Strong measurable continuous modifications of stochastic flows

This paper is devoted to the construction of stochastic flows of measurable mappings in a locally compact separable metric space (M, $\rho$). We propose a new construction that produces strong measurable continuous modifications for certain stochastic flows of measurable mappings in metric graphs.

Jan and Raimond 2020 and the complete corrected version that can be downloaded at arXiv:math/0203221v6; in our references to Le Jan and Raimond 2020 we use the numeration of Theorems from arXiv:math/0203221v6).To describe the problem we briefly review the characterization of stochastic flows of measurable mappings obtained in Le Jan and Raimond 2020.By B(M) we denote the Borel σ-field in the space M. For every n ∈ N, let (P (n) t , t ≥ 0) be a Feller transition function on M n .We will say that (P (n) • : n ∈ N) is a consistent sequence of "coalescing" Feller transition functions on M, if (TF 1) For any {i 1 , . . ., i k } ⊂ {1, . . ., n}, t ≥ 0, x ∈ M n and B ∈ B(M k ) t (π i 1 ,...,i k x, B), where π i 1 ....,i k : M n → M k is defined by π i 1 ,...,i k x = (x i 1 , . . ., x i k ); (TF 2) For any x ∈ M and t ≥ 0 The results of Le Jan and Raimond 2020 imply that to any consistent sequence (P (n) • : n ∈ N) of coalescing Feller transition functions on M one can associate a stochastic flow of measurable mappings ψ = (ψ s,t : −∞ < s ≤ t < ∞) in M (see Le Jan and Raimond 2020, Def.1.3.1).Let F be the space of all measurable mappings f : M → M equipped with the cylindrical σ-field.Denote by C 0 (M n ) the space of continuous functions f : M n → R that vanish at infinity.A stochastic flow of measurable mappings on M is a family ψ = (ψ s,t : −∞ < s ≤ t < ∞) of random elements in F defined on a probability space (Ω, A, P), such that (SF 1) For any s ≤ t, n ∈ N, x ∈ M n and f ∈ C 0 (M n ), Ef (ψ s,t (x 1 ), . . ., ψ s,t (x n )) = P (n) t−s f (x); (SF 2) There exists a family (J t : t ≥ 0) of measurable mappings J t : F × M → M, such that for all s ≤ t ≤ u and x ∈ M, J t−s (ψ s,t )(x) = ψ s,t (x) a.s., ψ s,u (x) = J u−t (ψ t,u ) • ψ s,t (x) a.s., and ψ s,s (x) = x; (SF 3) For any t 1 ≤ t 2 ≤ . . .≤ t n , the family (ψ t i ,t i+1 : 1 ≤ i ≤ n − 1) is a family of independent random mappings; (SF 4) For any f ∈ C 0 (M) and s ≤ t, lim (u,v)→(s,t) (SF 5) For any f ∈ C 0 (M), x ∈ M and s ≤ t, lim y→x E f (ψ s,t (y)) − f (ψ s,t (x)) 2 = 0 and lim y→∞ E f (ψ s,t (y)) 2 = 0.
Let ψ be a stochastic flow of measurable mappings in M and let ψ ′ be a family of random elements in (F, F ), such that ψ ′ s,s (ω, x) = x for all (s, x, ω) ∈ R × M × Ω, and P ψ ′ s,t (x) = ψ s,t (x) = 1 for all s ≤ t and x ∈ M, then ψ ′ is also a stochastic flow of measurable mappings in M (see Le Jan and Raimond 2020, Remark 1.3.2).If ψ ′ is such that the mapping (s, t, x, ω) → ψ ′ s,t (ω, x) is measurable, then the flow ψ ′ will be called a measurable modification of ψ.
Note that for all (s, x), the process ψ s,• (x) is Feller, and has a càdlàg modification.However, it is not obvious that there exists a measurable modification ψ ′ of ψ with càdlàg trajectories (i.e.such that ψ ′ s,• (x) is càdlàg for all (s, x, ω) ∈ R × M × Ω).We are interested in the existence of such measurable modifications.It is also natural to address the question of the existence of such modification to satisfy the following strong flow property that improves the flow property (SF 2): (SF 6) For all s ≤ t ≤ u and ω ∈ Ω (to simplify the notation, we have omitted the dependency on ω), ψ s,u = ψ t,u • ψ s,t .In this paper we only work with flows with continuous trajectories t → ψ s,t (x).In particular, we will always assume that the transition function (P A measurable modification ψ ′ of a stochastic flow of measurable mappings ψ on M will be called a strong measurable continuous modification of ψ if ψ ′ has continuous trajectories (i.e.ψ ′ s,• (x) is continuous for all (s, x, ω) ∈ R × M × Ω) and if ψ ′ satisfies the strong flow property (SF 6).A new method to construct such modification is given in this paper.This method requires an additional "compactness" condition (TF 4) on the sequence of n-point transition functions (P (n) • : n ∈ N) of ψ.This condition will be given in Section 3.1.3.
The existence of strong measurable continuous modifications of stochastic flows of measurable mappings is known in several cases.For a stochastic flow of solutions to a stochastic differential equation (SDE) with smooth coefficients in a finite-dimensional smooth manifold M, the existence of a strong measurable continuous modifications was proved in Kunita 1997.Strong measurable continuous modifications of certain instantaneously coalescing stochastic flows of measurable mappings in the real line were constructed in Riabov 2018.In Schertzer, Sun, and Swart 2014 it was proved that stochastic flows of kernels associated to the Brownian web possess strong measurable continuous modifications.We note that in Darling 1987 a strong stochastic flow associated to a consistent sequences of coalescing transition functions on M, even without the Feller property, was constructed.However, this flow is not measurable in either of the variables s, t, x.We apply our results to some stochastic flows in metric graphs.Stochastic flows of solutions to SDE's on metric graphs were studied in Hajri 2011, Le Jan and Raimond 2014, Hajri and Raimond 2014, Hajri and Raimond 2016.
The approach we propose is based on the analysis of families of deterministic continuous mappings θ s,• (x) : [s, ∞) → M, θ s,s (x) = x.In Section 2 we define the notion of skeleton.A skeleton ϕ is a sequence of continuous functions ϕ n : [s n , ∞) → M, n ∈ N, such that • {(s n , ϕ n (s n )) : n ∈ N} is dense in R × M; • if ϕ n (t) = ϕ m (t) then ϕ n (u) = ϕ m (u) for all u ≥ t; • for any compact L ⊂ R × M, the restrictions ϕ n to [s, ∞) for all n, s such that s n ≤ s and (s, ϕ n (s)) ∈ L form a relatively compact set in the space of all continuous paths on M. Given a skeleton ϕ we prove the existence of a family of continuous mappings θ s,• (x) : [s, ∞) → M, such that for all (s, x) ∈ R × M, θ s,s (x) = x and θ s,t (ϕ n (s)) = ϕ n (t) for all n ∈ N, s n ≤ t.We refer to the latter property as the preservation of the skeleton.In our construction each function θ s,• (x) is a limit point of the sequence (θ sn,• [s, ∞) : s n ≤ s) in the space of continuous paths.The existence of the family θ follows from the measurable selection Lemma 7.1 given in Appendix 7.
The strong flow property may fail due to the existence of the so-called bifurcation points of the skeleton ϕ.The set B(ϕ) of bifurcation points of the skeleton ϕ is defined in Section 2.3.The main idea of our approach is to achieve 1.1 by modifying functions θ s,• (x) when they hit the set of bifurcation points B(ϕ) (to be precise, when they hit some larger closed set F ⊃ B(ϕ)) in such a way that 1.1 holds.Theorem 2.8 gives a sufficient condition under which one can construct a new family of continuous mappings that preserves the initial skeleton ϕ and that satisfies 1.1.
In Section 3, we let ψ 0 be a stochastic flow of measurable mappings on M that is associated to a consistent sequence of coalescing Feller transition functions (P on M that satisfies conditions (TF 1), (TF 2), (TF 3) and (TF 4).A first measurable continuous modification θ of ψ 0 is given.The results of Section 2 are then applied to give sufficient conditions under which there exists a strong measurable continuous modification ψ of ψ 0 .These conditions are stated in Theorem 3.9.
In Section 4 we study stochastic flows and their skeletons that possess the instantaneous colasecing property.For such flows we give sufficient conditions under which they possess strong measurable continuous modifications (Proposition 4.9).
In Section 5 we study instantaneously coalescing stochatic flows on metric graphs.Corollary 5.5 gives a sufficient condition for the existence of strong measurable continuous modifications for stochastic flows of measurable mappings in a metric graph.
In the final Section 6 we give several applications of our approach.In Section 6.1 we prove the existence of a strong measurable continuous stochastic flow of measurable mappings in a metric graph, whose trajectories are coalescing Walsh Brownian motions independent before the meeting time.In Section 6.2 we prove the existence of a strong measurable continuous stochastic flows of mappings in R, whose trajectories are solutions to Tanaka's SDE.In Section 6.3 we prove the existence of a strong measurable continuous stochastic flow of mappings in R, whose trajectories are solutions to the Harrison-Shepp SDE for the skew Brownian motion (a Burdzy-Kaspi flow, see Burdzy and Kaspi 2004).In Section 6.4 we prove the existence of a strong measurable continuous stochastic flow of mappings in a metric graph M, whose trajectories are solutions to the Tanaka's SDE on M (see Hajri 2011, Hajri and Raimond 2014).

Flow extensions of skeletons on M
In Section 2.1 we introduce X, the space of continuous paths in M, and the notion of a skeleton in M. In Section 2.2 a measurable mapping Θ : R × M × S(M) → X that will allow us to construct for every skeleton ϕ a family of continuous mappings A family θ satisfying (i) and (ii) will be said to preserves the skeleton ϕ.
In Section 2.3 the set B(ϕ) of bifurcation points of a skeleton ϕ is defined.Out of θ we define a new family of mappings ψ s,• (x) : [s, ∞) → M that by construction satisfies the property ψ s,t • ψ r,s (x) = ψ r,t (x) at certain points (s, ψ r,s (x)) ∈ B(ϕ).In Theorem 2.8 we give a sufficient condition under which ψ is a family of continuous mappings that preserves the skeleton ϕ and satisfies the strong flow property (1.1).
2.1.The space of skeletons in M. As above, (M, ρ) is a locally compact separable metric space.Without loss of generality we will assume that ρ is complete and that all bounded subsets of (M, ρ) are relatively compact1 .For any interval I ⊂ R (possibly, unbounded) the space C(I : M) of continuous functions from I to M is equipped with the topology of uniform convergence on compact subsets of I. Let us introduce the space of continuous paths in M: Each element of X is a continuous function f : [s, ∞) → M. The starting time s will be denoted by i(f Let δ be a distance on C(R : M) associated with the topology of uniform convergence on compact subsets of R2 .The space R × C(R : M) equipped with the metric and is a complete separable metric space.Let us equip X with the distance d defined by d(f, g) = |i(f ) − i(g)| + δ(e(f ), e(g)).Then the mapping f → (i(f ), e(f )) from X onto X ′ is an isometry.As a consequence, (X, d) is a complete separable metric space and its Borel σ-field B(X) is the σ-field generated by the mappings f → e(f )(t) for all t ∈ R and f → i(f ).
For any f ∈ X and any interval I, we will denote by f I the restriction of e(f ) to I. Note that the mapping f → f I is a continuous (hence, Borel measurable) mapping from X to C(I : M).
Let ϕ = (ϕ n : n ∈ N) ∈ X N be a sequence in X.For n ∈ N, set s n = i(ϕ n ) and Definition 2.1.The sequence ϕ ∈ X N is called a skeleton if the following properties are satisfied: Let us denote by S(M) the space of all skeletons on M. The space X N , equipped with the product topology, is a separable completely metrizable space (see Srivastava 1998, Th. 2.4.3).Note that S(M) is a Borel subset of X N .We equip S(M) with the subspace topology induced from X N and its corresponding Borel σ-field.Thus, the σ-field on S(M) is generated by the mappings ϕ Let us prove the following lemma that is satisfied by every skeleton ϕ.Throughout the paper B(x, r) (resp., B(x, r)) will denote an open (resp., closed) ball in M with center x and radius r.
2.2.The measurable mapping Θ.From Lemma 7.1 stated in Appendix 7, it follows that there exists a measurable mapping ℓ : X N → X such that for any relatively compact sequence (f n : n ∈ N) in X, ℓ((f n : n ∈ N)) is a limit point of this sequence.We fix such a mapping from now on.
Let (ε k : k ∈ N) be a sequence of measurable functions from M into (0, ∞) such that for every x ∈ M, the sequence k < ∞ for all (s, x) ∈ R × M and all k ∈ N.This defines a sequence (ϕ s,x k : k ∈ N) in C([s, ∞) : M).Let us now define the mapping Θ such that for all (s, x, ϕ) ∈ R × M × S(M), Θ(s, x, ϕ) = ℓ (ϕ s,x k : k ∈ N) .Theorem 2.3.The mapping Θ : R × M × S(M) → X is measurable and satisfies the following properties for all (s, ϕ) ∈ R × S(m): • For all We now verify that Θ preserves the trajectories of the skeleton.Let us fix (s, ϕ) ∈ R × S(M) and let n ∈ I s .Set x = ϕ n (s).Then the sequence (n s,x k : k ∈ N) is stationary and Θ(s, x, ϕ) = ϕ n * [s, ∞) for some n * ∈ I s .We have that It remains to show that Θ is measurable.Let us fix k ∈ N. The mapping (s, x, ϕ) → n s,x k is measurable since for all m ∈ N, k is measurable, and the measurability of the mapping ℓ implies that Θ is measurable.
For s < t, ε > 0 and x ∈ M, denote by K s,t ε,x the closure of {ϕ n [s, t] : n ∈ I s , ϕ n (s) ∈ B(x, ε)} in C([s, t] : M).Assumption (Sk 3) implies that K s,t ε,x is a compact subset of C([s, t] : M) for all ε > 0. Lemma 2.2 implies that K s,t ε,x is non-empty.Therefore, K s,t x := ∩ ε>0 K s,t ε,x is a non-empty compact subset of C([s, t] : M).In fact, K s,t x contains the restriction θ s,• (x)[s, t].For s < t, set ν s,t x := #K s,t x (with ν s,t x = ∞, if K s,t x is infinite).Lemma 2.4.For all (s, x), the mapping t → ν s,t x is left-continuous and non-decreasing.Proof.Let us be given s < t < u and x , which shows that ν s,t x ≤ ν s,u x and so t → ν s,t x is non-decreasing.It remains to prove that t → ν s,t x is left-continuous.Let t > s and let m be an integer such that 1 ≤ m ≤ ν s,t x .There are at least m distinct functions f 1 , . . ., f m ∈ K s,t x .By continuity, for some r ∈ (s, t) the restrictions x are all distinct.Hence, x ≥ 2}.Definition 2.5.The set B(ϕ) := {(s, x) ∈ R×M : τ s x = s} is called the set of bifurcation points of the skeleton ϕ.
For every (s, x) ∈ R × M and every t ∈ (s, τ s x ) the set K s,t x contains a single function θ s,• (x)[s, t].
Lemma 2.6.If τ s x < ∞, then τ s x , θ s,τ s x (x) ∈ B(ϕ).Proof.For any t > τ s x we have ν s,t x ≥ 2 and there exist at least two distinct functions g 1 , g 2 ∈ K s,t x .Let (n 1 k : k ≥ 1) and (n 2 k : k ≥ 1) be two sequences in I s such that x .Consider a closed set F ⊃ B(ϕ).We define for every (s, Lemma 2.7.For all k ≥ 0, the mapping (s, x, ϕ) → (σ s x (k), z s x (k)) is measurable.Proof.For k = 0 the statement clearly holds.We proceed by induction on k.The inequality σ s x (k + 1) > t holds if and only if either σ s x (k) > t or σ s x (k) ≤ t and (r, θ σ s x (k) , r)(z s x (k)) ∈ F for all r ∈ (σ s x (k), u) with some u > t.Measurability of z s x (k + 1) follows from measurability of σ s x (k), σ s x (k + 1), z s x (k) and measurability of θ s,t (x) as a function of (s, t, x, ϕ).
Out of θ, we now define for all (s, The next theorem gives a sufficient condition under which ψ is a strong flow.
Theorem 2.8.Suppose that for any Then ψ is a family of continuous mappings that satisfies the strong flow property and preserves the skeleton ϕ, i.e.
(i) for all (s, Proof of Theorem 2.8.The fact that ψ preserves the skeleton ϕ follows from the fact that θ preserves the skeleton ϕ.To prove the theorem, it suffices to verify that ψ satisfies the strong flow property (iii), i.e. that for all (s, x) it holds that We claim that for all d ≥ 0 it holds that (2.3) is satisfied for all (s, x) ∈ S d .We prove this claim by induction on d.
Assume that (s, x) ∈ S 0 .Then ψ s,• (x) = θ s,• (x) and the assumption of the theorem implies that for all t > s, ψ s,t (x) = ϕ n (t) for some n ∈ I t .This implies that for all u > t . By construction, we obtain that (σ s x (j), z s x (j)) = (σ t y (j), z t y (j)) for all j ≥ 1.The definition of ψ then implies that ψ s,u (x) = ψ t,u (y) for all u ≥ σ s x (1).
We have thus proved that (2.3) is satisfied for all (s, x) ∈ ∪ ∞ d=0 S d = R × M, i.e. (iii) is satisfied.

Strong measurable continuous modifications of stochastic flows
Let ψ 0 be a stochastic flows of measurable mappings in M associated to a consistent sequence (P (n) • : n ∈ N) of coalescing Feller transition functions on M. We assume that the sequence (P : n ∈ N) satisfies properties (TF 1), (TF 2), (TF 3) and (TF 4) (the condition (TF 4) is introduced in Section 3.1.3).We define a random skeleton Φ and using the mapping Θ from Section 2.2, we construct a measurable continuous modification θ of ψ 0 .In Section 3.1.3we state sufficient conditions under which the construction of Section 2.3 is a.s.applicable to θ and produces a strong measurable continuous modification of ψ 0 .

A measurable modification of a stochastic flow. Let (P (n) •
: n ∈ N) be a consistent sequence of coalescing Feller transition functions on M that satisfies (TF 1), (TF 2) and (TF 3), and let ψ 0 be a corresponding measurable stochastic flow of mappings in M. For n ∈ N and x ∈ M n , we will denote by P ) is a Markov process with transition function P (n) and starting point x.When n = 1 and n = 2, we will denote X (1) and X (2) by X and (X, Y ), respectively.

3.1.1.
A first lemma.Before formulating condition (TF 4), we deduce some consequences of (TF 3) that will be used later.
Lemma 3.1.For any compact K ⊂ M and any r > 0, as t → ∞ Proof.The proof follows the one given in Riabov 2018 for the case M = R.Let us fix a compact K ⊂ M and r > 0. By (TF 3) for any α > 0 there exists δ > 0 such that for all t ∈ [0, δ] and x ∈ M with ρ(x, K) ≤ r, we have Then, for any x ∈ K we introduce the stopping time τ = inf{t > 0 : ρ(X t , x) ≥ r} and estimate x where we have used the strong Markov property at time τ and the fact that ρ(X τ , K) ≤ r when τ < ∞.
Lemma 3.2.Almost surely, for all t ∈ R the set {Φ n (t) : s n < t} is dense in M.
Proof.In this proof, we let {y i : i ∈ N} be a dense sequence in M.
For a < b, let Then for all δ > 0 and all n ∈ N such that t − 2δ < s n < t and such that ρ(x n , y) < r we have that ρ(Φ n (t), x n ) > r and as a consequence that sup s∈[0,2δ] ρ(Φ n (s n + s), x n ) > r.This implies that on Ω [a,b] there are y ∈ {y i : i ∈ N} and r ∈ Q * + such that for all δ > 0, there is denote by Ω δ,j y,r the event "(3.4) is satisfied for all n ∈ N".Then there is an n such that s n ∈ ((j − 1)δ, jδ] and ρ(x n , y) < r, and so Therefore
3.1.3.Assumption (TF 4).The next assumption is formulated in terms of P ∞ D : (TF 4) P ∞ D (S(M)) = 1.Another formulation of (TF 4) is to say that if a random variable Φ in X N is distributed as P ∞ D , then Φ is a skeleton a.s.Condition (TF 4) is equivalent to the following condition: Remark 3.3.The authors don't know whether condition (TF 4) follows from conditions (TF 1), (TF 2) and (TF 3).In Theorem 5.1 we prove this fact for instantaneously coalescing stochastic flows on metric graphs.

3.1.4.
A sequence of measurable mappings (ε k : k ∈ N).We recall that (X, Y ) denotes the canonical process on C([0, ∞) : M 2 ), so that under P (2) (x,y) the process (X, Y ) is a Markov process with transition function P (2) • and starting point (x, y).By d 0 we denote the restriction of the metric d from X to C([0, ∞) : M), and by d s we denote its shift: (xn,yn) → P (2) The second statement follows from the relation Lemma 3.5.The sequence (ε k : k ∈ N) is a non-increasing sequence of positive measurable functions, that converges uniformly towards 0 as k → ∞.
Proof.The facts that the sequence (ε k : k ∈ N) is non-increasing and converges to 0 for every x ∈ M easily follow from its definition.Lemma 3.4 implies that ε k (x) > 0 for all k ∈ N and x ∈ M. Let us now prove that for every k ∈ N, the mapping ε k is measurable.For r > 0 and k ∈ N, let Then g k is uppersemicontinuous (Lemma 3.4) and this implies that g r,k is also upper-semicontinuous.
For any a ∈ (0, 2 −k ], the set ε −1 k ((−∞, a)) has the following decomposition: Since g k is upper-semicontinuous, the set {x ∈ M : g r,k (x) ≥ s} is closed, and this proves that ε k is measurable.
Theorem 3.6.The family of random mappings θ is a measurable continuous modification of ψ 0 that preserves the skeleton Φ.
Proof.The facts that (s, t, x, ω) → θ s,t (ω, x) is measurable and that θ s,s (ω, x) = x follow immediately from Theorem 2.3.Let us now prove that θ is a modification of ψ 0 .Let us fix (s, x) ∈ R × M. Taking a continuous modification of ψ 0 s,• (x) we may assume that x k and n s,x k defined by (2.2).We denote by Φ s,x k the random variable in X defined by (2.2) with ϕ replaced by Φ.It holds that for each k ∈ N, ρ(Φ s,x k (s), x) < ε k (x), which implies that (see the definition of ε k (x) given in (3.8)) ) is a Markov process in M 2 with transition function given by P (2) .The Borel-Cantelli lemma shows that a.s.Φ s,x k → ψ 0 s,• (x) in X.This implies that a.s.Θ(s, x, Φ) = ψ 0 s,• and so that for all t ≥ s, a.s.θ s,t (x) = ψ 0 s,t (x), which proves that θ is a modification of ψ 0 .In particular, θ is a stochastic flow of measurable mappings in M (as it has been remarked in the Introduction).The fact that θ preserves the skeleton Φ follows from Theorem 2.3.
The measurable continuous modification θ possesses one useful feature: it a.s.satisfies the flow property at stopping times.
Then, for all y ∈ B(x, ε k (x)), we have g k (x, y) ≤ 2 −k .Finite point motions of the flow θ are Feller processes, in the sense that for any n ∈ N, (s 1 , x 1 ), . . ., (s n , x n ) ∈ R × M and s ≥ s 1 ∨ . . .∨ s n the process ((θ s 1 ,t (x 1 ), . . ., θ sn,t (x n )) : t ≥ s) is an (A θ t : t ≥ s)-Markov process in M n with transition function P (n) • .Below we will use the equality For any m ∈ N and k ≥ 1, let us introduce the event By the strong Markov property for the two-point motions of θ at time σ we find that The latter estimate follows from the fact that on the event E m,k we have ρ(θ sm,σ (x m ), ξ) < ε k (ξ).By the Borel-Cantelli lemma, a.s. on the set {s ≤ σ < ∞} the sequence (Φ σ,ξ k : k ≥ 1) converges to θ s,• (x)[σ, ∞).Hence, on this set a.s.θ σ,• (ξ) = θ s,• (x)[σ, ∞) and the lemma is proved.
Let F be a closed shell of the set of bifurcation points of Φ.We define stopping times σ s x (k), random variables z s x (k), k s x , and random functions ψ s,• (x) as in Section 2.3.Theorem 3.9.Assume that a.s.
• for all (s, Then ψ is a strong measurable continuous modification of ψ 0 .Proof.Using Theorem 2.8 one obtains that a.s.ψ is a random family of continuous mappings that satisfies the strong flow property and preserves Φ.It remains only to check that ψ is a modification of θ, which follows from Lemma 3.7 and the definition of ψ out of θ.

The instantaneous coalescence property
In this section we study skeletons ϕ in M that possess the instantaneous coalescing property (ICP).By ψ 0 we denote a stochastic flow of measurable mappings in M associated to a consistent sequence (P : n ∈ N) satisfies conditions (TF 1), (TF 2), (TF 3) and (TF 4).As in Section 3, we let D = {(s n , x n ) : n ∈ N} be a dense subset of R × M and let Φ = (Φ n : n ∈ N) be the skeleton of ψ 0 constructed in Section 3.1.2.Recall that for each n, Φ n is a continuous modification of ψ 0 sn,• (x n ).Let also θ be the measurable continuous modification of ψ 0 defined in Section 3.1.5such that θ preserves Φ a.s.In this section, it will be proves that if Φ satisfies the ICP a.s., then θ is a strong measurable continuous modification of ψ 0 .4.1.A sufficient condition for the strong flow property.We give a simple condition under which θ is a strong measurable continuous modification of ψ 0 .Note that if one takes F = R × M in Theorem 2.8, then k s x = 0 for all (s, x) ∈ R × M and ψ s,t = θ s,t for all s ≤ t.
Lemma 4.1.Assume that a.s., for any s < t and x ∈ M (4.10) Then θ is a strong measurable continuous modification of ψ 0 .
Proof.Let Ω 0 be an event of probability one on which Φ is a skeleton, θ preserves Φ and (4.10) is satisfied for all s < t and x ∈ M. Let us fix ω ∈ Ω 0 .Let s < t and x ∈ M. Then (4.10) with the fact that θ preserves the skeleton implies that for all u > t, This proves the lemma.
The condition given in Lemma 4.1 means that after any arbitrary short time, trajectories of θ meet trajectories of Φ.This happens under the instantaneous coalescence property we introduce in the next section.4.2.The instantaneous coalescence property.In this section we give a sufficient condition that allows to apply Lemma 4.1 and therefore to prove that θ is a strong measurable continuous modification of ψ 0 .Definition 4.2.A sequence ϕ ∈ X N is said to possess the instantaneous coalescence property (ICP) if for any s < t, the set {ϕ n (t) : n ∈ I s } is locally finite.
Lemma 4.3.Assume that Φ possesses the ICP almost surely.Then θ is a strong measurable continuous modification of ψ 0 .
Proof.We just have to verify that the condition of Lemma 4.1 is satisfied by θ.Let us place on an event of probability one on which Φ is a skeleton possessing the ICP and θ preserves Φ.Let t > s, x ∈ M and y = θ s,t (x).The set {Φ s,x k } k≥1 being relative compact, there is a subsequence {Φ s,x k i } i≥1 such that lim i→∞ Φ s,x k i = θ s,• (x) and in particular that lim i→∞ Φ s,x k i (t) = y.The ball B(y, ε) is relatively compact.The ICP implies that B(y, ε) ∩ {Φ n (t) : n ∈ I s } is a finite set and so there is i 0 such that for all i ≥ i 0 , Φ s,x k i (t) = y.Hence, there exists n ∈ I s such that for all The condition of Lemma 4.1 is verified and, as a consequence, θ is a strong measurable continuous modification of ψ 0 .4.3.Sufficient condition ensuring the a.s.ICP for Φ.The sufficient condition given in this section is an extension of the one given in Evans, Morris, and Sen 2013.We recall that (X, Y ) denotes the canonical process on C([0, ∞) : M 2 ), so that under P (2) (x,y) the process (X, Y ) is a Markov process with transition function P (2) • and starting point (x, y).Also, X (n) = (X 1 , . . ., X n ) denotes the canonical process on C([0, ∞) : M 2 ), so that under P (n) x the process X (n) is a Markov process with transition function P (n) • and starting point x.
Let us say that a compact set K ⊂ M satisfies P if there are positive and finite constants α, β, κ, p and C, such that ακ > 1 and such that conditions (P1) and (P2) given below are satisfied: (P1) For all ε > 0, (x, y) ∈ K 2 , it holds that (4.11) ρ(x, y) ≤ ε =⇒ P (2) We will use the following notation in this section: for s ≤ t and n ∈ I s , the point Φ n (t) will be denoted ϕ s,t (x) where x = Φ n (s).Then ϕ s, Let us first prove a localized ICP.For all s ≤ t and compact Proposition 4.4.If K ⊂ M is a compact set satisfying P, then a.s.A K s,t is a finite set for all t > s.
For the following lemmas 4.5, 4.6, 4.7 and in the proof of Proposition 4.4, we let K be a compact subset of M satisfying P and let α, β, κ, p and C be constants such that ακ > 1 and such that (P1) and (P2) are satisfied.For n ≥ 1, let σ n = inf{t : X (n) t ∈ K n } and for m ≤ n, let τ n m be the first time t such that #{X 1 t , . . ., X n t } ≤ m.
Lemma 4.5.Let x ∈ K n and k ≥ 1 be such that #{x 1 , . . ., x n } ≤ k.Then for all t ≥ 0 and ε = Ck −κ (4.13) k−1 = 0 and (4.13) holds.If #{x 1 , . . ., x n } = k, then there is i = j such that ρ(x i , x j ) ≤ Ck −κ .Therefore, since ε = Ck −κ , Using the Markov property at time t, we have Where we have used in the last inequality the fact that, on the event Proof.Applying Lemma 4.5 j times, one easily obtains (i).Now, using (i), we obtain This proves (ii).
Lemma 4.7.Let x ∈ K n and 1 ≤ m ≤ n.Then , using the strong Markov property and Lemma 4.6-(ii), we obtain The lemma is proved.
We have thus proved that for all s ∈ R, a.s.for all t > s, A K s,t is a finite set.This implies that a.s.for all s ∈ Q and t > s, A K s,t is a finite set.Let E be an event on which there is s < t such that #A K s,t = ∞.Then on E, there is a ∈ Q such that s < a < t and such that #A K a,t = ∞.This implies that P Remark 4.8.When the space M is compact, Proposition 4.4 shows that if M satisfies P, then a.s.for all s < t, {Φ n (t) : n ∈ I s } is a finite set, and Φ possesses the ICP.
When M is locally compact, then it is not clear that Proposition 4.4 implies that Φ possesses the ICP a.s.To prove the ICP, we need an additional assumption.Proposition 4.9.Let (K l : l ∈ N) be an increasing sequence of compact sets such that ∪ l∈N K l = M. Suppose that (i) for any l ∈ N, K l satisfies P; (ii) a.s.for all k ∈ N and s < t, there are l > k and r ∈ (s, t) such that for all n ∈ N, (4.15) Then a.s.Φ possesses the ICP.
Proof.For l ∈ N and s < t, we set A l s,t = A K l s,t and B l s,t = {Φ n (t) : n ∈ I s } ∩ K l .Let Ω 0 be an event of probability one on which for all k ∈ N and s < t, it holds that A k s,t is a finite set and that there are l > k and r ∈ (s, t) such that (4.15) is satisfied for all n ∈ N. On Ω 0 , which is a finite set.Therefore on Ω 0 , for all s < t and k ∈ N, B k s,t is a finite set.This implies that on Ω 0 , {Φ n (t) : n ∈ I s } is locally finite for all s < t.

Stochastic flows on metric graphs
Let (M, ρ) be a metric graph (metric graphs are defined in Section 5.1).Let ψ 0 be a stochastic flow of measurable mappings in M associated to a consistent sequence of coalescing Feller transition functions (P (n) • : n ∈ N) on M. It is assumed that (P (n) • : n ∈ N) satisfies conditions (TF 1), (TF 2) and (TF 3).As in Section 3, we let D = {(s n , x n ) : n ∈ N} be a dense subset of R × M and let Φ = (Φ n : n ∈ N) be the random sequence constructed out of ψ 0 in Section 3.1.2.Then Φ a.s.satisfies conditions (Sk 1) (by (TF 2)) and (Sk 2) (by construction).In subsection 5.2, we show that if Φ possesses the ICP a.s.then Φ a.s.satisfies (Sk 3), and a.s. is a skeleton.In subsection 5.3 a sufficient condition on Φ to satisfy the ICP a.s. is given.

Notations.
A locally compact separable metric space (M, ρ) is said to be a metric graph if there are a countable set (of vertices) V ⊂ M and a partition {E j } j∈J of M \ V (into edges) such that • For each j ∈ J, -the edge E j is an open subset of M; -there is an isometry e j : (0, the set J(v) = {j ∈ J : v ∈ ∂E j } is non-empty and finite; -the set {v} ∪ j∈J(v) E j is a neighborhood of v.For every j ∈ J, the isometry e j can be extended by continuity at 0 and at L j (when L j < ∞).By abuse of notation, this extension will also be denoted e j .Then e j (0) ∈ V and e j (L j ) ∈ V when L j < ∞.
For a vertex v ∈ V , the cardinality of J(v) is denoted by d(v) and is called the degree of V .For a point x ∈ M \V , the degree of x is defined by d(x) = 2.In this case, x = e j (t) for some j ∈ J and t ∈ (0, L j ), and the sets e −1 j ((0, t)) and e −1 j ((t, L j )) will be viewed as the two adjacent edges to x.
Local compactness of the space (M, ρ) implies d(v) < ∞ for every v ∈ V .Without loss of generality we will assume that M has no loops, i.e. if L j < ∞ then ∂E j contains exactly two vertices3 .
When x and y are two points belonging to the same edge E j , then there is (r, s) ∈ [0, L j ] 2 such that x = e j (r) and y = e j (s).When r ≤ s will denote by [x, y] (respectively [x, y), (x, y] and (x, y)) the set e −1 j [r, s] (respectively e −1 j [r, s), e −1 j (r, s] and e −1 j (r, s)).When r > s, we will denote by [x, y] (respectively [x, y), (x, y] and (x, y)) the set [y, x] (respectively (y, x], [y, x) and (y, x)).
A bounded connected open set O ⊂ M that contains at most one vertex v ∈ V and whose boundary contains no vertices v ∈ V (∂O ∩ V = ∅) will be called a simple open set.The closure of a simple open set is compact and will be called a simple compact set.
Let O be a simple open set.There are two possibilities: • V ∩ O = ∅ and there are j ∈ J and (u 1 , u 2 ) ∈ E 2 j such that O = (u 1 , u 2 ).In this case O will be called a simple neighborhood of x for any x ∈ O and will be denoted by U x {u 1 , u 2 } .• V ∩ O = {v} and for any j ∈ J(v), there is u j ∈ E j such that In this case O will be called a simple neighborhood of v will be denoted by Note that for any x ∈ M and all sufficient small ε > 0, the ball B(x, ε) is a simple neighborhood of x.For all x ∈ M, we denote by E x 1 , . . ., E x d(x) the edges adjacent to x.

Instantaneous coalescence property on metric graphs.
Theorem 5.1.On a metric graph, if Φ possesses the ICP a.s., then Φ is a skeleton a.s.
Before proving this theorem, let us prove the following simple lemma.
Lemma 5.2.Almost surely, Φ satisfies the following property: • For all x ∈ M, ε > 0 and s ∈ R, with ε < ρ(x, V \ {x}), there are δ = δ ε s,x > 0 and {n 1 , . . ., n d(x) } ⊂ I s−δ such that (i) Let Ω 0 be an event of probablility one on which Φ satisfies (SK 1), (SK 2) and on which it holds that for all t ∈ R the set {Φ n (t) : s n < t} is dense in M. Lemma 3.2 implies that such event exists.We now check that the property claimed in the lemma is satisfied by Φ on Ω 0 .Let us fix x ∈ M, ε > 0 and s ∈ R, with ε < ρ(x, V \ {x}).
Proof of Theorem 5.1.Let us place on an event Ω 0 of probability one on which Φ possesses the ICP and satisfies (Sk 1), (Sk 2) and the property given in Lemma 5.2.In order to prove that Φ is a skeleton on Ω 0 , it remains to verify that for any compact L ⊂ R × M, the set F L = {Φ n [s, ∞) : n ∈ I s , (s, Φ n (s)) ∈ L} is relatively compact in X.This will be done using Theorem 5.3 given below, which is a version of the Ascoli-Arzela theorem for the space X.
Let L ⊂ R × M be a compact set.Item (i) of Theorem 5.3 is satisfied by F L since for all f ∈ F L , there is (n, s) such that n ∈ I s , (s, Φ n (s)) ∈ L and f = Φ n [s, ∞), which implies that (i(f ), f (i(f ))) = (s, ϕ n (s)) ∈ L, and L is compact.
We now verify that item (ii) of Theorem 5.3 is satisfied by F L for all ω ∈ Ω 0 .We fix ε > 0 and let C < ∞ be such that s ≤ C for all (s, x) ∈ L. For any s,x (t).The set L being compact, there exists a finite covering of The family of mappings {Φ n l k } being finite, this family is uniformly equicontinuous and so there exists α > 0 such that for all (k, l) and (r for some l.Hence, using (Sk 1), Combining these two items, we obtain that for all (r 1 , r Item (ii) of Theorem 5.3 is verified and, by the Ascoli-Arzela theorem, the family F L is relatively compact in X for every compact L and the Theorem is proved.
Theorem 5.3 (Ascoli-Arzela theorem).A family of mappings F ⊂ X is relatively compact if and only if F is uniformly equicontinuous, i.e. if and only if the following items are satisfied: 5.3.Sufficient condition for the ICP on a metric graph.In this section we give a sufficient condition on P (2) • , the Feller transition function of the two-point motion of ψ 0 , under which Φ possesses the ICP a.s.
Proof.Using Proposition 4.9, we will prove that a.s.Φ possesses the ICP by verifying that simple compact sets satisfy P given in Section 4.3, i.e. (4.11) and (4.12) are satisfied, and by verifying Proposition 4.9-(i).Let K be a simple compact and let β, p > 0 be the constants given in Assumption (TF 5), taking α = 2 we obtain that (4.11) holds and so (P 1) is satisfied.Let us remark that on a metric graph, for any simple compact K there is C > 0 such that for all A ⊂ K and n ≥ 1, it holds that (5.17) This implies that (4.12) and so (P 2) holds with κ = 1.Since ακ = 2 > 1, the simple compact K satisfies P. We now verify Proposition 4.9-(i).Let (U i,k ) i≥1, k≥1 be a sequence of simple open sets such that . The space M being a locally compact separable metric space, such sequence exists.
For every k ≥ 1, let Let us place on an event of probability one such that for all for all t the set {Φ n (t) : s n < t} is dense in M. Lemma 3.2 ensures that this event exists.Let us fix k ≥ 1 and s < t.Then for all i ≤ k and j ∈ {1, . . ., d(x i )}, there exists n ij,k such that s n ij,k < t and such that Φ n ij,k (t) ∈ (u ij,k , u ij,k+1 ).By continuity of the trajectories Φ n ij,k , there exists r ∈ (s, t) such that for all i ≤ k and j ∈ {1, . . ., d( Let now n ≥ 1 be such that s n ≤ s and Φ n (t) ∈ K k .Then, there is i ≤ k such that Φ n (t) ∈ Ūi,k .Since s n < r, the coalescing property implies that for all t Applying Proposition 4.9, we prove that Φ possesses the ICP a.s.Applying now Theorem 5.1, we can conclude that Φ is a skeleton a.s.
Using the results of Theorem 3.6 and Lemma 4.3, we get the following corollary of Theorem 5.4.

Examples
6.1.Coalescing independent Walsh Brownian motions on a metric graph.Let M be a metric graph with vertices V , edges (E j : j ∈ J), and isometries e j : (0, L j ) → E j , 0 < L j ≤ ∞ (see Section 5.1).We equip M with the shortest path distance ρ.For all j ∈ J, denote g j = e j (0) and d j = e j (L j ) (with To each v ∈ V and j ∈ J(v) we associate a parameter p j (v) ∈ [0, 1], such that j∈J(v) p j (v) = 1.Denote by D the set of all continuous functions f : M → R, such that for every j ∈ J, f • e j ∈ C 2 ((0, L j )) with bounded first and second derivatives, and For f ∈ D and x = e j (t) set f ′ (x) = (f • e j ) ′ (t), f ′′ (x) = (f • e j ) ′′ (t), and for The operator A generates a continuous Feller Markov process on M (see, Freidlin and Wentzell 1993).We will call such process the Walsh Brownian motion (WBM) on M with transmission parameters p j (v), v ∈ V , j ∈ J(v).We are interested in the existence of a strong measurable modification of a stochastic flow of measurable mappings in M, whose trajectories are WBM's that are independent before meeting and coalesce at the meeting time.Let (P t : t ≥ 0) be the transition function of a WBM on M with transmission parameters p j (v), v ∈ V , j ∈ J(v).We define (P (n) • : n ∈ N) to be a unique consistent sequence of coalescing Feller transition functions on M obtained from (P ⊗n • : n ∈ N) (see Le Jan and Raimond 2020, Theorem 4.3.1).Let ψ 0 be a stochastic flow of measurable mappings in M associated to the consistent sequence of coalescing Feller transition functions (P (n) • : n ∈ N).Theorem 6.1.There exists a strong measurable continuous modification of ψ 0 .
Proof.According to Theorem 5.4 it is enough to verify (TF 5), i.e. for any simple compact K, there are β, p > 0 such that that (5.16) holds.Without loss of generality we will suppose that K is a neighborhood a vertex v ∈ V (in the case K doesn't contain any vertices, it suffices to add a vertex v ∈ K of degree 2 so that K is a compact neighborhood of v, and to set the transmission parameters to p 1 = p 2 = 1 2 ).Let M be a star graph that contains a simple compact isometric to K. Then M is a metric graph with only one vertex v and d = d(v) edges.We denote v by 0 and let the adjacent edges of 0 be E 1 , . . ., E d .Then for each j ∈ {1, . . ., d} there is a bijection e j : (0, ∞) → E j such that e j (0+) = 0.When x = e j (r), we set |x| = r and define the distance on M by: ρ(e i (r), e j (s)) = |r − s|, i = j r + s, i = j Denote by p 1 , . . ., p d ∈ [0, 1] the transmissions parameters associated to v and assign them to the edges of M .Let P x,y (= P x ⊗ P y ) be the distribution of (X, Y ) where X and Y are two independent WBM's on M started at x and y, respectively.Set T ∆ = inf{t : X t = Y t }.We note that if ρ(x, y) ≤ ǫ, then P (2) Hence, it is enough to show that for some β > 0, (6.18) inf The proof is separated into several lemmas.We first consider the case when y = 0 : Lemma 6.2.For any β > 0, there is p = p(β) > 0 such that for all x ∈ M, (6.19) Proof.The WBM on M is scaling invariant : if X is a WBM started at x, then for all λ > 0, the process X λ defined by X λ (t) = λ −1 X(λ 2 t) is a WBM started at λx.Using this scaling property with λ = |x| −1 , we obtain that where u = x/|x|.In other words, to prove the lemma it suffices to prove that for all j, we have that (6.20) P e j (1),0 [T ∆ ≤ β] > 0.
We now consider the case where x, y belong to two different edges : Lemma 6.3.There exists p > 0 such for all (x, y) ∈ E i × E j with i = j, (6.21) Then, using the strong Markov property at time τ Y , On the event {|X(τ Y )| ≤ |x| + |y|}, we have that where β = (|x|+|y|) 2 |X(τ Y )| 2 | ≥ 1 and p(1) is defined in Lemma 6.2.We therefore have that Since |X| is a BM reflected at 0, we have that under P x , |X| is distributed as ||x| + B| where B is a Brownian motion started at 0. Setting ϕ(t) = P [|B t | ≤ 1] for all t > 0, we thus have that We therefore have that, using again the scaling property for Y , which proves the lemma.
We finally consider the case where x, y both belong to the same edge : Denote by q the probability that two independent BMs started at distance one meet before time 1.Then q is a positive probability.Let ε > 0 be such that (6.22) Lemma 6.4.There exists p > 0 such for all (x, y) ∈ E i × E i , (6.23) The scaling property implies that, setting r = |x| |y| , P x,y T ∆ ≤ ρ(x, y) 2 = P i r,1 T ∆ ≤ (r − 1) 2 where P i r,1 = P e i (r),e i (1) .Further, where under Q r,1 , X and Y are two independent Brownian motions respectively started at r and 1.We thus have that We have that (using again the scaling property) Using the reflection principle, we have that continuous modification θ of ψ 0 as in Section 3.1.Let W n (t) = t sn sign(Φ n (s)) dΦ n (s), t ≥ s n .Note that a.s.
Lemma 6.5.Let L ⊂ R × R be a compact set.Then a.s. the set Proof.For a continuous real-valued function f denote its modulus of continuity on [a, b] by m a,b (δ; f ) : Hence, a.s.for any a, b, s n ≤ a < b, we have that m a,b (δ; The result follows from Theorem 5.3.Lemma 6.5 ensures that a.s.Φ is a skeleton.Lemma 6.6.F := R × {0} is a closed shell of B(Φ), the set of bifurcation points of Φ.
It is easy to see from (6.26) and (6.25) that K s,t x contains only the single function f defined by This proves that τ s x > t > s and that (s, x) ∈ B(Φ).Similarly, R × (−∞, 0) ⊂ B(Φ) c .Thus B(Φ) ⊂ F .Theorem 6.7.There exists a strong measurable continuous modification of ψ 0 .
Proof.Since Φ is a skeleton a.s., we can construct θ, the measurable continuous modification of ψ 0 , as in Section 3.1.
To prove this theorem we apply Theorem 3.9.We take for closed shell of B(Φ) the set F = R × {0} and as in Section 3.2 define out of θ the stopping times σ s x (k) and the random variables z s x (k), k s x .Note that z s x (0) = x and z s x (k) = 0 if k ≥ 1.To apply Theorem 3.9, we have to verify that a.s.for all (s, x) ∈ R × R, k s x < ∞, and if k s x = 0, then for every t > s there is n We note that with probability 1 for every (s, x) ∈ R × R and every n

Suppose that k s
x ≥ 3 for some (s, x) ∈ R × R and let n ∈ I s .Then s = σ s x (0) < σ s x (1) < σ s x (2) < σ s x (3).It follows that |θ σ s x (j),t (z s x (j))| > 0 for t ∈ (σ s x (j), σ s x (j + 1)), j ∈ {0, 1, 2}, and z s It follows that W n has two local minima at the level W n (s) − |x|, which is a.s.impossible (see Tanaka 2001).As a consequence, we have that a.s., k s x ≤ 2 for all (s, x) ∈ R × R. Suppose that k s x = 0 for some (s, x) ∈ R × R. Then σ s x (1) = s and x = 0.If Φ n (s) = 0 for some n ∈ I s , then for all t > s, θ s,t (0) = Φ n (t).Otherwise, there exists a sequence (n j : j ∈ N) in I s , such that lim j→∞ Φ n j [s, ∞) = θ s,• (0) in X.Without loss of generality we can assume that Φ n j (s) > 0 for all j ∈ N, and This is a.s.impossible (see Tanaka 2001).Hence, Φ n j 0 (r) = 0 for some j 0 ≥ 1 and r ∈ (s, t].It follows that Φ n j (r) = 0 = Φ n j 0 (r) for all j ≥ j 0 , and θ s, This verifies the condition of Theorem 3.9 and proves that ψ is a strong measurable continuous modification of ψ 0 .6.3.Burdzy-Kaspi flows.In this section we consider a stochastic flow of measurable mappings in R that consists of solutions to the Harrison-Shepp SDE for the skew Brownian motion (6.27) dX(t) = dW (t) + βdL(t), where W = (W (t) : t ∈ R) is a Brownian motion on R, L is the symmetric local time of X at zero and β ∈ [−1, 1].It is well-known that for all (s, x) ∈ R × R the equation (6.28) )dr, t ≥ s has a unique strong solution (see Harrison and Shepp 1981).Define where x ∈ R n , B ∈ B(R n ), t ≥ 0, and X i is the solution of (6.28) with initial condition X i (0) = x i , i ∈ {1, . . ., n}.The sequence (P (n) • : n ∈ N) is then a sequence of coalescing Feller transition functions that satisfies (TF 1), (TF 2) and (TF 3).Let ψ 0 be a stochastic flow of measurable mappings in R associated to (P (n) • : n ∈ N).Let D = {(s n , x n ) : n ∈ N} be a countable dense set in R × R, and let Φ be the random sequence constructed in Section 3.1.2.Recall that for each n, Φ n is a continuous modification of ψ 0 sn,• (x n ) with Φ n (s n ) = x n .By Burdzy and Kaspi 2004, Prop.2.1, the condition (TF 4) is satisfied and Theorem 6.8.There exists a strong measurable continuous modification of ψ 0 .
Proof.Similarly to the proof of Lemma 6.6 it can be checked that F = R × {0} is a closed shell of B(Φ), the set of bifurcation points of Φ.We then construct θ, the measurable continuous modification of ψ 0 , as in Section 3.1.As in Section 3.2 we define out of θ the stopping times σ s x (k) and the random variables z s x (k), k s x .Note z s x (0) = x and z s x (k) = 0 if k ≥ 1.To apply Theorem 3.9, we have to verify that a.s.for all (s, x) ∈ R × R, k s x < ∞, and if k s x = 0, then for every t > s there is n ∈ I t such that θ s,• (x)[t, ∞) = Φ n [t, ∞).
There exists a stochastic flow ψ 0 of measurable mappings in M, such that for all (s, x) ∈ R × M and all f ∈ D, f (ψ 0 s,t (x)) = f (x) + Theorem 6.9.There exists a strong measurable continuous modification of ψ 0 .
Proof.Similarly to the proof of Lemma 6.6 it can be checked that F = R × {0} is a closed shell of B(Φ), the set of bifurcation points of Φ.Let θ be the measurable continuous modification of ψ 0 defined from Φ and the mapping Θ.As in Section 3.2 we define out of θ the stopping times σ s x (k), and the random variables z s x (k), k s x .Consider the mapping G : M → R, G(x) = ε(x)|x|.To have notations consistent with Hajri 2011, we now suppose without loss of generality that D = Q × M Q , where M Q is a countable dense set in M. Out of the random skeleton Φ, we construct a family of random mappings Y such that Y n = G(Φ n ).The family Y is a random skeleton of a Burdzy-Kaspi flow (see Hajri 2011, in which ψ 0 is constructed with a Burdzy-Kaspi flow).Let us now define Y, the Burdzy-Kaspi flow, out of the skeleton Y in the same way as ψ 0 is constructed out of Φ (note that G(ψ 0 s,t (x)) = Y s,t (G(x))).The proof of Theorem 6.8 implies that a.s.for all (s, x) ∈ R × M, k s x ≤ 2 (the random variables k s x and k s ε(x)|x| are equal, where k s ε(x)|x| is defined out of Y ).Suppose that k s x = 0. Then x = 0 and σ s x (1) = s.The proof of Theorem 6.8 implies that a.s. the local time at zero L s,r of Y s,• (0) is positive for all r > s.Following the proof of Burdzy and Kaspi 2004, Prop.1.1(iii), for all small enough positive ε, {Φ n (t) : n ∈ I s , ρ(Φ n (s), 0) < ε} is a finite set.Since there is a sequence (n k : k ∈ N), such that Φ n k [s, ∞) converges towards θ s,• (0) in X, the sequence Φ n k [t, ∞) is stationary.This shows that there is n ∈ I t in such that θ s,u (0) = Φ n (u) for all u ≥ t.Hence, conditions of Theorem 3.9 are verified and there exists a strong measurable continuous modification of ψ 0 .
The following Corollary extends the result of Theorem 6.9 to general metric graphs.For the definition of the Tanaka SDE on a metric graph we refer to Hajri and Raimond 2014.

Institute of Mathematics of NAS of Ukraine
Email address: ryabov.george@gmail.com

•
: n ∈ N) of coalescing Feller transition functions on M. It is assumed that (P (n) • 0 s,r (x))f ′ (ψ 0 s,r (x))dW (r) + 1 2 t s f ′′ (ψ 0 s,r (x))dr, t ≥ s,where W is a Brownian motion on R Hajri 2011.Let D = {(s n , x n ) : n ∈ N} be a countable dense set in R × R. For each n ∈ N, we let Φ n be a continuous modification ofψ 0 sn,• (x n ) with Φ n (s n ) = x n .The proof of Lemma 6.5 shows that Φ is a.s. a skeleton.