; γ) = c2 (x; y)
e−φ2 (y−u) .
u∈γ\x
We have
1
KCx,y;z
= c1 (x, y; z) e(t(1)
z , ·),
that is
1
−1
Cx,y;z
= K −1 c1 (x, y; z) e(1 + t(1)
z − 1, ·) = c1 (x, y; z)K
X
e(t(1)
z − 1, ξ)
ξ⊂·
= c1 (x, y; z)K
−1
Ke(t(1)
z
− 1, ·) =
c1 (x, y; z) e(t(1)
z
− 1, ·).
Therefore
(3.12)
1
Cx,y;z
(η) = c1 (x, y; z) e(t(1)
z − 1, η).
Analogously we can derive
(3.13)
2
Cx;y
(η) = c2 (x; y) e(t(2)
y − 1, η).
Using the above, we can rewrite the operator L∆ . For convenience let us
denote the part of it corresponding to the coalescence, that is the terms
(3.6)–(3.9), as L∆
1 and the part corresponding to the jumps, that is the
60
K. Pilorz
terms (3.10)–(3.11), as L∆
2 . Substituting (3.12), we derive
Z Z X
X
1
c1 (x, y; z)
k(η\z ∪ ξ ∪ {x, y})
L∆
1 k(η) =
2
z∈η
ζ⊂η\z
(Rd )2 Γ0
× e(t(1)
z − 1, ξ ∪ ζ)λ(dξ)dxdy
Z Z X
X
1
−
c1 (x, y; z)e(t(1)
k(η ∪ ξ ∪ y)
z − 1, ξ ∪ ζ)λ(dξ)dydz
2
x∈η
ζ⊂η\x
(Rd )2 Γ0
−
1
2
Z
Z X
(Rd )2 Γ0
Z Z
y∈η
X
−
X
k(η ∪ ξ ∪ x)
c1 (x, y; z)e(t(1)
z − 1, ξ ∪ ζ)λ(dξ)dxdz
ζ⊂η\y
X
k(η ∪ ξ)
Rd Γ0 {x,y}⊂η
c1 (x, y; z)e(t(1)
z − 1, ξ ∪ ζ)λ(dξ)dz
ζ⊂η\{x,y}
and analogously using (3.13), we obtain
Z Z X
X
∆
L2 k(η) =
k(η\y ∪ ξ ∪ x)
c2 (x; y)e(t(2)
y − 1, ξ ∪ ζ)λ(dξ)dx
Rd Γ0
y∈η
ζ⊂η\y
Z Z
−
X X
k(η ∪ ξ)
c2 (x; y)e(t(2)
y − 1, ξ ∪ ζ)λ(dξ)dy.
x∈η ζ⊂η\x
Rd Γ0
Consider the first component of L∆
1 and denote it as
Z Z X
1
L∆
k(η\z ∪ ξ ∪ {x, y})
11 k(η) =
2
z∈η
(Rd )2 Γ0
×
X
c1 (x, y; z)e(t(1)
z − 1, ξ ∪ ζ)λ(dξ)dxdy.
ζ⊂η\z
Next, for a given η let us introduce C(η) = {ξ ∈ Γ0 : ξ ∩ η 6= ∅}. Then, because any configuration treated as a measurable subset of Rd is of Lebesgue
measure 0 and the empty configuration does not belong to C(η) for any
η ∈ Γ0 , we have λ(C(η)) = 0 for every η ∈ Γ0 . Indeed, using the characterization (2.6) of the integral w.r.t. the Lebesgue–Poisson measure, we
obtain
Z
Z
∞
X
1
(n)
(0)
IC(η) (x1 , . . . , xn )dx1 . . . dxn .
λ(C(η)) = IC(η) (ξ)λ(dξ) = IC(η) +
n!
n=1
Γ0
(Rd )n
(0)
First, notice that IC(η) = 0, as empty configuration cannot have common
part with any configuration. Then, because
(n)
(1)
(1)
(1)
IC(η) (x1 , . . . , xn ) ≤ IC(η) (x1 ) + IC(η) (x2 ) + · · · + IC(η) (xn )
A kinetic equation for repulsive coalescing random jumps...
we have for every n ∈ N
Z
(n)
IC(η) (x1 , . . . , xn )dx1 . . . dxn ≤ n
(Rd )n
Z
hZ
(Rd )n−1
61
i
(1)
IC(η) (x)dx dx1 . . . dxn−1 .
Rd
Taking into account that
Z
Z
(1)
IC(η) (x)dx = Iη (x)dx = l(η) = 0,
Rd
Rd
where l denotes the Lebesgue measure, one can clearly see that λ(C(η)) = 0.
Therefore, when integrating over Γ0 \C(η) instead of Γ0 , the result is the
same. However, all subconfigurations ζ of η are disjoint with any ξ ∈
Γ0 \C(η), which allows us to separate the product taken over ξ ∪ ζ into
one taken over ξ and another taken over ζ. Thus we can write
Z Z X
1
∆
L11 k(η) =
c1 (x, y; z)k(η\z ∪ ξ ∪ {x, y})
2
z∈η
(Rd )2 Γ0
× e(t(1)
z − 1, ξ)
X
e(t(1)
z − 1, ζ)λ(dξ)dxdy.
ζ⊂η\z
Recalling the definition (2.1) of the K-transform and its property (2.5), we
have
X
(1)
(1)
e(t(1)
z − 1, ζ) = K e(tz − 1, ·) (η\z) = e(tz , η\z).
ζ⊂η\z
Therefore we can rewrite the action of L∆
11 in the form
Z Z X
1
L∆
c1 (x, y; z)k(η\z ∪ ξ ∪ {x, y})
11 k(η) =
2
z∈η
(Rd )2 Γ0
(1)
× e(t(1)
z − 1, ξ)e(tz , η\z)λ(dξ)dxdy.
∆
Applying the same method for the rest of the L∆
1 and for the L2 , we obtain
the result.
4. The Vlasov scaling and the kinetic equation.
4.1. The Vlasov scaling. We follow the scaling technique described in
[8]. Let us introduce the scale parameter ∈ [0, 1] with = 1 corresponding
to the unscaled case (microscopic level) and → 0 to the fully rescaled case,
where the description of the system is based on the density of a medium
(mesoscopic level), instead of the corpuscular structure. Evolution of density
in time is characterized by the corresponding kinetic equation. The scaling
is performed in such a way that the evolution of the rescaled case preserves
poissonity of a correlation function. This allows us to identify above abstract
density of the medium with density of the Poisson measure, which is concrete
62
K. Pilorz
mathematical object. We alter the operator L∆ by scaling c1 → c1 , φ1 →
φ1 and φ2 → φ2 for ∈ (0, 1], which can be interpreted as weakening the
interactions between particles. Altered in such a way operator we denote
by L∆
. Next, we renormalize it, defining
|η| ∆ −|η|
Lren
k(η)).
k(η) = L (
Let us consider the first component of the operator Lren
. We have (cf.
Proposition 3.2)
Z Z X
1 |η|
ren
c1 (x, y; z)−|η\z∪ξ∪{x,y}| k(η\z ∪ ξ ∪ {x, y})
L11, k(η) = 2
z∈η
(Rd )2 Γ0
×
Y
(e−φ1 (z−u) − 1)
u∈ξ
Y
e−φ1 (z−u) λ(dξ)dxdy.
u∈η\z
Note that integrating over (Rd )2 \(η × η) instead of (Rd )2 and over Γ0 \(η ∪
{x, y}) instead of Γ0 does not influence the result, so we have
Z Z X
1
ren
L11, k(η) =
c1 (x, y; z)k(η\z ∪ ξ ∪ {x, y})
2
z∈η
(Rd )2 Γ0
×
Y
Y 1
e−φ1 (z−u) − 1
e−φ1 (z−u) λ(dξ)dxdy.
u∈ξ
u∈η\z
Let us pass with to the limit. Noting that
1 −φ1 (z−u)
e
− 1 = −φ1 (z − u),
lim
→0 we can write
Z Z X
1
ren
lim L11, k(η) =
c1 (x, y; z)k(η\z ∪ ξ ∪ {x, y})
→0
2
z∈η
(Rd )2 Γ0
×
Y
− φ1 (z − u) λ(dξ)dxdy.
u∈ξ
Let us denote V = lim Lren
. Calculating analogously as above, one derives
→0
Z Z X
1
V k(η) =
c1 (x, y; z)k(η\z ∪ ξ ∪ {x, y})
2
z∈η
(Rd )2 Γ0
×
Y
(−φ1 (z − u))λ(dξ)dxdy
u∈ξ
1
−
2
Z
Z X
(Rd )2 Γ0
x∈η
c1 (x, y; z)k(η ∪ ξ ∪ y)
Y
u∈ξ
(−φ1 (z − u))λ(dξ)dydz
A kinetic equation for repulsive coalescing random jumps...
1
−
2
Z
Z X
(Rd )2 Γ0
+
Z Z X
Rd Γ0
y∈η
y∈η
(−φ1 (z − u))λ(dξ)dxdz
Y
(−φ2 (y − u))λ(dξ)dx
u∈ξ
k(η ∪ ξ)
Rd Γ0
Y
u∈ξ
k(η\y ∪ ξ ∪ x)c2 (x; y)
Z Z
−
c1 (x, y; z)k(η ∪ ξ ∪ x)
63
X
c2 (x; y)
x∈η
Y
(−φ2 (y − u))λ(dξ)dy.
u∈ξ
Consider the following problem
d
(4.1)
rt = V rt ,
dt
in the Banach space
rt=0 = r0
Kθ = {r : Γ0 → R : ||r||θ < ∞},
where
||r||θ = ess sup eθ|η| |r(η)|.
η∈Γ0
As usual in the Vlasov scaling, we assume that the initial state is a Poisson
measure and hence r0 is of the form
Y
r0 (η) =
ρ0 (x).
x∈η
If rt is of the product form
rt (η) =
Y
ρt (x),
x∈η
then
d
X Y
d Y
d
rt (η) =
ρt (x) =
ρt (y)
ρt (x).
dt
dt x∈η
dt
x∈η
y∈η\x
Therefore, by expressing V rt in the form
X Y
V rt (η) =
ρt (y) v(ρt , x),
x∈η
y∈η\x
for some v(ρt , x) (which is done below), we can obtain a problem for ρt
corresponding to (4.1), namely a kinetic equation
d
ρt (x) = v(ρt , x), ρt=0 = ρ0 .
(4.2)
dt
If ρt is a solution of (4.2), then we can easily check that
Y
rt (η) =
ρt (x)
x∈η
is a solution of (4.1). Therefore by showing existence of solution of (4.2), we
obtain the existence of (4.1). As we show the local existence and uniqueness
64
K. Pilorz
of the kinetic equation (see Theorem 4.1 below), we obtain the local existence of (4.1). It is possible to show its uniqueness using the Ovsyannikov’s
method, but it is beyond the scope of this paper.
Let us denote the first component of V by V1 . We have
Z Z X
1
c1 (x, y; z)rt (η\z ∪ ξ ∪ {x, y})
V1 rt (η) =
2
z∈η
(Rd )2 Γ0
×
Y
(−φ1 (z − u))λ(dξ)dxdy
u∈ξ
=
Z
1
2
Z X
(Rd )2 Γ0
×
Y
Y
c1 (x, y; z)
z∈η
ρt (v)
v∈η\z∪ξ∪{x,y}
(−φ1 (z − u))λ(dξ)dxdy.
u∈ξ
Because l(η) = 0 and λ(C(η ∪ {x, y})) = 0, one can rewrite above as
Z
Z
1
X Y
ρt (v)
V1 rt (η) =
c1 (x, y; z)
2
z∈η
v∈η\z
× ρt (x)ρt (y)
(Rd )2 \(η×η) Γ0 \C(η∪{x,y})
Y
(−ρt (u)φ1 (z − u))λ(dξ)dxdy .
u∈ξ
Therefore
V1 rt (η) =
X Y
z∈η
ρt (v) v1 (ρt , z),
v∈η\z
where
v1 (ρt , z) =
1
2
Z
c1 (x, y; z)ρt (x)ρt (y)
Z Y
(−ρt (u)φ1 (z − u))λ(dξ)dxdy.
Γ0 u∈ξ
(Rd )2
Noting that for a(u) = −ρt (u)φ1 (z − u)
Z Y
Z
∞
X
1
a(x1 ) . . . a(xn )dx1 . . . dxn
a(u)λ(dξ) = 1 +
n!
n=1
Γ0 u∈ξ
=1+
∞
X
n=1
(Rd )n
1
n!
Z
n
Z
a(u)du ,
a(u)du = exp
Rd
Rd
we can reformulate the above, obtaining
Z
Z
1
v1 (ρt , x) =
c1 (y, z; x)ρt (y)ρt (z) exp − φ1 (x − u)ρt (u)du dydz.
2
(Rd )2
Rd
A kinetic equation for repulsive coalescing random jumps...
65
Calculating analogously as above, one can obtain explicit form of v and thus
the following kinetic equation
Z
Z
d
1
c1 (y, z; x) exp − φ1 (x − u)ρt (u)du ρt (y)ρt (z)dydz
ρt (x) =
dt
2
(Rd )2
1
2
−
Rd
Z
c1 (x, y; z) + c1 (y, x; z)
(Rd )2
Z
× exp −
(4.3)
φ1 (z − u)ρt (u)du ρt (x)ρt (y)dydz
Rd
Z
Z
c2 (y; x) exp −
+
φ2 (x − u)ρt (u)du ρt (y)dy
Rd
Rd
Z
Z
−
c2 (x; y) exp −
Rd
φ2 (y − u)ρt (u)du ρt (x)dy,
Rd
ρt=0 = ρ0 .
4.2. The kinetic equation. Let us rewrite the problem (4.3) as
(4.4)
d
ρt (x) = R1 (ρt , x) + R2 (ρt , x),
dt
ρt=0 (x) = ρ0 (x),
where
Z
1
R1 (ρt , x) = − ρt (x)
2
c1 (x, y; z) + c1 (y, x; z) ρt (y)dydz
(Rd )2
Z
− h(ρt , x)
c2 (x; y)dy
Rd
= − ρt (x)h(ρt , x)
for
Z
1
h(ρt , x) =
2
c1 (x, y; z) + c1 (y, x; z) ρt (y)dydz + hc2 i
(Rd )2
and
R2 (ρt , x) =
Z
1
2
Z
c1 (y, z; x) exp − φ1 (x − u)ρt (u)du ρt (y)ρt (z)dydz
(Rd )2
+
1
2
Rd
Z
(Rd )2
c1 (x, y; z) + c1 (y, x; z)
66
K. Pilorz
h
Z
× 1 − exp −
i
φ1 (z − u)ρt (u)du ρt (x)ρt (y)dydz
Rd
Z
Z
c2 (y; x) exp −
+
Rd
Z
+
φ2 (x − u)ρt (u)du ρt (y)dy
Rd
h
Z
i
c2 (x; y) 1 − exp − φ2 (y − u)ρt (u)du ρt (x)dy.
Rd
Rd
Note that from (4.4) we can obtain the equivalent integral equation
Zt
ρt (x) = ρ0 (x) exp −
h(ρs , x)ds
0
(4.5)
Zt
Zt
R2 (ρs , x) exp −
+
h(ρσ , x)dσ ds.
s
0
Theorem 4.1. Problem (4.4) with the initial condition ρ0 ∈ L∞ (Rd ), ρ0 ≥
0 has the unique local classical solution.
Consider XT = C([0, T ] → L∞ (Rd )), T > 0 with the norm
||ρ||T,γ = sup e−γhc2 it ||ρt ||L∞ .
t∈[0,T ]
Denote
BT,γ (r) = {ρ ∈ XT : ||ρ||T,γ ≤ r, ρt ≥ 0 ∀t ∈ [0, T ]},
BT,γ (r, ρ0 ) = {ψ ∈ BT,γ (r) : ψ0 = ρ0 },
where ρ0 ∈ L∞ (Rd ), ρ0 ≥ 0, r ≥ ||ρ0 ||L∞ and T, γ > 0.
Lemma 4.2. Given r > 0, there exist γ, T̃ > 0 such that F defined by the
RHS of (4.5) with the domain BT ∗ ,γ (r) ⊂ XT ∗ acts again to the BT ∗ ,γ (r)
for any T ∗ ∈ [0, T̃ ].
Lemma 4.3. Let ρ0 ∈ L∞ (Rd ), ρ0 ≥ 0 and r ≥ ||ρ0 ||L∞ . Let T̃ , γ satisfy
Lemma 4.2 for this r. We can choose T ∗ ∈ [0, T̃ ] in such a way that for
any ρ, ψ in BT ∗ ,γ (r, ρ0 ) the inequality ||F (ρ) − F (ψ)||T ∗ ,γ ≤ C||ρ − ψ||T ∗ ,γ
holds for some constant C < 1.
Proof of Theorem 4.1. Choose r > ||ρ0 ||L∞ and take corresponding γ, T̃
from Lemma 4.2. Take T ∗ as in Lemma 4.3. Define the sequence of Picard
iterations (ρ(n) )n∈N0 in the following way
(0)
(4.6)
ρt
= ρ0 ∀t ∈ [0, T ∗ ],
ρ(n) = F (ρ(n−1) ), n ∈ N.
A kinetic equation for repulsive coalescing random jumps...
67
Obviously, ρ(0) ∈ BT ∗ ,γ (r). Therefore, by Lemma 4.2, ρ(n) ∈ BT ∗ ,γ (r) for
all n ∈ N0 and from Lemma 4.3 we obtain
(n+k)
||ρ
(n)
−ρ
||
T ∗ ,γ
(1)
≤ ||ρ
(0)
−ρ
||
T ∗ ,γ
k
X
C n+i−1 ≤ ||ρ(1) − ρ(0) ||T ∗ ,γ
i=1
Cn
,
1−C
where C < 1 is a positive constant. Therefore ρ(n) n∈N0 defined by (4.6) is
a Cauchy sequence. As BT ∗ ,γ (r) is a closed subset of a Banach space, there
exists
lim ρ(n) = ρ ∈ BT ∗ ,γ (r).
n→∞
Clearly F (ρ) = ρ and therefore ρt satisfies the integral equation (4.5) for
t ∈ [0, T ∗ ]. Thus it is a local classical solution of (4.4).
Now suppose there is another local classical solution of this equation, say
ψ. Then ψ0 = ρ0 and for r, γ, T ∗ as above, there exists T ≤ T ∗ such that
ψ ∈ BT,γ (r). However, from Lemma 4.3 we have
||ρ − ψ||T,γ = ||F (ρ) − F (ψ)||T,γ ≤ C||ρ − ψ||T,γ
for C < 1, which means that
||ρ − ψ||T,γ = 0
and thus ρ is the unique local classical solution.
Proof of Lemma 4.2. Take arbitrary T, γ > 0 and ρ ∈ BT,γ (r). Note
that
h(ρt , x) ≥ hc2 i,
3
R2 (ρt , x) ≤ ||ρt ||2L∞ hc1 i + 2||ρt ||L∞ hc2 i,
2
ρt (x) ≤ ||ρt ||L∞ ≤ eγhc2 it ||ρ||T,γ .
(4.7)
It is obvious that F preserves positiveness of ρ. Furthermore, using above
estimates and the definition of BT,γ (r), we derive
Zt
F (ρ) (x) = ρ0 (x) exp − h(ρs , x)ds
t
0
Zt
Zt
R2 (ρs , x) exp −
+
s
0
≤ ||ρ0 ||L∞ e−thc2 i +
h(ρσ , x)dσ ds
Zt
0
R2 (ρs , x)e(s−t)hc2 i ds
68
K. Pilorz
Zt 3
≤e
||ρ||T,γ +
hc1 ie(2γ+1)hc2 is ||ρ||2T,γ
2
0
i
(γ+1)hc2 is
+ 2hc2 ie
||ρ||T,γ ds .
−thc2 i
h
Therefore, we obtain
F (ρ) t L∞
3hc1 ir
e(2γ+1)hc2 it − 1
2(2γ + 1)hc2 i
i
e(γ+1)hc2 it − 1 .
h
≤ e−thc2 i r 1 +
+
2
γ+1
Thus
F (ρ) T,γ
≤ r sup f (t),
t∈[0,T ]
where
3hc1 ir
e(2γ+1)hc2 it − 1
2(2γ + 1)hc2 i
i
e(γ+1)hc2 it − 1 .
h
f (t) = e−(γ+1)hc2 it 1 +
+
2
γ+1
Note that f (0) = 1. Additionally
h
f 0 (t) = − (γ + 1)hc2 ie−(γ+1)hc2 it 1 +
3hc1 ir
e(2γ+1)hc2 it − 1
2(2γ + 1)hc2 i
i
2 (γ+1)hc2 it
e
−1
(γ + 1)
h 3hc ir
i
1
+ e−(γ+1)hc2 it
e(2γ+1)hc2 it + 2hc2 ie(γ+1)hc2 it
2
+
and hence
f 0 (0) = −(γ + 1)hc2 i +
Choosing γ > 1 +
3hc1 ir
2hc2 i ,
3
2
hc1 ir + 2hc2 i .
we have f 0 (0) < 0, which guarantees existence of
T̃ such that sup f (t) = 1. Taking T = T ∗ for T ∗ ∈ [0, T̃ ] yields
t∈[0,T̃ ]
||F (ρ)||T ∗ ,γ ≤ r.
Therefore F (ρ) ∈ BT ∗ ,γ (r) for ρ ∈ BT ∗ ,γ (r).
A kinetic equation for repulsive coalescing random jumps...
Proof of Lemma 4.3. We have
Zt
F (ρ) − F (ψ) (x) = ρ0 (x) exp −
t
h(ρs , x)ds
0
Zt
+
Zt
R2 (ρs , x) exp −
s
0
(4.8)
h(ρσ , x)dσ ds
− ρ0 (x) exp −
Zt
h(ψs , x)ds
0
Zt
−
Zt
h(ψσ , x)dσ ds
R2 (ψs , x) exp −
s
0
Zt
= D1 +
D2 ds,
0
where
h
Zt
D1 = ρ0 (x) exp −
Zt
i
h(ρs , x)ds − exp − h(ψs , x)ds
0
0
and
D2 =
Zt h
Zt
R2 (ρs , x) exp −
h(ρσ , x)dσ
s
0
− R2 (ψs , x) exp −
Zt
h(ψσ , x)dσ
i
ds.
s
Take an arbitrary T ∗ ∈ [0, T̃ ]. We have
Zt
(4.9)
|D1 | ≤ ||ρ0 ||L∞ hc1 i
||ρs − ψs ||L∞ ds ≤ rhc1 iteγhc2 it ||ρ − ψ||T ∗ ,γ .
0
To estimate |D2 |, consider two cases. First, suppose
Zt s
h(ρσ , x) − h(ψσ x) dσ ≥ 0.
69
70
K. Pilorz
Then
h Zt i
|D2 | ≤ R2 (ρs , x) exp −
h(ρσ , x) − h(ψσ , x) dσ
s
h
− R2 (ψs , x) exp −
Zt i
h(ρσ , x) − h(ψσ , x) dσ s
h Zt i
+ R2 (ψs , x) exp −
h(ρσ , x) − h(ψσ , x) dσ − R2 (ψs , x)
s
≤ R2 (ρs , x) − R2 (ψs , x)
n
h Zt io
+ R2 (ψs , x) 1 − exp −
h(ρσ , x) − h(ψσ , x) dσ .
s
In the other case, when
Rt
h(ρσ , x) − h(ψσ , x) dσ < 0, we have analogously
s
|D2 | ≤ R2 (ρs , x) − R2 (ψs , x)
n
h Zt io
+ R2 (ρs , x) 1 − exp −
h(ψσ , x) − h(ρσ , x) dσ .
s
Note that both R2 (ρs , x) and R2 (ψs , x), as both belong to BT ∗ ,γ (r), satisfy
the same estimate (cf. (4.7))
3
R2 (ρs , x), R2 (ψs , x) ≤ hc1 ie2γhc2 is r2 + 2hc2 ieγhc2 is r,
2
which allows us to write
3
|D2 | ≤ R2 (ρs , x) − R2 (ψs , x) +
hc1 ie2γhc2 is r2 + 2hc2 ieγhc2 is r
2
(4.10)
io
n
h Zt × 1 − exp − h(ψσ , x) − h(ρσ , x)dσ .
s
We have
Zt i Zt 1 − exp − h(ψσ , x) − h(ρσ , x)dσ ≤ h(ψσ , x) − h(ρσ , x)dσ
h
s
=
1
2
Zt
s
Z
(Rd )2
s
c1 (x, y; z) + c1 (y, x; z)
ρσ (y) − ψσ (y) dydz dσ
A kinetic equation for repulsive coalescing random jumps...
Zt
≤
Zt
hc1 i||ρσ − ψσ ||L∞ dσ ≤
s
71
hc1 ieγhc2 iσ ||ρ − ψ||T ∗ ,γ dσ
s
≤ hc1 ie
γhc2 it
(t − s)||ρ − ψ||T ∗ ,γ ,
which yields
Zt i
(4.11) 1 − exp − h(ψσ , x) − h(ρσ , x)dσ ≤ hc1 iteγhc2 it ||ρ − ψ||T ∗ ,γ .
h
s
Let us estimate
R
(ρ
,
x)
−
R
(ψ
,
x)
2 s
2 s
Z
Z
1
c1 (y, z; x) exp − φ1 (x − u)ρs (u)du ρs (y)ρs (z)
≤
2
(Rd )2
Rd
Z
− exp −
φ1 (x − u)ψs (u)du ψs (y)ψs (z)dydz
Rd
+
Z
1
2
h
Z
i
c1 (x, y; z) + c1 (y, x; z) 1 − exp − φ1 (z − u)ρs (u)du
(Rd )2
Rd
h
Z
× ρs (x)ρs (y) − 1 − exp −
i
φ1 (z − u)ψs (u)du ψs (x)ψs (y)dydz
Rd
Z
+
Z
c2 (y; x) exp − φ2 (x − u)ρs (u)du ρs (y)
Rd
Rd
Z
− exp −
φ2 (x − u)ψs (u)du ψs (y)dy
Rd
Z
+
h
Z
i
c2 (x; y) 1 − exp − φ2 (y − u)ρs (u)du ρs (x)
Rd
Rd
h
Z
i
− 1 − exp − φ2 (y − u)ψs (u)du ψs (x)dy.
Rd
Denote by Ii the i-th component of the RHS of the above inequality for
i = 1, 2, 3, 4. Then estimating analogously as above, we derive
I3 , I4 ≤ hc2 i e2γhc2 is hφ2 ir + eγhc2 is ρ − ψ ∗ .
T ,γ
72
K. Pilorz
Moreover, noting that
ρs (y)ρs (z) − ψs (y)ψs (z)
1
1
≤
ρs (z) + ψs (z) ρs (y) − ψs (y) +
ρs (y) + ψs (y) ρs (z) − ψs (z),
2
2
we obtain
1
I1 ≤ hc1 i 2e2γhc2 is r + e3γhc2 is hφ1 ir2 ρ − ψ ∗ ,
2
T ,γ
2γhc2 is
3γhc2 is
2 I2 ≤ hc1 i 2e
r+e
hφ1 ir ρ − ψ ∗ .
T ,γ
Therefore,
h3
R2 (ρs , x) − R2 (ψs , x) ≤ hc1 ieγhc2 is 2eγhc2 is r + e2γhc2 is hφ1 ir2
2
i
(4.12)
γhc2 is
eγhc2 is hφ2 ir + 1 ρ − ψ ∗ .
+ 2hc2 ie
T ,γ
Substituting (4.11) and (4.12) into (4.10) and using it together with (4.9),
we obtain (cf. (4.8))
F (ρ) − F (ψ) (x) ≤ eγhc2 it f (t)ρ − ψ ∗ ,
t
T ,γ
where
h3
r2 hc1 ie2γhc2 it hc1 it + hφ1 i
2
i
+ reγhc2 it 2hc1 ihc2 it + 3hc1 i + 2hc2 ihφ2 i + 2hc2 i .
f (t) = t
Therefore,
F (ρ) − F (ψ)
T ∗ ,γ
≤ sup f (t)ρ − ψ T ∗ ,γ
t∈[0,T ∗ ]
.
Note that f (t) is continuous, increasing function of t and f (0) = 0. Thus,
there exists T ∗∗ > 0 such that f (T ∗∗ ) < 1 and f (t) ∈ [0, f (T ∗∗ )] for t ∈
[0, T ∗∗ ]. Choosing T ∗ = min(T ∗∗ , T̃ ), we obtain
F (ρ) − F (ψ) ∗ ≤ C ρ − ψ ∗
T ,γ
with C =
f (T ∗ )
≤
f (T ∗∗ )
< 1.
T ,γ
Acknowledgement. This work has been performed in the framework of
the joint Polish-German project No 57154469 “Dynamics of Large Systems
of Interacting Entities” supported by DAAD. The author gratefully acknowledges the support and the hospitality extended to him during the
stay at Bielefeld University where the main part of the work was done.
Special thanks the author wishes to express to Professor Yuri Kozitsky for
suggesting the problem and for many fruitful conversations, as without his
A kinetic equation for repulsive coalescing random jumps...
73
help this paper would not come into being. Finally, the author wishes to
express his gratitude to the referee for valuable remarks.
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74
Krzysztof Pilorz
Institute of Mathematics
Maria Curie-Skłodowska University
pl. M. Curie-Skłodowskiej 1
20-031 Lublin
Poland
e-mail: [email protected]
Received August 10, 2015
K. Pilorz