On Number of Particles in Coalescing-Fragmentating Wasserstein Dynamics
aa r X i v : . [ m a t h . P R ] F e b ON NUMBER OF PARTICLES INCOALESCING-FRAGMENTATING WASSERSTEIN DYNAMICS
VITALII KONAROVSKYI
Abstract.
Because of the sticky-reflected interaction in coalescing-fragmentatingWasserstein dynamics, the model always consists of a finite number of distinct par-ticles for almost all times. We show that the interacting particle system must admitan infinite number of distinct particles on a dense subset of the time interval if andonly if the space generated by the interaction potential is infinite-dimensional. Introduction
The coalescing-fragmentating Wasserstein Dynamics is a system of interacting parti-cles on the real line which intuitively can be described as follows. Diffusion particles startat some finite or infinite family of points move independently until their meeting. Everyparticle transfer a mass and its diffusion rate is inversely proportional to its mass. Whenparticles meet they sticky-reflect from each other according to some interaction potentialwhich is described by a non-decreasing right-continuous bounded function ξ : [0 , → R ,which we call an interaction potential . The evolution of the particle system is similarto the motion of particles in the Howitt-Warren flow [2]. The main difference is thatthe motion of particles in CFWD inversely-proportionally depends on their masses. Inparticular, particles with “infinitesimally small” mass have “infinitely large” diffusionrate. Let X ( u, t ) be the position of a particle labeled by u ∈ (0 ,
1) at moment t ≥
0. Theevolution of such a family of particles can be defined by the following SDE dX t = pr X t dW t + ( ξ − pr X t ξ ) dt, t ≥ ,X = g, (1)in the space L ↑ of all 2-integrable functions (classes of equivalences) f : (0 , → R which have a non-decreasing version, where W t , t ≥
0, is a cylindrical Wiener processin L = L ([0 , , du ) and pr f denotes the orthogonal projection in L onto its subspace L ( f ) of all σ ( f )-measurable functions. The initial condition g ∈ L ↑ describes the massdistribution µ of particles, that is defined as the image of the Lebesgue measure Leb on[0 ,
1] under the map g .In [4], the author shows the existence of a weak solution to equation (1) for any (2+)-integrable initial condition g , for some ε >
0. More precisely, if R g ε ( u ) du < ∞ forsome ε >
0, then there exists an L -valued cylindrical Wiener process W t , t ≥
0, anda continuous L ↑ -valued process X t , t ≥
0, both defined on the same filtered probabilityspace (Ω , F , ( F ) t ≥ , P ) such that E k X t k L < ∞ , t ≥
0, and X t = g + Z t pr X s dW s + Z t ( ξ − pr X s ξ ) ds, t ≥ . Let D ([0 , , C [0 , ∞ )) denote the Skorohod space of all c`adl`ag functions from [0 ,
1] tothe space C [0 , ∞ ) of real-valued continuous functions defined on [0 , ∞ ). If the initial Mathematics Subject Classification.
Primary 60K35, 60H05 ; Secondary 60H05, 60G44.
Key words and phrases.
Sricky-reflected particle system, modified massive Arratia flow, infinite di-mensional singular SDE.The work is supported by the Grant “Leading and Young Scientists Research Support” No.2020.02/0303. condition g and the interaction potential ξ are right-continuous and piecewise ( +)-H¨older continuous , then equation (1) admits a weak solution which have a modification { X ( u, t ) , t ≥ , u ∈ [0 , } from D ([0 , , C [0 , ∞ )), where X ( u, · ), u ∈ [0 , X t := X ( · , t ), t ≥
0, is aweak solution to equation (1) in the space L ↑ and the following properties holds:(R1) for all u ∈ [0 , X ( u,
0) = g ( u );(R2) for each u < v from [0 ,
1] and t ≥ X ( u, t ) ≤ X ( u, t );(R3) the process M ( u, t ) := X ( u, t ) − g ( u ) − Z t ξ ( u ) − m ( u, s ) Z π ( u,s ) ξ ( v ) dv ! ds, t ≥ , is a continuous square integrable martingale with respect to the filtration F t = σ ( X ( v, s ) , u ∈ [0 , , s ≤ t ), t ≥
0, where π ( u, t ) = { v : x ( u, t ) = X ( v, t ) } and m ( u, t ) = Leb π ( u, t );(R4) the joint quadratic variation of M ( u, · ) and M ( v, · ) equals h M ( u, · ) , M ( v, · ) i t = Z t I { X ( u,s )= X ( v,s ) } m ( u, s ) ds, t ≥ . The uniqueness of a solution to equation (1) remains an important open problem. Theexistence of reversible CFWD and their connection to Wasserstein diffusion [12] and thegeometry of the Wasserstein space of probability measures on the real line was studiedin [8].
Remark . Considering f ∈ L ↑ as a function, we will always take its right continu-ous version on (0 ,
1) which exists and is unique according to Proposition A.1 [5] andRemark A.6 ibid.We will denote by ♯f a number of distinct values of f ∈ L ↑ . By Lemma 6.1 [5], thesquare of the Hilbert-Schmidt norm of the orthogonal projection pr f coinsides with ♯f ,i.e.(2) k pr f k HS := ∞ X n =1 k pr f e n k L = ♯f, where { e n , n ≥ } is an orthonormal basis in L . Therefor, we can interpret the randomvariable k pr X t k HS = ♯X t as a number of distinct particles in CFWD at time t ≥
0. Inparticular, if X t = X ( · , t ), t ≥
0, where the random element { X ( u, t ) , t ≥ , u ∈ [0 , } in D ([0 , , C [0 , ∞ )) satisfies conditions (R1)-(R4), then ♯X ( · , t ) is exactly the number ofdistinct particles at time t ≥ X t , t ≥
0, is square integrable and ξ is bounded, Theorem 2.4 [1] and equality (2) imply Z t E ( ♯X s ) ds < ∞ for all t ≥
0. This yields that(3) P { ♯X t < ∞ for a.e. t ∈ [0 , ∞ ) } = 1 , i.e. the CFWD consists of a finite number of particles at almost all times. The goal ofthis paper is to show that with probability 1 there exists a (random) dense subset ofthe time interval [0 , ∞ ), where the CFWD has infinite number of particles if and onlyif ♯ξ = ∞ . We remark that the property ♯ξ = ∞ is equivalent to the fact that L ( ξ ) isinfinite dimensional, according to (2). Theorem 1.1. (i) If ♯ξ = + ∞ , then almost surely there exists a (random) densesubset R of [0 , ∞ ) such that ♯X t = ∞ , t ∈ R . There exist ε > ,
1] such that the functions are (cid:0) + ε (cid:1) -H¨oldercontinuous on each interval of the partition N NUMBER OF PARTICLES IN CFWD 3 (ii) If ♯ξ < ∞ , then P { ♯X t < ∞ , t ∈ [0 , ∞ ) } = 1 . We remark that the CFWD coincides with the modified massive Arratia flow [5, 6, 7,9, 10] for ξ = 0. In this case, the claim of the theorem was proved in [5, Proposition 6.2].2. Auxiliary statements
Let C ([ a, b ] , L ↑ ) denote the space of continuous functions from [ a, b ] to L ↑ endowedwith the distance of uniform convergence. We recall that the map h
7→ k pr h f k L from L ↑ to R is lover semi-continuous for each f ∈ L , that is,(4) k pr h f k L ≤ lim n →∞ k pr h n f k L , as h n → h in L ↑ . The proof of this fact can be found in [4, Lemma A.4]. By Fatou’s lemma, the map h
7→ k pr h k HS is lover semi-continuous as well.The following lemma is needed for the measurability of events which will appear inthe proof of Theorem 1.1. Lemma 2.1.
For each [ a, b ] , the map f sup t ∈ [ a,b ] k f t k HS from C ([ a, b ] , L ↑ ) to R ∪{ + ∞} is measurable.Proof. We first note that the map f
7→ k pr f t k HS from C ([ a, b ] , L ↑ ) to R is lover semi-continuous for each t ≥ C ([ a, b ] , L ↑ ) ∋ g g t ∈ L ↑ and the lower semi-continuouous map L ↑ ∋ h
7→ k pr h k HS ∈ R . This yields theclaim of the lemmas due to the measurability of f
7→ k pr f t k HS and the equality { f : sup t ∈ [ a,b ] k f t k HS ≤ c } = \ t ∈ [ a,b ] ∩ Q (cid:8) f : k pr f t k HS ≤ c (cid:9) , for all c ≥ (cid:3) The following lemma directly follows from the lover semi-continuity of the map t pr f t k HS for every f ∈ C ([0 , ∞ ) , L ↑ ). Lemma 2.2.
For every f ∈ C ([0 , ∞ ) , L ↑ ) , c ≥ and ≤ a < b the set A f,a,bc := { t ∈ [ a, b ] : k pr f t k HS ≤ c } is closed in [0 , ∞ ) . We will also need a property of a function f ∈ C ([0 , ∞ ) , L ↑ ) if the Hilbert-Schmidtnorm k pr f t k HS , t ∈ [0 , ∞ ), is constant on an interval. Lemma 2.3.
Let f belong to C ([0 , ∞ ) , L ↑ ) and k pr f t k HS , t ∈ [ a, b ] , be a constant forsome ≤ a < b . (i) For every u ∈ (0 , there exist u < u < u and α < β from [ a, b ] such that f t is a constant on [ u , u ) and [ u , u ) for each t ∈ [ α, β ] . (ii) For u = 0 (resp. u =1) there exist u > u (resp. u < u ) and α < β from [ a, b ] such that f t is a constant on [ u , u ) (resp. on [ u , u ] ) for each t ∈ [ α, β ] .Proof. Since k pr f · k HS is a constant on [ a, b ], the function f t takes a fixed number ofdistinct values, denoted by n , for each t ∈ [ a, b ], by equality (2). Let f t = n X k =1 x k ( t ) I [ q k − ( t ) ,q k ( t )) , t ∈ [ a, b ] , where x ( t ) < . . . < x n ( t ) and 0 = q ( t ) < q ( t ) < . . . < q n ( t ) = 1.We first check that the functions x k and q k are continuous on [ a, b ] for each k in [ n − t n ∈ [ a, b ], n ≥ t . We can choose a subsequence N ⊂ N VITALII KONAROVSKYI such that x k ( t n ) → y k and q k ( t n ) → p k along N for each k ∈ [ n − y k = −∞ ( y k = + ∞ ), then p k = 0 (resp. p k = 1) due to k f t k L < ∞ , t ∈ [ a, b ]. We set h := n X k =1 y k I [ p k − ,p k ) . Then, it is easy to see that f t n → h a.e. along N . But, by the continuity of f , f t n → f t in L . Thus, f t = h , that implies the equalities y k = x k ( t ) and p k = q k ( t ) for all k ∈ [ n ]. Thus, the needed continuity holds.If there exists l ∈ [ n ] such that(5) u ∈ ( q l − ( t ) , q l ( t )) for some t ∈ ( a, b ) , then one can take u < u < u and α < β from [ a, b ] satisfying u , u in ( q l − ( t ) , q l ( t ))for all t ∈ [ α, β ], by the continuity of q k , k ∈ [ n − l satisfying (5) does not exist, then u = q l ( t ) for some l ∈ [ n ] ∪ { } andall t ∈ [ a, b ], which also yields the statement. (cid:3) Proof of Theorem 1.1
In order to show that with probability 1 there exists a dense subset R of [0 , ∞ ) suchthat ♯X t = ∞ for all t ∈ R , it is enough to prove that P ( sup t ∈ [ a,b ] ♯X t = ∞ ) = 1 , where the measurability of sup t ∈ [ a,b ] ♯X t follows from Lemma 2.1 and equality (2). In-deed, this will imply P {∃ R dense in [0 , ∞ ) such that ♯X t = ∞ , ∀ t ∈ R } = P \ a . Setting A a,bn := (cid:8) t ∈ [ a, b ] : k pr X t k HS ≤ n (cid:9) and using equality (2), we can concludethat P ( ∞ [ n =1 A a,bn = [ a, b ] ) > . By Lemma 2.2 and the Baire category theorem, we have P (cid:8) ∃ a < b from [ a, b ] and n ∈ N such that k pr X t k HS ≤ n ∀ t ∈ [ a , b ] (cid:9) > . Consequently, we can find non-random a < b from [ a, b ] and k ∈ N such that P (cid:8) k pr X t k HS ≤ k ∀ t ∈ [ a , b ] (cid:9) > . Since, P ((cid:8) k pr X t k HS ≤ k ∀ t ∈ [ a , b ] (cid:9) ∩ k [ k =1 n A a ,b k \ A a ,b k − = ∅ o!) > , there exists k ≤ k satisfying P n(cid:8) k pr X t k HS ≤ k ∀ t ∈ [ a , b ] (cid:9) ∩ n A a ,b k \ A a ,b k − = ∅ oo > , N NUMBER OF PARTICLES IN CFWD 5 where A a ,b := ∅ . Next, since A a ,b k \ A a ,b k − is open in A a ,b k and non-empty withpositive probability, we can find non-random a < b from [ a , b ] satisfying P (cid:8) k pr X t k HS = k ∀ t ∈ [ a , b ] (cid:9) > . Next, due to the equality ♯ξ = ∞ , there exists u ∈ [0 ,
1] such that ξ takes an infinitenumber of distinct values in [ u , u ) for all u < u or in [ u , u ) for all u > u . UsingLemma 2.3 and the monotonisity of X t ( ω ) for all t and ω , one can find non-random a < b from [ a , b ] and u < v such that u = u or v = u , ξ takes an infinite numberof distinct values on [ u, v ] and P {∀ t ∈ [ a , b ] X t is a constant on [ u, v ) } > . Let h := I [( u + v ) / ,v ) − I [ u, ( u + v ) / . Since X t , t ≥
0, solves equation (1), one has that( X t , h ) L , t ≥
0, is a continuous non-negative process such that M h ( t ) = ( X t , h ) L − Z t (cid:0) ξ − pr X s ξ, h (cid:1) L ds, t ≥ , is a continuous square integrable ( F t )-martingale with quadratic variation h M h i t = Z t k pr X s h k ds, t ≥ . Thus, using the equalities ( X t , h ) L = 0, pr X s h = 0 and(pr X s ξ, h ) L = ( ξ, pr X s h ) L = 0for all s ∈ [ a , b ] on the event A := {∀ t ∈ [ a , b ] X t is a constant on [ u, v ) } , we have(6) M h ( t ) = Z a ( ξ − pr X s ξ, h ) L ds + Z ta ( ξ, h ) L ds and [ M h ] t = Z a k pr X s h k ds on A for all t ∈ [ a , b ]. The equality for the quadratic variation of M h and the rep-resentation of continuous martingales as a time shift of a Brownian motion (see [3,Theorem II.7.2’]) imply that M h ( t ) = M h ( a ), t ∈ [ a , b ], on A . But according toequality (6), M h ( t ), t ∈ [ a , b ], is strictly increasing on the event A because ( ξ, h ) L > P { A } >
0, we get a contradiction. This finishes the proof of the first part of thetheorem.We next prove claim (ii). Due to ♯ξ < ∞ , there exists a finite partition π k , k ∈ [ n ], ofthe interval [0 ,
1) by intervals of the form [ a, b ) such that ξ ( u ) = n X k =1 ξ k I π k ( u ) , u ∈ [0 , . In order to prove (ii), it is enough to show that almost surely X t takes a finite numberof distinct values on every interval π k . We fix k ∈ [ n ] and consider the countable familyof functions h u,v := I [( u + v ) / ,v ) − I [ u, ( u + v ) / from L , u, v ∈ π k ∩ Q , denoted by R .We first remark that for every h ∈ R the process ( X t , h ) L , t ≥
0, is a non-negative con-tinuous supermartingale. Indeed, the non-negativity follows from the fact that ( f, h ) L ≥ f ∈ L ↑ and h ∈ R . In order to show that ( X t , h ) L , t ≥
0, is a supermartin-gale, we use the fact that it is a weak martingale solution to equation (1). Hence for each h ∈ R ( X t , h ) L = M h ( t ) + Z t (cid:0) ξ − pr X s ξ, h (cid:1) L ds = M h ( t ) − Z t (cid:0) pr X s ξ, h (cid:1) L ds, t ≥ , where M h is a martingale. According to Lemma A.2 [4], the orthogonal projectionpr f maps the space L ↑ into L ↑ for every f ∈ L ↑ . Hence, pr X s ξ ∈ L ↑ and, therefore, (cid:0) pr X s ξ, h (cid:1) ≥
0. This implies that ( X t , h ) L , t ≥
0, is a continuous supermartingale.
VITALII KONAROVSKYI
For every h ∈ R we defineΩ h = (cid:26) for every t ∈ [0 , ∞ ) the equality ( X t , h ) L = 0implies ( X t , h ) L = 0 for all s ≥ t (cid:27) . By Proposition II.3.4 [11], P { Ω } = 1. Thus, the event Ω ′ := T h ∈R Ω h has the prob-ability 1. Take ω ∈ Ω ′ , u, v ∈ (0 , t ≥ X t ( u, ω ) = X t ( v, ω ). Thenfor every h ∈ R one has ( X t ( ω ) , h ) L = 0 and, consequently, ( X s ( ω ) , h ) L = 0, by thechoise of ω . Using the right continuity of X s ( · , ω ) (see Remark 1.1), it is easily seen that X s ( u ) = X s ( v ).Combining the coalescing property of X t , t ≥
0, on every interval π k , k ∈ [ n ] withequality (3), we get claim (ii) of the theorem. References [1] Leszek Gawarecki and Vidyadhar Mandrekar,
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