Random jumps and coalescence in the continuum: Evolution of states of an infinite particle system

The dynamics is studied of an infinite collection of point particles placed in \begin{document}$ \mathbb{R}^d $\end{document} , \begin{document}$ d\geq 1 $\end{document} . The particles perform random jumps with mutual repulsion accompanied by random merging of pairs of particles. The states of the collection are probability measures on the corresponding configuration space. The main result is the proof of the existence of the Markov evolution of states for a bounded time horizon if the initial state is a sub-Poissonian measure. The proof is based on representing sub-Poissonian measures \begin{document}$ \mu $\end{document} by their correlation functions \begin{document}$ k_\mu $\end{document} and is done in two steps: (a) constructing an evolution \begin{document}$ k_{\mu_0} \to k_t $\end{document} ; (b) proving that \begin{document}$ k_t $\end{document} is the correlation function of a unique sub-Poissonian state \begin{document}$ \mu_t $\end{document} .


1.
Introduction. The dynamics of infinite particle systems in the course of which the constituents can merge attracts considerable attention. The Arratia flow introduced in [1] provides an example of the system of this kind. In recent years, it has been being extensively studied, see [4,7,8,12] and the works quoted in these publications.
In Arratia's model, an infinite number of Brownian particles move in R independently up to their collision, then merge and move together as single particles. Correspondingly, the description of this motion (and its modifications) is performed in terms of diffusion processes. In this work, we propose and study an alternative model of this kind, in which the constituents -unlike to Arratia's model -interact with each other. In view of the infinite number of them, the construction of the corresponding stochastic process for this model is far beyond the technical possibilities available in this domain. Thus, we are content with a more modest resultdescribing the evolution of states by solving an appropriate Fokker-Planck equation.
Similarly as in [3], in our model point particles perform random jumps with repulsion in R d , d ≥ 1. Additionally, two particles (located at x and y) can merge into a particle (located at z) with intensity (probability per time) c 1 (x, y; z). Thereafter, this new particle participates in the motion. The phase space of such a system is the set Γ of all locally finite configurations γ ⊂ R d , see [3,5,9,10], and the states of the system are probability measures on Γ the set of which will be denoted by P(Γ).
The description of their evolution µ 0 → µ t is based on the relation µ t (F 0 ) = µ 0 (F t ) where F 0 : Γ → R is supposed to belong to a measure-defining class of functions, µ(F ) := F dµ and the evolution F 0 → F t is obtained by solving the Kolmogorov equation  Then the collection {F ω : ω ∈ Ω} is a measure-defining class. The set of measures (called sub-Poissonian) P exp ⊂ P(Γ) we will work with is defined by the condition that its members enjoy the following property: the map Ω ω → µ(F ω ) ∈ R can be continued to an exponential type entire function defined on L 1 (R d ). Then, for µ ∈ P exp , we set B µ (ω) = µ(F ω ) and derive L from L according to the rule ( LB µ )(ω) = µ(LF ω ). Thereafter, we construct the evolution B µ0 → B t by solving the corresponding evolution equation. The next (and the hardest) part of this scheme is to prove that B t = B µt for a unique µ t ∈ P exp .
In Section 2, we outline the mathematical background of the paper. In Section 3, we introduce the model and present the results in the form of Theorems 3.2 and 3.3. Their proof is performed in Sections 4 and 5, respectively.

2.
Preliminaries. As mentioned above, we work with the phase space where | · | denotes cardinality. It is equipped with the vague (weak-hash) topology see e.g., [10]) and the corresponding Borel σ-field B(Γ). In this interpretation, configurations γ ∈ Γ are considered as Radon measures, and the vague topology is the weakest topology that makes continuous all the maps γ → f (x)γ(dx) = x∈γ f (x), f ∈ C 0 (R d ) -the set of all compactly supported continuous functions. The set of all finite configurations is denoted by Γ 0 . It is the union of the sets Γ (n) = {γ ∈ Γ : |γ| = n}, n ∈ N 0 . Γ 0 is endowed with the topology induced by the vague topology of Γ, that coincides with the usual weak topology that makes continuous all the maps γ → x∈γ f (x) with bounded continuous f : R d → R.
Then the corresponding Borel σ-field B(Γ 0 ) is a sub-field on B(Γ).
Then the action of L can be transferred to B µ by means of the rule This allows one to pass from (1.1) to the following evolution equation The advantage of using P exp is that, for each of its members, the function B µ admits the representation Here every k with one and the same C > 0 for all n ∈ N. In the second line of (2.7), we use the The function k µ is called the correlation function of the state µ, whereas k (n) µ is its n-th order correlation function. k µ completely characterizes µ ∈ P exp . For instance, k π (η) = e( ; η) for the Poisson measure π with density : R d → [0, +∞). On the other hand, the following is known, see [10, Theorems 6.1, 6.2 and Remark 6.3]. Proposition 1. A function k : Γ 0 → R is a correlation function of a unique measure µ ∈ P exp if and only if it satisfies the conditions: (a) k(∅) = 1; (b) the estimate in (2.8) holds for some C > 0 and all n ∈ N; (c) for each G ∈ B * bs , the following holds Notably, the cone {G ∈ B bs : G(η) ≥ 0} is a proper subset of B * bs . Corollary 1. An exponential type entire function B : L 1 (R) → R satisfies (2.4) for a unique µ ∈ P exp if and only if it admits the expansion as in (2.7) with k satisfying the conditions of Proposition 1.
Having in mind the latter facts we will look for the solutions of (2.6) in the form where L ∆ is to be obtained from L (and thus from L) according to the rule, cf.
(2.5) and (2.11), For µ ∈ P exp and a compact Λ ⊂ R d , the projection of µ defined in (2.3) is absolutely continuous with respect to the Lebesgue-Poisson measure λ. Let R Λ µ be its Radon-Nikodym derivative. It is related to the correlation function k µ by (2.14) One of our tools in this work is based on the Minlos lemma according to which, cf.
[5, eq. (2. 2)], holding for appropriate G, H : Γ 0 → R. By taking here and then by (2.2) we obtain its following special case Analogously, for 3. The results. Our model is specified by the operator L the action of which on an observable F : Γ → R is Here c 1 ≥ 0 is the intensity of the coalescence of the particles located at x and y into a new particle located at z. Note that c 1 does not depend on the elements of γ other than x and y. For simplicity, we assume that c 1 (x, y; z) = c 1 (y, x; z) = c 1 (x + u, y + u; z + u) for all u ∈ R d . For a more general version of this model, see [15]. The second summand in (3.1) describes jumps performed by the particles. As in [3], we setc with φ and c 2 being the repulsion potential and the jump kernel, respectively. By these assumptions the model is translation invariant. The functions c 1 , c 2 and φ take non-negative values and satisfy the following conditions: Now we pass to the equation in (2.12). The corresponding operator L ∆ is to be calculated from (3.1) by (2.5) and (2.13). It thus takes the form, cf. [15], is the part responsible for the coalescence whereas L ∆ 2 = L ∆ 21 + L ∆ 22 describes the jumps. Their summands are: In view of (2.8), the Banach spaces for (2.12) ought to be of L ∞ type. Thus, we set By this definition it follows that each k ∈ K θ satisfies |k(η)| ≤ e θ|η| k θ .
Let us now define L ∆ in a given K θ . To this end, we set Then, similarly as in (3.7) -(3.9), we obtain that both L ∆ 1 and L ∆ 2 map the elements of D θ into K θ . Let L ∆ θ denote the operator (L ∆ , D θ ). Then, in the Banach space K θ , the problem in (2.12) takes the form Definition 3.1. A classical solution of (3.11) on a given time interval [0, T ) is a continuous function [0, T ) t → k t ∈ D θ that is continuously differentiable in K θ on (0, T ) and is such that both equalities in (3.11) hold.
As is typical for problems like in (3.11), in view of the complex character of the corresponding operator it might be unrealistic to expect the existence of classical solutions for all possible k 0 ∈ D θ . Thus, we will restrict the choice of k 0 to a proper subset of the domain (3.10). For θ > θ, we have that K θ → K θ , i.e., K θ is continuously embedded in K θ . Similarly as in [3,5] we will solve (3.11) in the scale {K θ } θ∈R . By means of the estimates in (3.7) -(3.9) one concludes that L ∆ can be defined as a bounded linear operator from K θ to K θ whenever θ > θ. We shall denote this operator by L ∆ θ θ . By this estimate one also gets that K θ ⊂ D θ , θ > θ, (3.12) and Theorem 3.2. For each α 0 ∈ R and α * > α 0 , and for an arbitrary k 0 ∈ K α0 , the problem in (3.11) has a unique classical solution k t ∈ K α * on [0, T (α * , α 0 )).
A priori the solution k t described in Theorem 3.2 need not be a correlation function of any state, which means that the result stated therein has no direct relation to the evolution of states of the system considered. Our next result removes this drawback. Theorem 3.3. Let µ 0 ∈ P exp be such that k µ0 ∈ K α0 . Then, for each α * > α 0 , the evolution k µ0 → k t described in Theorem 3.2 has the property: for each t < T (α * , α 0 )/2, k t is the correlation function of a unique state µ t ∈ P exp . By Theorem 3.3 we also have the evolution B µ0 → B t = B µt = e(·; ·), k t , where B t solves (2.6), cf. (2.11). Along with its purely theoretical value, this result may serve as a starting point for a numerical study of the random motion of this type, cf. [14], including its consideration at different space and time scales [2,16]. To this end one can use kinetic equations related to the model specified in (3.1), see [15].

Proof of Theorem 3.2. The solution in question will be obtained in the form
where the family of bounded operators Q α * α0 (t) : where the differentiation is taken in the classical sense in the Banach space of all bounded linear operators L(K α0 , K α * ). Additionally, Q α * α0 (0) is considered as the embedding operator, and hence k t given in (4.1) satisfies the initial condition up to this embedding. Each Q α * α0 (t) is constructed as a series of t-dependent operators, convergent in the operator norm topology for t < T (α * , α 0 ). In estimating the norms of these operators we crucially use (3.7) -(3.9). As the right-hand sides of (3.7) and (3.8) contain different powers of |η|, it is convenient to split L ∆ = A + B with A = L ∆ 14 . By A θ and B θ we denote the unbounded operators (A, D θ ) and (B, D θ ), respectively. Likewise, we introduce A θ θ and B θ θ , θ > θ. Their operator norms are to be estimated by means of (3.7) -(3.9) and the following inequalities After some calculations we then get with β(θ) given in (3.14). Now, for θ > θ and t > 0, we define a bounded linear (multiplication) operator S θ θ (t) : K θ → K θ by the formula and by S θ θ (0) we will mean the corresponding embedding operator. Then, for each that readily follows by (4.3). Note that the multiplication operator by exp(−tΨ) acts from K θ to K θ for any θ; hence, S θ θ (t) : K θ → D θ , see (3.12). We define it, however, as above in order to have the continuity secured by the estimate in (4.5). By (4.4), for any θ ∈ (θ, θ ), we have that Also by (4.4) it follows that Let O be an operator acting in each K θ such that: (a) O : D θ → K θ ; (b) O : K θ → K θ is a bounded operator whenever θ > θ. As in the case of A and B, we define the operators O θ = (O, D θ ) and O θ θ . Similarly as in (3.13), for these operators, we have where the second equality holds for all θ ∈ (θ, θ ). Now we can turn to constructing the resolving operators Q α * α0 (t), see (4.1). For a given n ∈ N and q > 1, we introduce In particular α 2n+1 = α * . For these α l , l = 0, . . . , 2n + 1 and Similarly as in obtaining the second equality in (4.8), we conclude that π (n) α * α0 (t, t 1 , ..., t n ) is independent of the particular choice of the partition of (α 0 , α * ) into subintervals (α l , α l+1 ). In view of (4.6), we have that holding for all α ∈ (α 0 , α * ). For the same α, by setting in (4.10) t 1 = t we obtain . . , t n ), see (4.8). By (4.7) and the second estimate in (4.3) we get the following estimate of the operator norm of (4.10) (4.13) (4.14) Then by (4.13) it follows that For each τ < T (α * , α 0 ), by using (3.14) we conclude that there exist q > 1 and α ∈ (α 0 , α * ) such that qτ < T (α, α 0 ). Then by the above estimate it follows that, uniformly on [0, τ ], {Q (n) αα0 (t)} is a Cauchy sequence with respect to the operator norm. Let Q αα0 (t) be its limit. Clearly, this also applies to the sequence {Q (n) α * α0 (t)}, which therefore converges to Q α * α0 (t) in the same sense. By this we have that: (a) The latter follows by (3.12) and (4.15). In the sequel, we will use the following estimate that readily follows by (4.13). For n ∈ N and α ∈ (α 0 , α * ), by (4.11) and (4.12) we obtain from (4.14) that Fix τ < T (α * , α 0 ) and then pick α ∈ (α 0 , α * ) such that qτ < T (α, α 0 ). By the arguments used above the right-hand side of (4.17) converges as n → +∞, uniformly on [0, τ ], to This completes the proof that k t given in (4.1) is a solution of the problem in (3.11) in the sense of Definition 3.1. The uniqueness stated in the theorem can be obtained similarly as in the proof of the same property in [3, Lemma 4.1].
5. Proof of Theorem 3.3. In this case, the proof is much longer and will be done in several steps. In view of (3.3), the solution described by Theorem 3.2 has the property k t (∅) = k 0 (∅) for all t < T (α * , α 0 ) since (L ∆ k)(∅) = 0. By the very choice of the spaces (3.5) this solution satisfies condition (b) of Proposition 1. Thus, it remains to prove that it has the positivity property defined in (2.9). To this end we make the following. First, in subsection 5.1 we introduce an auxiliary model, described by L σ with some σ > 0. For this model, by repeating the proof of Theorem 3.2 we obtain the evolution k 0 → k σ t in K θ -spaces. In subsection 5.3, we prove that holding for all G ∈ B bs , cf. (2.9) and (2.10). In the proof, we use the predual evolution constructed in subsection 5.2. To show that k σ t has the positivity property (2.9) we construct its approximations (subsection 5.4). As we then show, these approximations coincide with the directly obtained local correlation functions, see (5.31) and Corollary 2, that have the required positivity by construction. Finally, in subsection 5.4.4 we eliminate the approximation and thus obtain the desired positivity of k σ t .

Predual evolution.
To prove (5.1) we allow G to evolve accordingly to the rule G t , k 0 = G 0 , Q α * α0 (t)k 0 The proper context to this is to construct the corresponding evolution in the space predual to K α * , which ought to be of L 1 -type. For θ ∈ R, we introduce Obviously, for θ > θ, we have that G θ → G θ . Notably, G ∈ B bs lies in G θ with an arbitrary θ ∈ R. Indeed, let M be the bound of |G| and N and Λ be as in Definition 2.1. Then we have Lemma 5.1. Let Q α * α0 (t), α 0 ∈ R, α * > α 0 , t < T (α * , α 0 ), see (3.14), be the family of bounded operators constructed in the proof of Theorem 3.2. Then there exists the family H α0α * (t) : G α * → G α0 , t < T (α * , α 0 ) such that: (a) the norm of H α0α * (t) satisfies (4.16); (b) for each G ∈ G α * and k ∈ K α0 , the following holds is continuous in the operator norm topology.
Proof. Clearly, the most challenging part is the continuity stated in (c). Thus, we start by deriving the corresponding generating operator. To this end, we use the rule It can be shown, see [15], that it has the following form where Ψ is as in the last line of (3.3). Then define, cf. (3.10), Like in the dual spaces K θ , cf. (3.12), here we have that bothÂ andB mapD θ in G θ . This allows one to introduce the operatorsÂ θ = (Â,D θ ) andB θ = (B,D θ ) as well as bounded operatorsÂ θθ andB θθ mapping G θ to G θ for θ > θ. Their operator norms satisfy the same estimates as the norms of A θ θ and B θ θ , respectively, see (4.3). For such θ and θ , we also setŜ θθ (t) : G θ → G θ to be the multiplication operator by the function exp(−tΨ(η)). Similarly as in (4.5) one shows that which yields the continuity of the map [0, +∞) t →Ŝ θθ (t) in the operator norm topology. By the very construction of these operators we have that, for each G ∈ G θ and k ∈ K θ , the following holds where the second equality holds for all t ≥ 0. Now, for a given n ∈ N, α l , l = 0, . . . , 2n + 1 defined in (4.9) and t 1 , . . . , t n as in (4.10), we set Then we define Since the operator norms of allŜ andB satisfy the same estimates as the norms of respectively S and B, the operator norm of (n) α0α * * satisfies (4.13). Hence, the series in (5.11) converges in the norm topology, uniformly on compact subsets of [0, T (α * , α 0 )), which together with (5.9) yields the continuity stated in claim (c) and the bound stated in (a). In view of the convergence just mentioned, to prove (5.7) it is enough to show that, for each n ∈ N and 0 ≤ t n ≤ t n−1 ≤ · · · ≤ t 1 ≤ t, the following holds which is obviously the case in view of (5.10).

5.3.
Taking the limit σ → 0. Our aim now is to prove the following statement, cf. (5.1).

Approximations.
Our aim now is to prove that k σ t = Q σ α * α0 (t)k 0 has the positivity property defined in (2.9) whenever k 0 is the correlation function of a certain µ 0 ∈ P exp . Then, for t < T (α * , α 0 )/2, the same positivity property of k t = Q α * α0 (t)k 0 will follow by Lemma 5.2. Similarly as in [3,9], the main idea of proving the positivity of k σ t is to approximate it by a correlation function of a finite system of this kind, which is positive by Proposition 1. Thereafter, one has to prove that the positivity is preserved when the approximation is eliminated. 5.4.1. The approximate evolution. For a compact Λ ⊂ R d , let µ Λ 0 be the projection of the initial state µ 0 , see (2.3). Then its density R Λ µ0 and the correlation function k 0 are related to each other in (2.14). For N ∈ N, we set ≤ k µ0 , so that k µ0 ∈ K α0 implies k Λ,N 0 ∈ K α0 . Then by Theorem 3.2, is the unique classical solution of the problem on the time interval [0, T (α * , α 0 )).
, t < T (α * , α 0 ) be as in (5.28). Then, for each G ∈ B * bs and t < T (α * , α 0 ), the following holds The proof of this statement will follow by Corollary 2 proved below in which we show that Here R Λ,N t is the (non-normalized) density obtained from R Λ,N 0 given in (5.26) in the course of the evolution related to L σ . By this fact, q Λ,N t satisfies (5.29), which will yield the proof. According to this, we proceed by constructing the evolution R Λ,N 0 → R Λ,N t , which will allow us to use q Λ,N t defined in (5.31). The next step will be to prove (5.30).

5.4.2.
The local evolution. As just mentioned, the evolution R Λ,N 0 → R Λ,N t is related to the local evolution of the auxiliary model described by L σ , see subsection 5.1.
Here local means the following. Assume that the initial state ν 0 is such that ν 0 (Γ 0 ) = 1. That is, the system is finite and hence local. Assume also that it has density R ν0 = dν0 dλ . Then the evolution related to the Kolmogorov equation with L σ can be described as the evolution of densities by solving the corresponding Fokker-Planck equation where L † is related to L σ according to the rule Our aim now is to show that L † is the generator of a stochastic semigroup S † = {S † (t)} t≥0 , which we will use to obtain R Λ,N t in the form S † (t)R Λ,N 0 . In doing this, we follow the scheme developed in [3, Sect. 3.1].
The semigroup S † is supposed to act in the space G 0 , see (5.5). Along with this space we will also use Clearly, for each θ ∈ R and θ > θ, we have that Note that these are AL-spaces, which means that their norms are additive on the corresponding cones of positive elements These cones naturally define the cones of positive operators acting in the corresponding spaces. It is also convenient to relate this property to the following linear functionals Then ϕ(G) = |G| 0 and ϕ f ac θ (G) = |G| fac,θ for G ∈ G + 0 and G ∈ G f ac,+ θ , respectively. To formulate (5.32) in the Banach space G 0 , we have to define the corresponding domain of L † . To this end, we write and then set The construction of the semigroup S † is performed by means of the perturbation technique developed in [17]. We formulate here the corresponding statement borrowed from [3,Proposition 3.2] in the form adapted to the present context.

YURI KOZITSKY AND KRZYSZTOF PILORZ
Hence, to prove that all the conditions mentined in (ii) are met we have to show the continuity of the map t → (S † 0 (t)G)(η) ∈ G f ac θ . That is, we have to show that Γ0 1 − exp (−tE σ (η)) |G(η)|e θ|η| |η|!λ(dη) → 0, as t → 0 + , which obviously holds by the Lebesgue dominated convergence theorem since G ∈ G f ac θ . To check that B † : D † θ → G f ac θ , for G ∈ D † θ ∩ G + 0 , we apply (2.15) and obtain from (5.34) that This yields B † : D † θ → G f ac θ . In the same way, one shows that that completes the proof of item (iii).
To prove that (iv) holds, by (5.39), (5.35) and (5.44) we obtain Then the inequality in item (iv) can be written in the form where W (c, ε; ∅) = 0 and Since both first and second summands in (5.45) are bounded from above in η, one can pick c > 0 big enough to make W (c, ε; η) ≤ 0 for all η ∈ Γ 0 . This completes the proof of the lemma.
By (5.26) and (5.38) we have that Hence, R Λ,N 0 ∈ G f ac θ for any θ ∈ R. By the same arguments we also have that |R Λ,N 0 | 0 ≤ 1. Then, for all t > 0, we have that The latter means that one can apply L ∆,σ to q Λ,N t pointwise, and then calculate the integral with e(ω; ·). At the same time, for each θ ∈ R, we have Keeping in mind (5.47) and (5.48) let us define L ∆,σ in a given G f ac θ . To this end, we set, cf. (5.8), , it belongs to U σ α0 and to G fac β0 with To prove the former, by (5.27) we readily get Our aim now is to prove that both evolutions q Λ,N 0 = k Λ,N 0 → k Λ,N t and q Λ,N 0 → q Λ,N t take place in U σ α * . To this end, we define L ∆,σ in U σ θ and split it L ∆,σ = A σ + B σ , as we did in (5.3). Then set Like above, one can show that U σ θ ⊂ D σ θ whenever θ < θ, cf. (3.12). Thus, A σ : D σ θ → U σ θ . Likewise, B σ : D σ θ → U σ θ , and hence we can define in U σ θ the unbounded operators A σ u,θ = (A σ , D σ θ ), B σ u,θ = (B σ , D σ θ ) and L ∆,σ u,θ = (A σ +B σ , D σ θ ). Note that L ∆,σ u,θ satisfies, cf. (5.57), where the latter operator is the same as in (5.4). Likewise, by (5.58) we have with θ satisfying the bound in (5.58). That is, L ∆,σ u,θ = L ∆,σ θ | D σ θ , that holds for each θ ∈ R; as well as, L ∆,σ u,θ = L f ac,∆,σ θ | D σ θ , holding for all θ and θ satisfying (5.55). Let us now consider the problem d dt Its solution is to be understood according to Definition 3.1.
By means of these estimates we define a bounded operator (B σ u ) θ θ acting from U σ θ to U σ θ . Its norm satisfies the corresponding estimate in (4.3) with the same right-hand side. Then the proof follows in the same way as in the case of Theorem 3.2.

YURI KOZITSKY AND KRZYSZTOF PILORZ
Proof. We take and obtain by (5.55), (5.56) and (5.54) that is a unique classical solution of the problem in (5.4) with the initial condition k Λ,N 0 . Let u t , t < T (α * , α 0 ) be the solution of the problem in (5.61). Then the map t → u t ∈ U σ α * ⊂ K α * , cf. (5.64), is continuous and continuously differentiable in K α * as the corresponding embedding is continuous. By (5.59) u t satisfies also (5.4) with the initial condition q Λ,N 0 = k Λ,N 0 , and hence u t = k Λ,N t = Q σ α * α0 (t)q Λ,N 0 in view of the uniqueness of the solution of (5.4). Likewise, by (5.60) and Lemma 5.5 one proves that u t = q Λ,N t , t < T (α * , α 0 ), where q Λ,N t is the unique solution of (5.52) in G f ac β * with β * as in (5.63).
Since k 0 ∈ K α0 and G

|G
(p) [3,5,9], the construction of a Markov process corresponding to the stochastic evolution of our model is beyond the technical possibilities existing at this time. In view of this, we restricted ourselves to constructing the evolution of states in Theorem 3.3. Unlike to [3], in this work we failed to continue the evolution to all values of t > 0, which one would expect to be possible in view of the nature of the motion. The main technical reason for this is the presence of a positive part L ∆ 11 (see (3.3)) of L ∆ 1 , which could not be estimated in a way that would allow to construct such a continuation. Perhaps, one has to elaborate a more sophisticated method for this. Another direction in which we will continue studying the model introduced here is to get its mesoscopic limit, cf. [2], by constructing Poisson approximations of the states µ t and thereby by deriving the corresponding kinetic equation. We also plan to study our model numerically, mostly by means of the mentioned kinetic equation, to get a more detailed information on the properties of the solution the existence of which was proved here.