On asymptotic behavior of the modified Arratia flow

We study asymptotic properties of the system of interacting diffusion particles on the real line which transfer a mass [arXiv:1408.0628]. The system is a natural generalization of the coalescing Brownian motions. The main difference is that diffusion particles coalesce summing their mass and changing their diffusion rate inversely proportional to the mass. First we construct the system in the case where the initial mass distribution has the moment of the order greater then two as an $L_2$-valued martingale with a suitable quadratic variation. Then we find the relationship between the asymptotic behavior of the particles and local properties of the mass distribution at the initial time.


Introduction
In the paper we study local properties of the modified Arratia flow. The flow is a variant of the Arratia flow [3,11,25] for a system of Brownian motions on the real line which move independently up to their meeting and then coalesce. The fundamental new feature is that particles carry mass which is aggregated as particles coalesce and which determines the diffusivity of the individual particle in an inverse proportional way. The modified Arratia flow was first constructed in [19] (see also [22,18,24,23]), as a physical generalization of the system of coalescing Brownian motions, in the case where particles start from integer Universität Leipzig, Fakultät für Mathematik und Informatik, Augustusplatz 10, 04109 Leipzig, Germany; e-mail: konarovskiy@gmail.com Mathematics Subject Classification (2010): Primary 82B21, 60K35; Secondary 60D05 points with unit masses. Later in [20] the modified Arratia flow for a system of particles which start from all points of the interval [0, 1] with zero mass (the distribution of the mass of particles at the initial time is the Lebesgue measure on [0, 1]) was constructed as a scaling limit. The first main result of the paper is the generalization of the model constructed in [20] to the case of any mass distribution of particles at the start. Using martingale methods, we prove the following theorem. where m(u, t) = Leb{w : ∃s ≤ t X(u, s) = X(w, s)} and τ u,v = inf{t : X(u, t) = X(v, t)} ∧ T .
The process X describes the evolution of particles with the mass distribution µ at the start, where µ is the push forward of the Lebesgue measure on [0, b], i.e µ = g # Leb [0,b] . (1. 2) The following lemma explains that. is well-defined and satisfies (A1) for all x ∈ A the process Z(x, ·) is a continuous square integrable martingale with respect to the filtration σ(Z(x, s), x ∈ A, s ≤ t), t ∈ [0, T ]; (A2) for all x ∈ A, Z(x, 0) = x; (A3) for all x < y from A and t ∈ [0, T ], Z(x, t) ≤ Z(y, t); (A4) for all x, y ∈ A the joint quadratic variation where m µ (x, t) = µ{z : ∃s ≤ t Z(z, s) = Z(x, s)} and τ µ x,y = inf{t : Z(x, t) = Z(y, t)} ∧ T .
Proof. Since for g(u) = g(v) we have X(u, ·) = X(v, ·) (it follows from Remark 6.5 and propositions 6.2 and 2.3 below), the process Z is well-defined. Moreover, if x = g(u), So, we see that interpreting Z(u, t) as the position of the particle at time t starting from u, the family of processes {Z(u, ·), u ∈ A} is a description of the system of particles which start from almost all points of supp µ with the mass distribution µ. Although it seems that Z gives a simpler description of the model, it is easier to work with the process X. Firstly, the values of the random variable X(·, t) are functions defined on the interval (0, b), where the interval is independent of the support of the initial distribution µ (it only depends on the total mass of the system). Consequently, the particle system can be approximated by finite subsystems on the same state space. Secondly, X is an L ↑ 2 -valued continuous martingale with the quadratic variation X t = t 0 pr X(s) ds, where L ↑ 2 is the set of all non-decreasing functions from L 2 and pr f h denotes the projection of h in L 2 on the subspace of σ(f )-measurable functions. Moreover, we will show that each L ↑ 2valued continuous martingale X with the quadratic variation X t = t 0 pr X(s) ds has a modification that satisfies the same properties as X (see Theorem 6.4). Thus, to construct the modified Arratia flow it is enough to construct an L ↑ 2 -valued continuous martingale with the needed quadratic variation.
The second main result of the paper is a relationship between local properties of the distribution of particle mass at the start and asymptotic behavior of individual particles and its masses for small time. Using estimations of the expectations of particle mass and particle diffusion rate (see Section 7) and also the law of the iterated logarithm for the Wiener process, we prove the following statements. Theorem 1.3. Let α > 1 2 , u 0 ∈ (0, 1) and there exist C > 0 and δ > 0 such that the following assumptions hold Then for all ǫ > 0 Theorem 1.4. Let u 0 ∈ (0, 1), α > 1 2 and there exist δ > 0 and C > 0 such that then g also satisfies assumptions (i), (ii) of Theorem 1.3.
In particular, theorems 1.3, 1.4 imply that the modified Arratia flow constructed in [20] (where g(u) = u, u ∈ [0, 1]) has the following behavior for all u ∈ (0, 1) (see Remark 8.1). We note that the asymptotic behavior of each particle in the Arratia flow {a(u, t), u ∈ R, t ≥ 0} is as follows since each process a(u, ·) is a Brownian motion with unit diffusion rate. Moreover, the process ν(t) = Leb{u : ∃s ≤ t a(u, s) = a(0, s)}, t ≥ 0, that describes the cluster size (it corresponds to the particle mass in our case), has the following behavior [8] Comparing the behavior of particles and their masses in the modified Arratia flow with the behavior of particles in the Arratia flow, we see that asymptotics are completely different, since the diffusion rates of particles in the first case grow to infinity and make particles to fluctuate more and more intensively for small time.
Here we would like to note that many methods which work for studying of local properties of the Arratia flow do not work in our case, since they are based on the fact that every system of particles can be considered separately from the whole system. Therefore, the Arratia flow can be investigated just by studying of its finite subsystems (see, e.g [10,28,5,9]). There is an opposite situation for studying of the modified Arratia flow, where every finite subsequence cannot be considered as a separate system.
The modified Arratia flow has a connection with the Wasserstein diffusion, constructed by M.-K. von Renessse and T. Sturm in [31] (see also [2,29,30]). In fact, in [21] V. Konarovskyi and M.-K. von Renesse proved that the process describing the evolution of particle mass in the modified Arratia flow solves a SPDE that is similar to the SPDE for the Wasserstein diffusion and also showed via a large deviation analysis that the flow satisfies the Varadhan formula with the square of the Wasserstein distance as the rate function.
is a weak solution to the equation where d W denotes the (quadratic) Wasserstein metric on the space of probability measures on R. Basically we believe that the same form of the short time behavior of the particle system with the initial particle distribution µ is valid for any probability measure µ (with 1] . Thus, the process constructed in the present paper can be considered as a candidate for an intrinsic Brownian motion on the Wasserstein space of probability measures. Consequently, the question of existence such a process is important and its local properties is of interest.

Organization of the article.
In section 2 we introduce the main notation and formulate some statements about L ↑ 2valued continuous martingales. In section 3 a finite system of particles is defined as a continuous martingale taking values in L ↑ 2 . The main estimations for the particle system is obtained in section 4. Section 5 is devoted to the construction of an L ↑ 2 -valued continuous martingale X which starts from a function g ∈ L 2+ε and has the quadratic variation X t = t 0 pr X(s) ds. In section 6 we prove that the martingale X has a modification from the Skorohod space D((0, b), C[0, T ]) that satisfies similar properties as the flow constructed in [20]. Section 7 is the key section of the paper. There we obtain estimations of the expectations of mass and diffusion rate of individual particles which allow to state the asymptotic behavior of the particle system in section 8.

The main definitions 2.1 Some notation
For p ≥ 1 we denote the space of p-integrable functions (more precisely equivalence classes) from [a, b] to R by L p [a, b] or L p and · Lp is the usual norm on L p . Also (·, ·) denotes the inner product in Since each function f from L ↑ p has a unique modification from D ↑ (see Remark A.6), considering f as a map from [a, b] to R, we always take its modification from D ↑ .
For each f ∈ L ↑ p , let Π f denote the class of sets {v ∈ [a, b] : f (v) = c}, c ∈ R, of the positive length (as we agreed, f ∈ D ↑ ). Since f is a non-decreasing function, elements of , then |Π f | denotes the number of elements in Π f , otherwise |Π f | = +∞. Let us introduce the partial order for Π · . We write Π g ≤ Π f if for each π ∈ Π f there exists π ′ ∈ Π g such that Int π ⊆ π ′ , where Int π denotes the interior of π. Let |π| denote the length of π for π ∈ Π f . Remark 2.1. From definitions of Π · and |Π · | it follows that the inequality Π g ≤ Π f implies |Π g | ≤ |Π f |.

L ↑ 2 -valued martingales
Let (F t ) t∈[0,T ] be a right continuous filtration on a probability space (Ω, F , P). An L ↑ 2valued continuous random process X(t), t ∈ [0, T ], given on (Ω, F , P), is called an , then we will call X just a square integrable martingale.
It is well-known that two real-valued continuous martingales T ] coincide after their meeting. This property implies that Π X(t) , t ≥ 0, decreases a.s.
We define the quadratic variation X t , t ∈ [0, T ], of X as an (F t )-adapted continuous process starting from zero, with values in the space of nonnegative definite trace-class operators on L 2 , such that for all h, g ∈ L 2 the joint quadratic variation of the martingales For more details we refer to [13].

A finite system of particles
In this section we construct an L ↑ 2 [a, b]-valued square integrable martingale with the suitable quadratic variation that describes the evolution of a finite system of coalescing diffusion particles. Let the system of processes . . , m 0 d . Such a system of processes has been constructed e.g. in [22] and satisfies the following properties Moreover, (F 1) − (F 4) uniquely determine the distribution of the system that is stated in the following lemma.
Proof. The proof is similar to the proof of Lemma 3 [19].
Let us construct an L ↑ 2 -valued process that corresponds to the system and is a martingale. Consequently, X is a square integrable martingale. Let us evaluate its quadratic variation. Denote the projection of h in L 2 on the subspace of σ(g)-measurable functions by pr g h. If g is defined by (3.1), then Using properties (F 1) − (F 4), similarly to [21] one can show that By the polarization formulas for the inner product (·, ·) and the joint quadratic variation ·, · · , we obtain for h, f ∈ L 2 Thus, we have shown that X is an L ↑ 2 -valued continuous square integrable martingale with the quadratic variation We note that t 0 pr X(s) ds is a trace-class operator, since X is a square integrable martingale [13, Lemma 2.1]. It follows also from the fact that pr X(s) is a projection on a space with dimension smaller or equal than d for all s ∈ [0, T ].
Next we prove the inverse statement.
, be as above, g be defined by (3.1) and X be an L ↑ 2 [0, b]-valued continuous square integrable martingale with the quadratic variation · 0 pr X(s) ds. Then there exists a system of processes We denote the modification also by X.
Then by Proposition 2.3, We evaluate It finishes the proof.
Lemmas 3.1 and 3.2 immediately imply the following result.
-valued continuous square integrable martingale with the same quadratic variation that starts from g, then the distributions of X and Y coincide in We denote the distribution of the L ↑ 2 -valued continuous square integrable martingale with quadratic variation (M ′ ) starting from g in the space C([0, T ], L ↑ 2 ) by P g . We will consider the set of step functions St as a topological subspace of L ↑ 2 with the induced topology. Let P be the space of all probability measures on C([0, T ], L ↑ 2 ), endowed with the weak topology. Since the system of processes can be constructed by coalescence of Wiener trajectories (see e.g. [19,22]) and X can be defined by (3.2), it is easy to see that the map P · : St → P is measurable. Consequently, the probability measures is well-defined for any random element ξ in St with the distribution Ξ.
Proof. The statement follows from the existence of regular conditional distribution of X given σ(X(0)) (see Theorem 1.3.1 [14]) and Proposition 3.3.

The main estimations
In this section we will suppose that Y It should be noted that in this section we do not claim that the process Y exists, here we only study properties of Y if it exists.
We will interpret Y as the description of the evolution of particles on the real line which coalesce and change their masses and diffusion rates. Since m(u, t) is the mass of particle at time t that starts from g(u), the inequality m(u, t) < r implies that the particles starting from g(u) and g(u + r) (g(u − r)) have not coalesced by t. Moreover, the particle, which starts from g(u), has diffusion rate grater then 1 r . Consequently, P{m(u, t) < r} can be estimated by P{the Wiener process starting from g(u + r) − g(u) with diffusion 1 r does not hit 0 by time t}. This is the main idea of the proof of the following lemma that is the key statement that allows to prove the existence of a martingale with the quadratic variation (M ′ ) which starts from g ∈ L ↑ p and to study its asymptotic behavior.

Remark 4.2.
The lemma also is true if the assumption 0 < r < b − u is replaced by 0 < r < u − a and the function G(u, r) = g(u + r) − g(u) by G(u, r) = g(u) − g(u − r).
Proof of Lemma 4.1. The proof is similar to the proof of Lemma 2.16 [20]. Let 0 < r < b − u. We denote and Note that M(·) is a continuous square integrable martingale with the quadratic variation Taking Next, since M(·) is a continuous square integrable martingale, there exists a Wiener process w(t), t ≥ 0, such that by Theorem 2.7.2' [14]. We set It is easy to see that (4.1) implies Note that if ω ∈ A t , then τ (ω) > t and hence, by the last inequality, Now we are ready to estimate the probability of A t . So, It finishes the proof.
Proof. Without loss of generality, we assume that a = 0. Using Lemma 4.1 and Hölder's inequality, we can estimate The proposition is proved.
Proof. Without loss of generality, let X be defined by (3.2). Using the Burkholder-Davis-Gundy inequality and Proposition 4.3, we obtain The proposition is proved.
Since for all h ∈ L 2 the process (X n (·), h) is a continuous square integrable martingale with the quadratic variation and pr Xn(t) h L 2 ≤ h L 2 , t ∈ [0, T ], the Aldous tightness criterion (see e.g. Theorem 3.6.4. [7]) easily implies (J2). It completes the proof of the proposition.

Some limit properties
In this subsection we show that under the assumption {|Π gn |, n ≥ 1} is bounded, each limit point of the set {P gn } n≥1 is P g for some g ∈ St.
Proof. Let Π gn = {π n 1 , . . . , π n qn }, where elements of π n k are less then elements of π n k+1 , k ∈ [q n − 1]. Since {q n } n≥1 is bounded, there exist an infinite sequence {n ′ } and q ∈ N such that q n ′ = q for all n ′ . Without loss of generality, we may assume that q n = q for all n ≥ 1. Next, setting m 0,n k = |π n k |, k ∈ [q], n ≥ 1, and using the boundedness of Again, without loss of generality, we assume that n ′ = n. Set x 0,n k = 1 m 0,n k π n k g n (u)du, k ∈ [q], n ≥ 1.
Since m 0,n k → m 0 k > 0, k ∈ I, and { g n L 2 } n≥1 is bounded, it is easy to see that {x 0,n k } n≥1 is also bounded for all k ∈ I. Thus, there exists a sequence {n ′ } such that x 0,n ′ k → x 0 k for all k ∈ I. Let again n ′ = n.
Next, let and for all π ∈ Π g there exist π i 1 , . . . , π i l ′ such that π = l ′ j=1 π ′ i j . We set h n k = 1 m 0,n k I π n k , k ∈ [q], n ≥ 1, By the construction of m k i and π ′ i , i ∈ [l], we have h n k i → h i in L 2 for all i ∈ [l]. Next, using Skorohod's theorem (see Theorem 3.1.8 [12]), we may assume that We note that, by Proposition 2.2, for all t ∈ [0, T ] x n k (t)I π n i a.s., n ≥ 1, It is easy to see that for all i ∈ [l] We can estimate the second moment of x n k i (t), i ∈ [l], as follows where C is a constant that is independent of n, t and k i . By Fatou's lemma Ex 2 i (t) ≤ C for all t ∈ [0, T ] and i ∈ [l]. Therefore, Proposition 9.1.17 [15] implies that x i is a continuous (F t )-square integrable martingale for any i ∈ [l]. To finish the proof of the lemma, we show that the joint quadratic variation of x i and x j satisfies (F 4).
By Lemma 2.10 [20], for each i, j ∈ [l], τ n k i ,k j → τ i,j in probability. Since we can choose a sequence {n ′ } such that τ n ′ k i ,k j → τ i,j a.s. for all i, j = 1, . . . , l, without loss of generality, we may suppose that τ n k i ,k j → τ i,j a.s. Let us denote It is easily seen that Leb{R} = 0 and for all t ∈ R c = [0, T ] \ R Note, that in the first term of the right hand side tends to m i (t) = l j=1 m 0 k j I {τ i,j ≤t} a.s. and the second term tends to zero. So, m n k i (t) → m i (t) a.s. for all i = 1, . . . , l and t ∈ R c . Since is bounded uniformly by t for all i ∈ [l]. Hence, by the dominated convergence theorem, we obtain Thus, Lemma B.11. [6] implies that The lemma is proved.
Proof. By Proposition 5.2 and Lemma 5.3, every subsequence of {P gn } n≥1 has a subsubsequence converging to P g . It proves the proposition.

Existence in the general case
In this section we construct an L ↑ 2 -valued continuous square integrable martingale with the quadratic variation (M ′ ) starting from g ∈ L ↑ 2+ε as a weak limit of processes with distributions P gn , g n ∈ St. Proof. We set S n = σ k−1 2 n , k 2 n , k ∈ [2 n ] , n ≥ 1, and g n = E Leb (g|S n ), where E Leb denotes the conditional expectation on the probability space ([0, 1], B([0, 1]), Leb). Since g n → g in L 2+ε (see [1]), the sequence { g n L 2+ε } n≥1 is bounded. Therefore the sequence {P gn } n≥1 is tight in P, by Proposition 5.2.
On the other hand, P Xn(r k ) → Law{X(r k + ·)} and consequently, X(r k + ·) has the distribution P X(r k ) . So, from Proposition 3.4 it follows that X(r k + ·) is an L ↑ 2 -valued continuous square integrable martingale with the quadratic variation for all h ∈ L 2 . Since for each h ∈ L 2 , (X n (·), h) → (X(·), h) a.s. and one can show that (X(·), h) is a continuous square integrable martingale and Therefore, Making r k ′ → 0, we obtain The theorem is proved.

Coalescence in a finite number of points
We will prove that any L ↑ 2 -valued continuous square integrable martingale X(t), t ∈ [0, T ], with the quadratic variation (M ′ ) takes values from St and for all t ∈ (0, T ] Let us prove an auxiliary lemma.  Proof. We suppose that g ∈ St and prove (6.1). Let Π g = {π k , k ∈ [q]}. Then pr g e n = q k=1 1 |π k | π k e n (u)duI π k and consequently, Hence, Next, suppose that (6.1) holds. Then pr g is a Hilbert-Schmidt operator. Since pr g is a projection on the subspace of σ(g)-measurable functions H g ⊂ L 2 , it is easy to see that H g is a finite dimensional Hilbert space. Therefore σ(g) is generated by a finite number of sets. This implies that g ∈ St. The lemma is proved.

Estimations of the expectation of mass and diffusion rate
Throughout this and the next sections we will suppose that {X(u, t), u ∈ (0, 1), t ∈ Proof. To prove the proposition, we will use the estimation of P{m(u, t) < r} (see Lemma 4.1). Assume that t ∈ (0, T ] is fixed and g(u 0 + u) − g(u 0 ) ≤ Cu α , u ∈ [0, δ], for some C > 0 and δ > 0. For the case g(u 0 ) − g(u 0 − u) ≤ Cu α the proof is similar. We estimate and note that x t strictly decreases to zero for each t. Consequently, there exists the inverse map r t (x) = max{r : where c = (g(u 0 + δ) − g(u 0 )) √ δ. Hence, interchanging of integrations, we obtain Next, for fixed x ′ ∈ 0, c/ √ t we denote By assumption of the proposition, for all r ≥ 1 Thus, using the inequality g(u 0 +1/r ′ )−g(u 0 ) √ r ′ ≥ x ′ √ t and (7.1), we have Let us come back to the estimation of E 1 m(u 0 ,t) . So, since the integral in the brackets {·} is finite for α > 1 2 . The proposition is proved.

Estimation of the expectation of mass
Let u 0 ∈ (0, 1) be fixed. In this subsection we estimate the expectation Em(u 0 , t) in the case where the function g(u 0 + ·) − g(u 0 ) is locally (at zero) similar to |u| α , u ≤ δ. To get the estimation we use the rescaling property of X. The following statement holds. Lemma 7.3. Let X ρ be defined by (7.2) with q = −u 0 and there exists C > 0 such that Em ρ (0, T ) ≤ C for all ρ ∈ (0, 1]. Then Proof. By (7.3), It implies the needed estimation if t = T ρ γ ∈ (0, T ].
Next, we will estimate Var M ρ (u, T ). By (C4), where ξ ρ (u) = T 0 E 1 mρ(u,s) ds. It should be noted that ξ ρ (0), ρ ∈ (0, 1], is bounded. Indeed, inserting ς = u − u 0 in (i), we can see that Thus, by Proposition 7.1, we have the estimation Hence, for all ρ ∈ (0, 1] where c is a constant. Consequently, Note that if ξ ρ (u) was bounded with respect to ρ and u (e.g. it is true if g(u) = u, u ∈ (0, 1)), then the integral would be bounded with respect to ρ. It would prove the proposition. In general, we should not expect that ξ · (·) is bounded, since it depends on local properties of g ρ at each point. So, in order to prove the boundedness of the integral, we will use Lemma 4.1.
Since 1 mρ(u,s) ∈ [ρ, ∞) for all s ∈ (0, T ] and u ∈ (b, b/ρ), we can estimate Let bρ < δ. Then by (i) and (ii), Here the integral b δ +∞ θ ρ g(u+u 0 )−g(u+u 0 −1/r) √ r dr du is estimated similarly as in the proof of Proposition 4.3. Since α > 1 2 , it is easy to see that the right hand side of the latter inequality is bounded by a constant that is independent of ρ. It finishes the proof of the proposition.
Proof. The statement follows from the Cauchy-Schwarz inequality. Indeed, Proof of Proposition A.1. We note that if g ∈ D ↑ , then it is easily seen that (A) holds. So, we need to show that (A) implies that g has a modification from D ↑ . From the previous lemmas the sequence {(g, h u,ε )} ε>0 is bounded and decreasing for all u ∈ (a, b). Consequently, there exists a limit g(u) = lim ε→0+ (g, h u,ε ) for all u ∈ (a, b).
From (A) it follows that g(u), u ∈ (a, b), is increasing. So, we can set g(a) = lim u→a+ g(u) and g(b) = lim u→b− g(u).
Let us show that g belongs to D ↑ . Take r > 0, u ∈ (a, b) and a sequence {u n } n≥1 ⊂ (a, b) such that u n ↓ u. Then there exists ε ∈ (0, b − u) such that (g, h u,ε ) − g(u) < r 2 .
We note that U is dense in (a, b) and since g is a monotone function, U is also countable.
By the monotonicity of X(u, t) in u, we have Similarly, Hence, | X(u, t)| = X + (u, t) + X − (u, t) ≤ Next, if s < t, then E ( X ε (u, t)| F s ) = X ε (u, s) and the monotone convergence theorem imply E X(u, t) F s = X(u, s). The proposition is proved.