SPLITTING FOR SOME CLASSES OF HOMEOMORPHIC AND COALESCING STOCHASTIC FLOWS

. The splitting scheme (the Kato–Trotter formula) is applied to stochastic flows with common noise of the type introduced by Th.E. Harris. The case of possibly coalescing flows with continuous infinitesimal covariance is considered and the weak convergence of the corresponding finite-dimensional motions is established. As applications, results for the convergence of the associated pushforward measures and dual flows are given. Similarities between splitting and the Euler-Maruyama scheme yield estimates of the speed of the convergence under additional regularity assumptions


Introduction
Introduced (with zero drift) in [1], Harris flows are families of random transformation of the real line that represent the joint movement of one-dimensional interacting Brownian particles whose pairwise correlation depends on the distance between them and is given via so-called infinitesimal covariance φ.Since coalescence is possible, such transformations are not, in contrast to the case of diffeomorphic flows obtained as solutions to SDEs with sufficiently smooth coefficients [2], necessarily continuous.One natural and straightforward extension of the notion of the Harris flow is to add drift to affect the motion of particles in a way similar to the case of the Arratia flow in [3].This brings us closer to biological and physical models that use potentials of different forms [4,5] while introducing common noise as in [6,7] including one that forces particles to collide.
The main goal of the paper is to apply the well-known method of splitting in the stochastic setting [8][9][10][11][12][13][14][15][16] to Harris flows, so that the actions of the semigroups generated by the corresponding driftless Harris flow and the ordinary ODE are separated.
The formal definition of a Harris flow with drift adopted in the paper is based of the definition of a driftless Harris flow from [17] (see also [1,18,19]), with a minor modification as in [20].Let D ↑ (R) be a separable metrizable topological space of non-decreasing càdlàg functions on R equipped with the Skorokhod J 1 topology [21,22].Since for any f, g ∈ D ↑ (R) the composition f • g ∈ D ↑ (R) [23], Lemma 13.2.4,and, for D ↑ (R)−valued random elements ξ, η, we have where f is the càglàd generalized inverse of f, the composition ξ • η defines a random element in D ↑ (R).The space of R d −valued càdlàg functions with non-decreasing coordinates endowed with the J 1 −topology is denoted by D ↑ (R d ), and standard Skorokhod spaces of functions on [0, T ], T > 0, with values in R d are denoted D([0, T ], R d ).
The following splitting scheme is used.Let 0 = t 0 < t 1 < . . .< t m = T for some T. Define recursively piecewise continuous processes (u, y) such that for any k = 0, m − 1 with y 0− = x.We establish the weak convergence of finite-dimensional motions in Skorokhod spaces in the general case of continuous φ as the size of a partition tends to 0 (Thm.5.1).This result is used to derive the convergence of the pushforward measures under the actions of the corresponding flows under an additional assumption that guarantees the initial flow to be a coalescing one (Thm.6.1).As a second application, the convergence of the associated dual flows in the reversed time is established (Thm.7.5).
If a Harris flow admits a representation as the unique strong solution of an SDE (see Sect. 3 for details), which corresponds to the additional assumption of √ 1 − φ being Holder continuous of order β ≥ 1 2 , one can (almost) mechanically transfer proofs and conclusions for the Euler-Maruyama scheme [10,12] into our setting (Thm.8.1).We emphasize the highly derivative nature of the results in this case.To formulate the results, the Wasserstein distance on the space of distributions of random measures is chosen (Thm.8.5).
Example 2.6.The Brownian web (the Arratia flow) [25] can be seen as an extreme example of the Harris flow with φ(x) = 1I[x = 0].In this case particles are independent before a collision.
Example 2.7.Consider the following example of φ ∈ Φ 3/2 that does not satisfy the Holder condition of any positive order.Given positive Then for sufficiently large m ∈ N and x ∈ (e −2(m+1) , e −2m ] On the other hand, for any k ∈ N and x m = e −km , m ∈ N, Theorem 2.8.Suppose φ ∈ Φ * and a is measurable and of linear growth.Then the Harris flow X φ,a exists and is unique in distribution.
Theorem 2.9.Suppose that φ ∈ Φ α and a Either result is essentially an extension of the corresponding theorem in [1].Still, original proofs need to be modified to accommodate the presence of non-zero drift, so we present a brief description of the necessary changes in Appendix A.

Harris flows as solutions to SDEs
This section describes the approach of [17] that provides the representation of X φ,a as a solution to a SDE w.r.t a cylindrical Wiener process.
Let H φ be the separable Hilbert space obtained as the completion of span k=1,n We denote such W as W(X φ,a ).
Assume that e n , n ∈ N, is an orthonormal basis in H φ .Then given where the last integral is understood in the sense of [2] and σ(x), x ∈ R, are Hilbert-Schmidt operators from H φ to R : 17], p. 356).Under the same assumptions as in Proposition 3.1 for all x ∈ R and s ≥ 0 with probability 1 Here where ∥ • ∥ HS is the corresponding Hilbert-Schmidt norm.
Both propositions are formulated in [17] without proofs and in the case of a = 0, so we need to justify them for nontrivial drift.Sketches of the corresponding proofs are presented in Appendix B.
We denote the space of Holder continuous functions of order β on R by H β (R) henceforth.
, a ∈ Lip(R), then for every x ∈ R and s ≥ 0 the process X φ,a s,• (x) is the unique strong solution of (3.1).

The splitting scheme and the example of the Brownian web
Only a ∈ Lip(R) is considered hereinafter unless stated otherwise explicitly.T > 0 is fixed.We define the composition of m ∈ N functions by Consider a sequence T = ({t n j | j = 0, N n }) n∈N of partitions of [0, T ] : and set Let F t (x), t ≥ 0, be the solution to For all n ∈ N set y n 0− (x) = x and define processes (u n , y n ) ∈ D([0, T ], R 2 ) such that for k = 0, N n − 1 and t ∈ I n k (cf.[26], equation (2.1) for the Brownian web) ) is equivalent in distribution to the following system.Let W = W(X φ,a ).Formally define Borrowing notation from [12] and setting we can rewrite both (4.1) and (4.2) as where w n (x) are standard Wiener processes.In the case of (4.2) The collection (u n t (•), y n t (•)), t ∈ [0, T ], can be considered as a D ↑ (R 2 )−valued random process in either case.
Definition 4.1.We denote ((u n , y n )) n∈N by Spl(X φ,0 ; a; T ) in the case of (4.1) and by Spl(X φ,a ; a; T ) in the case of (4.2), respectively.
We summarize the discussion of the well-posedness of (4.1) and (4.2) as follows.
, additionally, then for any x ∈ R and n ∈ N the pair (y n • (x), u n • (x)) from Spl(X φ,a ; a; T ) is the unique strong (F X φ,a 0,s ) s∈[0,T ] −adapted solution to the system (4.2);moreover, Spl(X φ,0 ; a; T ) and Spl(X φ,a ; a; T ) are identically distributed.
The proof of the following lemma uses reasoning similar to that in [27], pp.171-172 and is therefore omitted.
For any x ∈ R and n ∈ N there exists, possibly on an extension of the original probability space, a standard Wiener process b n (x) such that Define 1.For any x ∈ R and p ≥ 1 there exists C = C(p, x, T ) such that 2. For p ≥ 2 there exists 3. For any x ∈ R and p ≥ 2 there exists 4. There exists 5. For any x ∈ R there exists Proof.We drop the argument x which is assumed to be fixed.
(1) The inequality follows trivially by the Gronwall lemma.
(2) We have for some (3) is a corollary of ( 1) and ( 2). ( 4) Set Then we need to estimate and, for a standard Wiener process w and k = 0, 2 dv, we get, iterating conditioning, using the inequality k=1,m and standard estimates for the Gaussian distribution, that given α = 2δ n log 1 δn we have for sufficiently large n and some absolute constants so the application of (1) yields the desired estimate.Let M p (R) be the metric space of probability measures on R with finite p−th moment and let W p be the corresponding Wasserstein distance [28].For fixed p, we define the Wasserstein distance between probability measures L 1 , L 2 on M p (R) as where the infinum is taken over the set of pairs of M p (R)−valued random elements Define random pushforward measures and their distributions as measures on M p (R) The rest of the section describes the example of splitting for the Brownian web.Let B be a Brownian web.The corresponding counterpart with drift B a is defined and constructed in [3], Chapter 7 as a family {B a • (x) | x ∈ R} of coalescing semimartingales.One defines the associated splitting Spl(B; a; T ) via (4.1) by replacing X φ,0 with B. It is worth noting that the limit in Proposition 3.1 does not exist due to [26], Proposition 1.5.
Theorem 4.6 ([29], Thm.2.1).Assume that the sequence {nδ n } n∈N is bounded by K and a ∈ L ∞ (R).Then for every p ≥ 2 there exist C = C(p, K, T ) > 0 such that Remark 4.7.The formulation of [29], Theorem 2.1 is erroneously missing the second logarithm due to a calculational error in the end of the proof.

Weak convergence
Let ((y n , u n )) n∈N = Spl(X φ,0 ; a; T ) for some T .The main result of this section is the following theorem.
Theorem 5.1.Assume that φ ∈ Φ * and a ∈ Lip(R).For any m ∈ N and any The proof of Theorem 5.1 is split into a series of lemmas.Recall to be standard Wiener processes.We denote the modulus of continuity by ω and the Lipschitz constant for a by C a .
Remark 5.2.Proceeding exactly as in the proof of [26], Proposition 2.2, one can show that for Proof.By [21], Theorem 15.2, it is sufficient to prove that for any ε > 0 and any k = 1, m where for f ∈ D([0, T ], R) For fixed k and ε consider s 1 , s 2 such that 0 < s 2 − s 1 < κ.We drop argument x to simplify notation.Since we have for some C > 0 Let w be a standard Wiener process.By Lemma 4.4, for some fixed constant K Since ω(•, κ) ≤ ω(•, 2κ) for κ ≪ 1, (5.1) follows.
Lemma 5.4.For any weak limit ξ = (ξ 1 , . . ., ξ m ) of the sequence (y n (x)) n∈N and for any pair i, j ∈ {1, . . ., m}, i ̸ = j, Proof.We adopt the idea from [26], Proposition 3.8, referring to the aforementioned proof for those calculations that are shared between the proofs.Define Assume that i, j are fixed and that x j > x i .It is sufficient to show that for any κ lim inf where ∆y n = y n (x j ) − y n (x i ).Put One can show that for ε ≪ κ ∆y n r ≤ ε; we consider only the sum Thus which yields (5.3).
Lemma 5.5.Let C m be the set of elements of C(R + , R m ) whose coordinates merge after a meeting.Any weak limit ξ of the sequence (y n (x)) n∈N is a C m −solution in the sense of Definition A.1 in Appendix A to the martingale problem on R m for the operator Proof.W.l.o.g.we can suppose that y n (x) ⇒ ξ, n → ∞.Using [21], Theorem 15.5 and (5.2) we can check that ξ is continuous a.s.. Recalling (4.3), define in probability in D([0, T ], R m ) in the uniform metric and therefore in the J 1 topology.Thus, in probability and therefore Proceeding as in Lemma 5.3, we can check that the sequence is weakly relatively compact in D([0, T ], R 3m ).Hence, by the Skorokhod representation theorem and (5.6) we can assume w.l.o.g that in D([0, T ], R 3m ).By the Itô lemma and Proposition 3.2, for any bounded ) for some M ∈ N. Then for arbitrary s, t, s ≤ t and s 1 , . . ., s M ≤ s so the process is a martingale w.r.t. the filtration generated by y n (x).
Proof.C m −solutions are unique by Proposition A.2.This finishes the proof of Theorem 5.1.
Remark 5.7.The splitting scheme and Theorem 5.1 can be extended to some classes of non-Lipschitz a as follows.Assume that a satisfies the one-sided Lipschitz condition: for some C a(x) − a(y) ≤ C(x − y), x ≥ y.
Then the unique flow X φ,a exists and has finite moments of any order.For any s ∈ [0, T ) consider a SDE where ε n → 0+, n → ∞, and w is a Wiener process on [0, T ] independent of X φ,0 .Such SDEs have unique strong solutions.At each step of the splitting procedure, replace u n on I n k in (4.1) with Then the new splitting scheme Spl(X φ,0 ; a; T ) is well defined.Moreover, Lemmas 4.4 and 5.3 follow immediately.For Lemma 5.4, note that general comparison theorems for SDEs [30][31][32] imply that F n t is monotone mapping for any t and n so we can use the Gronwall lemma to establish analogs of (5.4) and (5.5) for conditional expectations w.r.t. the extended filtration instead of (F X φ,0 0,s ) s∈[0,T ] , which yields the conclusion of Lemma 5.4 for such a.Lemma 5.5 is also valid for such drift.This establishes the claim.See also [33] for the Euler-Maruyama scheme for SDEs with discontinuous coefficients and [13][14][15][16] for splitting schemes for SPDEs with non-Lipschitz coefficients.

Convergence of pushforward measures
((u n , y n )) n∈N = Spl(X φ,0 ; a; T ) is considered in this section.Let R(R) be the set of Radon measures on R. Theorem 6.1.Assume that φ ∈ Φ α for some α < 2 and Then for any t ∈ [0, T ] ν n t , ν t ∈ R(R) a.s. and in R(R) under the vague topology.
For the pushforward measures defined in (4.4), the following conclusion holds.
We need the following lemma whose proof is postponed until Appendix A. 2. Assume that φ ∈ Φ α for some α < 2. Then for any p ≥ 2 and ε ∈ (0; Proof of Theorem 6.1.Using Lemma 6.3, we can repeat the reasoning in [20], proof of Theorem 1, pp. 87-90 as soon as we have proved the following two claims: for arbitrary κ > 0 and compactly supported f there exists and for arbitrary For that, it is sufficient to show that for any S > 0 lim M →∞ |x|≥M P X φ,a 0,t (x) ≤ S dν 0 (x) = 0, ( Then on {inf t∈[0,T ] η n t (x) > 0} for t ∈ I n k for some k Here where The concentration inequality for a supremum of a Gaussian process [34], Theorem 2.1.1,implies where M n ε is the smallest number of balls of size ε that cover [0, T ] in the intrinsic metric of the process η n (0).Note that η n (0) is a Wiener process on every I n k and for .
We assume T = 1 for the rest of the proof.The diameter of [0, T ] in ρ n does not exceed 2C so M n ε = 1 for ε ≥ C for all n.
Assume ε > 2C(δ n ) 1/2 .For a unit interval, consider a ε 2 4C 2 −net A in the Euclidean metric.Then for any s , and small, otherwise.We construct an ε−net as follows.Let B be a ε 2 C1 −net for [0, 1] in the Euclidean metric where For each large I n k let B k be a net of the same size such that t n k ∈ B k , also in the Euclidean metric.Set Since for large intervals 4C 2 δ n k ε 2 ≥ 1, we have then the intervals I n m1 , . . ., I n m2 are small; 2. either m 1 = 0 or I n m1−1 is large; 3. either m 2 = N n − 1 or I n m2+1 is large; 4. at least one of I n m1−1 and I n m2+1 is large.
Consider the case Then there exists s ∈ B ∩ j=m1,m2 I n j such that |t − s| ≤ ε 2 C1 and for some j, m 1 ≤ j ≤ m 2 , we consider two possibilities.If I n m1−1 is large, then If m 1 = 0, then t n m2+1 = L, so This proves the claim.Estimating the integral in (6.4) we obtain for some for sufficiently large absolute x 0 and C 0 .Since the same estimate can be obtained for P(sup t∈[0,T ] η n t (x) ≥ S) for n ∈ N and negative x, (6.2) follows.
For the limit process, we get for x ≫ S P inf where w being a Wiener process.Since where the last term is a continuous square integrable martingale with bounded quadratic variation, and P(X φ,a 0,t (x) ≥ −S) can be estimated similarly, (6.1) follows.

Convergence of dual flows
We assume that ((u n , y n )) n∈N = Spl(X φ,0 ; a; T ) and establish the convergence of so-called dual flows [1,19,20,36].Here given a coalescing stochastic flow X the dual flow X is defined via and is again a collection of D ↑ (R)−valued random elements.
To start, we need one extension of the splitting scheme (4.1).Let l(s) equal the unique k such that s For any s ∈ [0, T ) and any f ∈ D([s, T ], R) we extend f onto [0, T ] by setting For instance, y n,e s,• (x), u n,e s,• (x) are random elements in D([0, T ], R).Since constructing the families {y n s,• (x) | x ∈ R} uses the single flow X φ,0 for all s, the mappings {y n s,t | 0 ≤ s ≤ t ≤ T } are consistent and form a coalescing flow, so using (7.1), one defines the corresponding dual flow y n = { y n s,t | 0 ≤ s ≤ t ≤ T }.We use the idea from [20], where a precise construction of the dual flow as a function that preserves the weak convergence is given.
Consider the set {(s n , x n ) ∈ [0, T ] × R | n ∈ N} containing all points whose coordinates are dyadic numbers.The corresponding version of Theorem 5.1 implies that for any m ∈ N Denote by P k the projector on the first k coordinates in the space D([0, T ], R) ∞ endowed with the product topology and by Q k the projector on the k−th coordinate in the same space.Consider the s1,• (x 1 ), . . ., y n,e sm,• (x m ) , P m ( X) = X φ,a,e s1,• (x 1 ), . . ., X φ,a,e sm,• (x m ) , n, m ∈ N.
As in [20], p. 86, we obtain Consider a mapping I : Then one can check that I(Y n ) = Y n a.s., n ∈ N, and I(X) = X a.s..
Proof.Proceeding as in [20] p. 87, one uses (A.1) in the proof of Proposition A.3 and the fact that the set {X φ,a s,t (x) | x ∈ R} is a.s.locally finite by Theorem 2.9 for any s, t, s < t.Combining Lemmas 7.3 and 7.4 yields the following result.
It is possible to refine the order of convergence for u n .
Proposition 8.2.For any M ≥ 0 and some Proof.Dropping the x argument and setting we get by Lemma 4.4 for some The order of convergence in Proposition 8.2 cannot be improved, as shown by the following example.
Remark 8.8.Following assumptions in [10], consider a = a 1 + a 2 , a 1 ∈ Lip(R), a 2 ∈ H α (R) for some α ∈ (0, 1) and assume that a 2 is non-increasing.Theorems 8.6 and 8.7 can be extended to such a as follows.Consider the Proof of Proposition A.3.We modify the original proof.Define, for fixed T > 1 and N > 1 It is sufficient to show for any ε > 0 Proceeding as in [1], Proof of Lemma 4.5 and using Lemma A.
In what follows we refer to [27], Section 5.5 and [40], Chapter 16 for the theory of Feller's one-dimensional diffusions and a classification of boundaries and to [41] for basic facts about Bessel processes including those with negative dimension.
Example A.7.Let ρ(x) = 1 and α < 1.Then the situation of Example A.6 repeats.In particular, given y > x the difference X φ,a 0,• (y) − X φ,a 0,• (x) goes to infinity with a positive probability in either example.
The next lemma is proved with a standard localization argument.
Proof.By Lemma A.8 it is sufficient to prove (A.4) for the diffusion ξ.We follow the idea of [1] of switching to a squared Bessel process.Since ξ is not in the natural scale, one step is added and the coefficients are distorted, so we provide necessary details.
Remark A.11. [42], Example 6.4 uses [43], Exercise XI.1.22that is stated for only positive dimensions.However, the claim of [43], Exercise XI.1.22 is known to hold for Bessel processes with arbitrary dimensions (after restricting to trajectories that do not hit 0).
Remark A.12.In [18], the coalescing property is established and estimates for the number of surviving particles are obtained under weaker assumptions on φ and for zero drift by studying eigenfunction expansions of the corresponding transitional densities.
Proof of Lemma 6.3.(1) follows from taking expectation in (5.4). ( The flow X φ,a is a coalescing flow by Theorem 2.9.Define and  Since uncorrelated continuous martingales with independent increments are independent, the proposition is proved. Proof of Proposition 3.2.By definition of the stochastic integral,   f n,m X φ,a s,t n,k (x), X φ,a s,s n,k m,j (x) ,