Time-Reversal of Coalescing Diffusive Flows and Weak Convergence of Localized Disturbance Flows

We generalize the coalescing Brownian flow, aka the Brownian web, considered as a weak flow to allow varying drift and diffusivity in the constituent diffusion processes and call these flows coalescing diffusive flows. We then identify the time-reversal of each coalescing diffusive flow and provide two distinct proofs of this identification. One of which is direct and the other proceeds by generalizing the concept of a localized disturbance flow to allow varying size and shape of disturbances, we show these new flows converge weakly under appropriate conditions to a coalescing diffusive flow and identify their time-reversals.


Introduction
This paper is a contribution to the theory of stochastic flows in one dimension, specifically the study of inhomogeneous flows and their time-reversals.
We provide two proofs of our main result which is Theorem 4.1 which says that the time-reversal of a coalescing diffusive flow with drift b and diffusivity a is (provided the spatial derivative a ′ of a is Lipschitz) given by a coalescing diffusive flow of drift −b + a ′ 2 and diffusivity a. Theorem 5.4 which establishes convergence of certain families of inhomogeneous disturbance flows to coalescing diffusive flows may also be of independent interest.
A disturbance flow, introduced in [5] is a composition of independent random maps of the circle to itself. Unlike [5], we do not require that our maps are identically distributed or that their distributions are invariant under conjugation by a rotation of the circle. For a pair of suitably smooth a, b, we consider limits where the maps F are close to the identity, well localized and have mean of F (x) − x close to hb(x) and variance of F (x) − x close to ha(x) as h → 0. We prove convergence of individual paths to diffusion processes and of the flow as a whole to the coalescing diffusive flow with diffusivity and drift given by a and b. We also describe the time-reversal of the disturbance flows and use this to describe the time-reversal of a coalescing diffusive flow.
The coalescing diffusive flow consists of a diffusion process starting from each point in space-time each with drift and diffusivity given by the same functions of space and time, they evolve independently until they collide at which point they coalesce. The idea of such an object with standard Brownian motions instead of diffusions has been studied widely, starting with Arratia in 1979 [1]. One approach to this is to define a family of random measurable functions (φ ts : s ≤ t ∈ R) satisfying the flow property φ ts • φ sr = φ tr , r ≤ s ≤ t and such that every finite collection of trajectories (φ ts (x) : t ≥ s) performs coalescing Brownian motions, this is the approach taken in Arratia [1], Le Jan and Raimond [4] and Tsirelson [8]. A problem however with this approach is that the φ ts cannot be chosen to be right-continuous. An alternative approach that avoids this problem is given by Fontes et al. [3] based on completing the set of trajectories to form a compact set of continuous paths, this completion can be done in multiple ways leading to multiple objects known as Brownian webs. Another way around the problem was introduced by Norris and Turner in [5] based on the idea of considering pairs {φ − , φ + } of left and right continuous modifications of the Arratia flow, this set up doesn't store the information of the value of φ ts at a jump and as a result the flow property must be relaxed to a weak flow property (definition in Section 3 of this paper), the space of weak flows with the metric appearing in [5] provides a useful space for studying weak convergence as it contains flows without continuous trajectories such as disturbance flows, this is the approach that this paper builds on. The paper is structured as follows. Section 2 proves existence and uniqueness of a simplified version of the coalescing diffusive flows which consists of only countably many paths. Section 3 defines the metric spaces that our flows take values in and proves existence and uniqueness of the coalescing diffusive flows (Theorem 3.1). Section 4 defines the time-reversal of a flow and provides the statement of our main result (Theorem 4.1) which identifies the time-reversal of a coalescing diffusive flow. At this point the reader has the option of skipping straight to section 7 which will not require sections 5 or 6. Section 5 defines the notion of a disturbance flow and shows convergence of paths from the flow to diffusions and of countable collections of paths to the simplified flow from section 2. Section 6 shows convergence of the disturbance flows to coalescing diffusive flows, identifies their time-reversals and uses this to provide a proof of Theorem 4.1. Section 7 provides an alternative proof of Theorem 4.1 that does not require the use of disturbance flows, it also contains as an intermediate weaker version (requiring more smoothnes of a and b) Theorem 7.1.
The disturbance flow based approach to our main result is based on [5], much of the notation is taken from there and some of the proofs are very similar however there are multiple places where new ideas are required to handle the generalization. While [5] allows the distribution of disturbances to be random only in that the location of the disturbance is chosen uniformly at random from around the circle we allow the disturbances to vary in size and shape both randomly and with location in space and time, the shape and size is also allowed to vary a lot more as we take the limit to small disturbances than is allowed in [5]. The new ideas in the proofs are first evident in the proof of Theorem 5.1, showing that individual trajectories of suitable disturbance flows converge weakly, where the proof of tightness requires bounds that hold despite the possibly varying drift and diffusivity. The time reversal results in Section 6 are generalizations of those in [5] however the statement of our main result Theorem 4.1, is not something that you would obviously expect and the proof had to be modified substantially to deal with the more general disturbance flows.
The proof in section 7 is original in idea as well as in detail. It is about the same length as the disturbance flow based proof, however the weaker version of our main result Theorem 7.1 (which is identical except it assumes that a and b are Lipschitz in time as well as space) is proved with a substantially smaller amount of work (about 5 pages after the statement has been made rigorous rather than 18) and might suffice for future applications. In particular it provides a short proof, without the use of disturbance flows, of the Brownian case which is Corollary 7.2 of [5].

Countable Collections of Coalescing Diffusions
In this section we recall uniqueness of law for weak solutions of SDEs, then define a metric space, D E , whose elements consist of countable collections of cadlag paths. Finally we identify certain elements of D E , using a martingale problem in the style of [7], specifically those corresponding to a countable family of coalescing diffusion processes that are independent until collision. Given functions a : R 2 → R >0 and b : R 2 → R measurable, bounded uniformly on compacts in the first variable and L-Lipschitz in the second, then the SDE has uniqueness of law for weak solutions [6], i.e. given e = (s, x) and a triple (X, W ) t≥s , (Ω, F , P), (F t ) t≥s , such that a) (Ω, F , P) is a probability space with (F t ) t≥s as a filtration satisfying the usual conditions b) X is adapted to (F t ), X is continuous and W is an (F t )-Brownian motion c) X s = x d) Almost surely, both X and the quadratic variation of X are bounded on each compact time interval and e) Almost surely then the law of X is determined by a,b and e. We will write this law as µ a,b e , and say that X is a diffusion process with drift b and diffusivity a. Throughout we will assume that a and b have period 1 in the second variable (as well as the properties above) and X will be considered as a diffusion process on the circle R/Z.
We will in several proofs use the notation where I is an compact interval of time that contains all the times relevant to the given context. It will only be important that in any given context these numbers are finite and a * > 0.
Let D e = D x ([s, ∞), R) be the space of cadlag paths starting from x at time s. Write d e for the Skorokhod metric on D e . Given a sequence E = (e k : k ∈ N) in R 2 , set The T jk are the collision times of the paths considered in R/Z. The following is a generalization of a reformulation in [5] of a result of Arratia in [1]. Proposition 2.1. Given a, b measurable and bounded uniformly on compacts in time and L-Lipschitz in space as in (1), there exists a unique Borel probability measure µ a,b E on D E under which, for all j, k ∈ N, the processes t≥s j ∨s k are both continuous local martingales.
We give the following proof sketch. For existence, one can take independent diffusion processes, with coefficients a and b, from each of the given time-space starting points and then impose a rule of coalescence on collision, deleting the path of larger index. The law of the resulting process has the desired properties. On the other hand, given a probability measure such as described in the proposition, on some larger probability space, one can use a supply of independent Brownian motions to build diffusions continuing each of the paths deleted at each collision. Then the martingale problem characterization of diffusion processes given in [7] can be used to see that one has recovered the set-up used for existence, this gives uniqueness.

Existence and Uniqueness of Coalescing Diffusive Flows
We now introduce the space of continuous weak flows C • (R, D) and the space of cadlag weak flows D • (R, D) both introduced in [5]. We will then identify certain elements of C • (R, D) as coalescing diffusive flows, again using a martingale problem. C • (R, D) is sufficient for stating our main result and understanding the proof that doesn't use disturbance flows, however we will need D • (R, D) to deal with the fact that the disturbance flows are not continuous in time. The following explanation of notation follows [5] very closely, and all the claims made in italics are proved in [5]. We consider non-decreasing, right-continuous functions f + : R → R with the degree 1 property Let us denote the set of such functions by R and the set of analogous leftcontinuous functions by L. Each f + ∈ R has a left-continuous modification given by f − (x) = lim y↑x f + (y). Let D denote the set of corresponding pairs We will write f in place of f ± when the choice is irrelevant for the purpose at hand, especially in the case when f + = f − i.e. f + is continuous.
Firstly we define a metric on D. Associate to each function f a function Figure 1: The graph of f × can be formed from the graph of f by rotating the axes by π 4 and scaling both axes up by √ 2 as shown in Figure 1. We can define a complete locally compact metric Consider φ = (φ I : I ⊆ R), with φ I ∈ D and I ranging over all non-empty bounded intervals. φ is a weak flow if given I a disjoint union of intervals I 1 and I 2 , Here the convergence of funcitons is in the uniform norm (also note that this definition is left-right symmetric, we call it cadlag to match previous work).
is the set of cadlag weak flows. We set φ ∅ = id. Given {I n : n ∈ N} and I bounded intervals, write I n → I if [s,t] ∀s < t, denoting these all by φ ts we define C • (R, D) to be the set of all such (φ ts : s, t ∈ R, s < t). For φ, ψ ∈ C • (R, D) and n ≥ 1, define d and then let Under this metric C • (R, D) is complete and separable.
In the interests of defining a metric on D • (R, D), for λ an increasing homeomorphism of R we define and let χ n be the cutoff function given by We can now define for φ, ψ ∈ D • (R, D) and n ≥ 1, where the infimum is taken over the set of increasing homeomorphisms λ of R. Then define to D e , by setting Z e,± (φ) = (φ ± (s,t] (x) : t ≥ s). The maps, t → Z e,± t (φ) are continuous when φ ∈ C • (R, D). Finally define a σ-algebra F and a filtration (F t ) t∈R on C • (R, D) by F = σ(Z e t : e ∈ R 2 , t ≥ s(e)), F t = σ(Z e r : e ∈ R 2 , r ∈ (−∞, t] ∩ [s(e), ∞)).
Then F t is generated by the random variables Z e r with e ∈ Q 2 and r ∈ (−∞, t] ∩ [s(e), ∞), and F is the Borel σ-algebra of the metric d C .
The following theorem states the existence of coalescing diffusive flows. The proof given follows the line of argument for the less general result Theorem 3.1 in [5], the italicized assertions in the proof below are proved in [5]. Generalizing the argument requires generalized versions of results from [5] which are Proposition 2.1 and Proposition 8.1. and are continuous local martingales wrt (F t ) t∈R . Moreover, for all e ∈ R 2 we have µ a,b A -almost surely Z e,+ = Z e,− .

Proof. Fix an enumeration
Then the sets C •,± E are measurable subsets of C E and, by Proposition 8.1, , which we denote by Φ E,± , are measurable. Write Z E for Z E,+ and Φ E for Φ E,+ . Then, on C •,+ E , for all j, k ∈ N, we have and for all t ∈ R and B ∈ F t we have 1 A , for all j, k ∈ N, taking e = e j and e ′ = e k makes the processes (3) and (4) into continuous local martingales for (F t ) t∈R .
On the other hand, for every probability measure µ on C • (R, D) having this property, under the image measure µ • (Z E ) −1 on C E , for all j, k ∈ N, taking e = e j and e ′ = e k makes the processes (3) and (4) into continuous local martingales for E by Proposition 2.1, and so µ = µ a,b A . Given e −1 , e 0 ∈ R 2 , all the assertions above hold when E is replaced by the sequence E ′ = (e −1 , e 0 , e 1 , e 2 , ...). We repeat the steps taken to obtain a probability measure µ ′a,b Then, under µ ′a,b A , taking e = e −1 and e ′ = e 0 makes the processes (3) and (4) into continuous local martingales for (F t ) t∈R . But also, under µ ′a,b A , for all j, k ∈ N, taking e = e j and e ′ = e k makes the processes (3) and (4) into continuous local martingales for E ′ -almost surely, and so Z e,− = Z e,+ , µ a,b A -almost surely, as claimed. We will often write µ A instead of µ a,b A in order to simplify notation.

Time Reversal
In this section we quote some definitions and observations from [5] and then state our main theorem. For f + ∈ R and f − ∈ R, define the left-continuous inverse from R to L and the inverse operation right-continuous inverse respectively as follows Note that these operations are distributive over concatenation. The inverse of f ∈ D is given by The time-reversalφ of a flow φ is given bŷ As before let a and b be the diffusivity and drift of a diffusive flow with law µ A . We require that a and b satisfy the smoothness requirements of section 2 and further require that a is differentiable with respect to x with derivative a ′ (x, t), which is L-Lipschitz in x and measurable and bounded uniformly on i.e. let it be the law of a disturbace flow with drift and diffusivity given by b ν and a ν . Finally writeμ A for the image measure of µ A under time-reversal.

Disturbance Flows from Countably Many Points on a Circle
This section lays the ground work for section 6, the reader may skip to section 7 at this point if they only wish to read the direct proof. We start this section by defining the notion of a disturbance flow on the circle, this is based on a notion of disturbance flow which was given in [5] but is more general so as to allow for our disturbance flows to have drift and varying diffusivity. We will then proceed to state and prove two propositions and deduce a theorem. Firstly that undera ppropriate conditions a sequence of single paths from disturbance flows can converge to a diffusion process, secondly that a sequence of countable families of paths from disturbance flows can converge to a countable family of coalescing diffusions and finally combining these propositions with a result from [5] we conclude that disturbnce flows can converge to coalescing diffusive flows.
We specify a disturbance flow by a family of probability distributions on D written The parameters of the family are, h > 0 which corresponds to the size of the disturbance (the limit for our convergence later will be taking h to 0 while making disturbances more frequent) and time t which allows our flow to be inhomogeneous in time, we require that η be measurable as a function of t.
is not in general an element of D, however so long as f 1 sends no interval of positive length to a point of discontinuity of f 2 we will have f 2 • f 1 ∈ D. To avoid this issue we will only consider families of probability distributions on D such that, if We denote the set of such families by D * , we assume from here on that We extend the inverse functions of section 4 to families of probability distributions F ∈ D * by setting where the inverse on the right hand side is being taken with respect to the x argument (as opposed to the implicit ω argument).
We can now define the discrete disturbance flow Φ ± for a fixed h > 0 and fixed (t n : n ∈ Z, t n < t n+1 ). Take the sequence (F n = F h,tn : n ∈ Z), then define for m, n ∈ Z with m < n and Φ ± n,n = id. Note that if Φ n−1,m ∈ D then it must have at most countably many intervals of constancy, and by condition (5) we have that a.s. Φ n,m = {Φ − n,m , Φ + n,m } ∈ D, thus a.s. ∀m ≤ n we have Φ n,m ∈ D.
To embed this disturbance flow into continuous time we proceed as follows. Let N be a Poisson random measure on R of intensity h −1 and set These definitions can be extended to intervals closed on the left (and/or open on the right) by replacing s −→ N s by its left-continuous modification. Write Φ for the family of maps Φ I wgere I ranges over all bounded intervals in R.
We call Φ the Poisson disturbance flow or just the disturbance flow and write µ η A for the distribution of Φ in D.
Fixing e = (s, x) ∈ R 2 we define 2 processes X e,± t by setting X e,± and thus by right continuity of X e,± we have a.s. that ∀t ≥ s, Thus we drop the ± and write simply X e . Write µ η e for the distribution of X e on the Skorokhod space D e . Similarly, for E = (e k ∈ R 2 : k ∈ N), (X e k : k ∈ N) is a random variable in D E and we write µ η E for its distribution on D E .
Given a family η ∈ D * , and coefficients a and b as in section 2, we define the functionsF The following three conditions will be important for the next proposition and consequently for the rest of the results: Proposition 5.1. Suppose a and b are coefficients as in section 2, and that η is such that conditions (6), (7) & (8) hold then we have µ η e → µ a,b e weakly on D e , as h → 0.
Proof. Let (X n ) n∈N be a sequence of processes distributed according to µ η e with h → 0 as n → ∞. By the definition of the Skorokhod metric it suffices to show that for any T > s the restrictions of X n to [s, T ] converge weakly to a solution of the SDE on [s, T ], for the remainder of this proof we consider X n to be restricted to [s, T ]. We then take e = (s, x) = (0, 0) and T = 1, without loss of generality.
Firstly will shall calculate (up to an error that is small for small h) 2 expected values. We shall then prove a characterization of tightness of the sequence, which will require us to use these calculations to show that the process can't vary too much on a given interval, then deduce the existence of a subsequential limit of each subsequence by Prokhorov's theorem. Finally we will identify the distribution of every subsequential limit as a weak solution of equation (1) using again the 2 expectation calculations, then we will conclude the proof using the uniqueness of law for such solutions .
Let F n t be the completion of the filtration generated by Where the approximation errors E i can be bounded as follows Breaking the interval (s, t] into a large number of small intervals and taking the limit as the interval sizes go to 0, we have that: The characterization of tightness that we shall use is given in Billingsley 1968 [2] Theorem 15.3, it says that tightness is equivalent to the following 2 conditions holding: Note that B h and A h going to 0 as n → ∞ means that |b| and |a| are bounded uniformly in n, x and t ∈ [0, 1], we call the bounds B and A respectively.
The first condition can be shown as follows, where T K is the first time t such that X n t ≥ K.
where in the final inequality we have used Chebyshev's inequality. This bound goes to 0 as K → ∞ uniformly in h. Combining with a corresponding bound for inf X n t gives the first condition. Note that for the second condition it suffices to show the following stronger statement, where I δ is the set of subintervals of [0, 1] of length δ.
∀ǫ > 0 there exists δ ∈ (0, 1) and N ∈ N such that which is in turn weaker than the following, where I ′ δ is the set of intervals of length δ with endpoints that are multiples of δ/2.
There are only 2 δ elements in I ′ δ , so using a union bound it suffices to show that for sufficiently small h and some δ we have where a factor of 4 has been included purely for convenience later. We present the proof for I = [0, δ] but the same argument and bound will hold for all I ∈ I ′ δ . We have that We will bound the first term on the right with a bound that will also apply to the second term by symmetry. Unfortunately, Chebyshev is not strong enough to bound the first term sufficiently tightly. We will apply the Azuma-Hoeffding inequality which requires the following set-up. Let X ′n t = X n t − tB and note that this is a super-martingale. Fix 0 < α < 1 2 , let R 0 = 0 and for i ≥ 1 let R i be the first time t such that |X ′n by the same argument used in the first condition we have the following for l < ] for sufficiently small h. An application of the Azuma-Hoeffding Inequality to uniform random variables gives the following.
We will bound the first term on the right of the last inequality, and note the second term can be bounded similarly. Let X ′′n i = X ′n R i − iM h note that this is a discrete super-martingale with step size bounded by M α h + M h .
for sufficiently small h (12) where we have used the Azuma-Hoeffding inequality again. Bringing these bounds together gives that for a given δ we have for sufficiently small h that Thus, by choosing δ so that the second term is less than δǫ 4 and then choosing N such that ∀n ≥ N we have that h is sufficiently small that the bound (12) holds and that the first term is less than δǫ 4 , we can conclude that the second condition holds and the sequence µ η e is tight. By Prokhorov's theorem we now know that every subsequence has a weakly convergent subsequence and by standard arguments it suffices to show that the limit of every such sequence is µ a,b e (restricted to [0, 1]). Let µ be the limit of such a subsequence and X be distributed according to µ.
We now show that X is a solution of the SDE (1). Now let (F t ) t≥s be the completion of the filtration generated by X and let W be defined as follows.
Note continuity of X follows from the bound (11) and so F is rightcontinuous and thus satisfies the usual conditions. X s = x is immediate Equation (2) holds by the definition of W .
The identities (9) and (10) show in the limit n → ∞ that both X and the quadratic variation of X are a.s. bounded on each compact interval. The same argument used to get these identities can also be used to find that From the definition of W and the continuity of X we can deduce W is continuous a.s., putting this together with the above expectations we can conclude by Lévy-Characterization that W is a (F t )-Brownian motion. Thus X solves (1) and has the required law.
Define λ h (f ) to be the infimum of λ such that, Proof. We write X k for X e k . The family of laws on D E is tight as each family of marginal laws on D e k is tight. Let µ be a weak limit law for µ η E , then for all j, k and all t > s ≥ s j ∨ s k , letting E * (·) = E(· | t Ns+1 ≤ t < t Ns+2 , F s ) we have: Where we have (by the same method used to bound E 3 in Proposition 5.1) So for (t − s) Hence breaking [s j ∨ s k , ∞) into intervals of length t − s and taking the limit as t − s and h go to 0 gives that the following process stopped at time T jk is a martingale.
Further this process must be continuous because Proposition 5.1 tells us that Z j t and Z k t are continuous. We know from Proposition 5.1 that, under µ, both are continuous local martingales. It remains to show that Z j t − Z k t is constant for t ≥ T jk after which the result follows from Proposition 2.1. Let Y t = Z j t − Z k t and assume w.l.o.g that Y 0 > 0 and Y T jk = 0. Y inherits the property of not changing sign as our disturbances are order preserving. Given R ∈ R and ǫ > 0 localize Y using the stopping time S = inf{t : Y t > 1 or t > R} and note that: where L is the Lipschitz constant of b. So, by Gronwall's inequality, E|Y S T jk +t | is identically 0 up to time t = R. So Y t = 0 for all t > T jk a.s. and we are done.
Let E = (e k : k ∈ N) be an enumeration of Q 2 . Write Z E,± for the maps D • (R, D) → D E given by Z E,± = (Z e k ,± : k ∈ N). Write Z E = Z E,+ . The following result is a criterion for weak convergence on D • (R, D), and is Theorem 5.1 of [5].
Theorem 5.3. Let (µ n : n ∈ N) be a sequence of Borel probability measures on D • (R, D) and let µ be a Borel probability measure on C • (R, D) Assume that Z E,− = Z E,+ holds µ n -almost surely for all n and µ-almost surely. Assume further that µ n • (Z E ) −1 → µ • (Z E ) −1 weakly on D E . Then µ n → µ weakly on D • (R, D).
The following result is immediate from Proposition 5.2 and Theorem 5.3.

Proof of Theorem 4.1 using Disturbance Flows
In this section we identify the time-reversal of a generic disturbance flow. We then apply this identification to an explicit sequence of flows and as the limit of the reversals must be the reversal of the limit we can deduce Theorem 4.1.
The following proposition is a generalization of the first half of Proposition 7.1 of [5] which can be recovered by assuming that b h ≡ 0 and a h ≡ 1.
Proof. The proof is very close to the second half of the proof of proposition 7.1 of [5]. Set m and n to be the minimal and maximal values taken by N t in I and −n and −m to be the minimal and maximal values taken by N t in −I. Then we can define a disturbance flow Φ with disturbance F h , by . By the properties of the Poisson process (−t −m , . . . , −t −n ) is equal in distribution to (t m , . . . , t n ), soΦ is a disturbance flow with disturbance G h .
In [5] it is then shown that for a ≡ 1 and b ≡ 0 we have that µ A is invariant under time-reversal, we generalize this result to Theorem 4.1.
Theorem 4.1. If a has spatial derivative a ′ and a, b and a ′ are uniformly bounded on compacts in time and Lipschitz in space then Proof. The proof is based on the fact that given a family (F h ) h>0 (satisfying the conditions of proposition 5.1) we have that: µ It thus suffices to show for some specific family (F h ) h>0 that µ h,−t ) h>0 satisfies the conditions that we put on F but with a and b replaced by a ν and b ν . Letâ h ,â,b h andb be defined from F −1 as a h and b h are defined from F . We will consider the family of disturbances given by letting θ = θ h,t be i.i.d. uniform random variables on [0, 1], otherwise.
An example from this family is graphed in Figure 2. Note that λ → 0 for both f and f −1 (The disturbance of size r θ,t is negligible in computing λ as it is O(h 3 ) in the definition of λ). The first 3 cases in the above definition also contribute nothing to either lim h→0 a h or lim h→0âh and their contribution to lim h→0 b h is exactly the negative of their contribution to lim h→0bh so it suffices to prove that the proposition holds for the case b = a ′ i.e. the case where r θ,t ≡ 0.
We write w ± for the largest offsets from x a disturbance can have whilst not mapping x to itself. For sufficiently small h they are given by the following implicit equation: . gives where unless otherwise specified a and a ′ are evaluated at (x, t). We can now By Taylor and binomial expansion we also get Which allows us to calculate, So the result holds.
The following corollary is similar to Corollary 7.3 of [5] (and with an almost identical proof) in that it gives weak convergence for paths running both forward and backward from a given sequence of points. First we define the notation for this result.
Given e = (s, x) ∈ R 2 , defineD e = {ξ ∈ D(R, R) : ξ s = x} and for ThenZ e,± (φ) ∈D e and extends Z e,± (φ), from [s, ∞) to the whole of R. For all e ∈ R 2 , we haveZ e,− almost everywhere on D • (R, D) for both µ A and µ f A , for every disturbance function f . So we drop the ±. Denote byμ f E the law of (Z e k : k ∈ N) onD E under µ f A and byμ a,b E the corresponding law under Proof. Given φ with law µ a,b A , we have that almost surelȳ uniformly on R as δ → 0. We also have φ ∈ C • (R, D) almost surely and it follows thatZ (s,x),+ is continuous at φ almost surely. Thus the result holds as we already know the convergence holds component wise.

Proof of Theorem 4.1 without Disturbance Flows
In this section we first prove a version of Theorem 4.1 with the extra hypothesis that a and b are Lipschitz in time. Then we use an approximation argument to show Theorem 4.1 in the general case.
Theorem 7.1. If a has spatial derivative a ′ and a, b and a ′ are Lipschitz in both time and space thenμ Proof. Let φ ∼ µ A . It suffices to show that the restriction ofφ to E given by Z E,+ (φ) which we shall callφ E has distribution ν E , for each countable set E ⊂ R × R. The distribution ν E is characterised by its restriction to two point motions by Theorem 3.1. Coalescence of two motions follows immediately from the definition of time-reversal. As does the continuity of a single motion.
As φ ts and φ su are independent for s ∈ (u, t) we have the Markov property. Thus by Donsker's Invariance Principle we can identify the two point motion from just the mean and covariance matrix of small increments.
First we consider each one point motion separately. We will proceed by relating the backward and forward flows. Then noting that increments of the forward process are small, we approximate a and b on an interval that the forward process almost surely won't leave in such a way as to make exact calculations possible. Then we check that the incurred error is small using that a and b are Lipschitz in time and that the exact calculations give the required answer. Finally we will show that the increments of each process are independent conditional on an event of large probability and so the covariances are small.
We have the relation, which we can use to determine the distribution ofφ t+h,t (y) if we first understand the distributions of the variables φ −t,−t−h (x).
To study these variables we first show that the forward paths are localised.
Where we have written a for a(t, y), a ′ for a ′ (t, y) and b for b ′ (t, y).ã andb are then extended toL-Lipschitz andL-Lipschitz differentiable functions on the circle for someL. Note that a =ã(s, y), a ′ =ã ′ (s, y) and b =b(s, y) this will turn out to make them sufficiently good approximations. We now approximate the diffusion process φ t+δt,t (x) for each x ∈ [y − h 1 2 −ǫ , y + h 1 2 −ǫ ] by a diffusion process X δt started from x with driftb and diffusivityã but driven by the same Brownian motion B δt as φ t+δt,t (x). Let G be the event and note the second event in this union has probability bounded like the first and that on this event X and φ t+δt,t (x) stay within the interval we explicitly definedã andb on. Note also that P(G) = 1 − O e −Ch 2ǫ .
On this event the error in the approximation is given by We have that if E h := max δt<h |∆ δt | then Finally consider G ′′ = {E ′′ δt < h 3 2 −3ǫ } and note that the probability of this event is 1 − O(e −Ch −2ǫ ). Thus we can conclude that This result suffices to control the error of the approximation.
Next we calculate the distribution of X h . Note that on the event G we have, for some Brownian motion W, that Where we have written a for a(y), a ′ for a ′ (y) and b for b ′ (y). Define ). An application of Itō's lemma gives that The choices forã,b and f were made so that this equation has constant coefficients, thus f (X h ) is normally distributed with mean and variance h. So we can calculate and thus and for y = 0 and |x| < h We can relate this toφ by Finally we use this to compute, and similarly that Var φ t+h,t (y) = ah + O(h 2−8ǫ ).
Thus the single point motions are diffusion processes with the required drift and diffusivity.
Next, we will show that the motions started from y 1 and y 2 have zero covariation until they coalesce and thus are independent until they coalesce. This follows immediately from the fact that for y 1 = y 2 Cov φ t+h,t (y 1 ),φ t+h,t (y 2 ) = o(h).

To establish this fact consider the events
On the intersection of these events we know that theφ t+h,t (y i ) are independent as the forward flows on [−t − h, t] × y i − |y 2 −y 1 | 2 , y i + |y 2 −y 1 | 2 are independent and each determines the corresponding A i andφ t+h,t (y i ). Thus, writing B for the complement of A 1 ∩ A 2 , Cov φ t+h,t (y 1 ),φ t+h,t (y 2 ) = Cov 1 Bφt+h,t (y 1 ), 1 Bφt+h,t (y 2 ) The final integral converges to 1 by dominated convergence and so the covariance is O(h 3 2 ). This establishes the result.
Finally we relax the restriction that a and b are Lipschitz in time.
Theorem 4.1. If a has spatial derivative a ′ and a, b and a ′ are uniformly bounded on compacts in time and Lipschitz in space then Proof. Define approximations a n and b n by a n = a * K n and b n = b * K n where * denotes convolution in time, and K is a smooth, non-negative function supported on [−1, 1] with supremum and integral equal to one. Let φ n ∈ C • (R, D) be the coalescing diffusive flow driven by a n and b n and let b * k = sup We define A N to be the subset of φ ∈ C • (R, D) such that ∀k both In Proposition 8.2 we prove that A N is compact and in Proposition 8.3 we prove that φ n ∈ A N with high probability in N uniformly in n thus we can deduce that the φ n are tight. Let φ be a weak sub-sequential limit of φ n , we will show that φ ∼ µ A and thatφ ∼ ν A which establishes the theorem.
We present here only the proof that φ ∼ µ A , the proof that lim n→∞φ n ∼ ν A is identical but consideringφ n and −b + a ′ 2 instead of φ n and b, it then follows thatφ ∼ ν A as time reversal is an isometry. By Theorem 3.1 it suffices to show that The proof of these two statements are very similar so we will only provide the more complicated second one here.
Proposition 8.4 says that E (M st (x 1 , x 2 , b, a, φ)) is a continuous function of x 1 and x 2 thus it suffices to show that where E x averages over values of x 1 and x 2 in a pair of intervals I 1 and I 2 respectively and E φ is the same as E on previous lines. Proposition 8.6 says which is used in the calculation below. Writing D n for M st (x 1 , x 2 , b, a, φ n ) − M st (x 1 , x 2 , b n , a n , φ n ) we can calculate, using Proposition 7.1 in the fourth equality, that It remains only to show that E φ n (D n ) goes to 0 uniformly in x as n → ∞.
a n (r, φ n rs (x 1 )) − a(r, φ n rs (x 1 ))dr Each of these terms has expectation tending to 0, we will prove this for the first term, the second term is very similar and the third term is even simpler so the same argument works. We firstly rearrange each half of the first term separately. where v = r + u and I 1 = φ n v−u,s (x 1 )b(v, φ v−u,s (x 2 ))K n (u). The first two of these integrals are over an area that is O(n −2 ) and the integrand I 1 = O(n) so only the final integral will contribute to the limit. where I 2 = φ n rs (x 1 )b(r, φ rs (x 2 ))K n (u). Again the first two terms are O(n −1 ) so only the last term will contribute to the limit. Combining these 2 rearrangements together and discarding small terms we find that where ) .

Appendix
The following result is required to prove the existence of the coalescing diffusive flows as stated in Theorem 3.1. It is a generalization of Proposition 8.10 of [5] and has a similar proof.
Proposition 8.1. Let E be a countable subset of R 2 containing Q 2 and let a, b be measurable and uniformly bounded on compacts in time and Lipschitz in space. Then, taking Proof. Following the proof of Proposition 8.10 in [5] we will verify each of the following 5 conditions hold a.s. and as they characterize C • E inside C E [5] the result follows. Firstly: is defined to make the diffusivity of this process 1.
For sufficiently large n this can be bounded above by a Brownian motion B τ started at 1/n and So P(A) = 1 and the conditions hold. The final condition is that for all ǫ > 0 and all n ∈ N, there exists δ > 0 such that Define for δ > 0 and e = (s, x) ∈ E, Then, letting B be a standard Brownian motion, for sufficiently small δ and large n P(V e (δ) > nδ) ≤ 2P sup Consider, for each n ∈ N the set 3 ) 2 ≤ |E n |e −n 5 4 , so n P(A n ) < ∞, so by Borel-Cantelli, almost surely there exists some N < ∞ such that V e (2 −n ) ≤ n2 −n for all e ∈ E n , for all n ≥ N.
The rest of the propositions in this appendix are used in the direct proof of Theorem 4.1.

Proposition 8.2. A N is compact
Proof. A N is a closed subset of C • (R, D) and so is complete. Therefore by a diagonal argument it suffices to show that for all ǫ > 0 and for all sequences S in A N there exists a subsequence S ′ that is contained in a ball of radius ǫ.
To this end take M such that ∞ m=M +1 2 −m < ǫ 2 then we have that Thus it suffices to find a subsequence S ′ where, for m = 1 to M, we have As d We will take the S ′ corresponding to where K = max ⌈ 6 ǫ ⌉, M . It remains to show from (14) that (13) holds for m = M, i.e.
By the definition of d D this is the same as saying there exists s, t, φ, ψ as above such that ∀x and We will show the first of these the other follows by symmetry. Given s, t, φ, ψ as in (15), there exists and by the equicontinuity condition in the definition of A N Putting these together with (14) we get This is equation (16) and so we are done.
Proof. Throughout W t is a standard Brownian motion. We start by showing that w.h.p. the condition that gives uniform boundedness on compact intervals holds.
Now we will show that w.h.p. the equicontinuity requirement on compact intervals holds. Let The below calculation says that with high probability for all k paths from each of these points will not move more than 1 3k from their stating point within time 2δ k,N and the non-crossing property then implies the required equicontinuity.
As the maximum can be bounded by a polynomial in k, N, a * k and b * k and Φ(. . . ) is decreasing exponentially in all of those variables we can conclude by use of a union bound that Proof. We will show that ) < δ as δ → 0. We start by decomposing M st (x 1 , x 2 , b, a, φ) into the integrals up to time s + δ and the rest. The integrals up until time s + δ are − s+δ s φ rs (x 1 )b(r, φ rs (x 2 )) + φ rs (x 2 )b(r, φ rs (x 1 ))dr and − (T (s,x 1 )(s,x 2 ) ∧t)∨(s+δ) T (s,x 1 )(s,x 2 ) ∧t a(r, φ rs (x 1 ))dr.
Taking expected value w.r.t. φ and exchanging order of integration leaves two integrals with length at most δ and integrands bounded by ) and a * respectively. As φ rs (x i ) is uniformly integrable for r ≤ t these integrals contribute only O(δ) to M, thus they can be neglected.
We will use M δ st to mean M st minus the integrals we have just shown are O(δ). Note that and by the strong Markov property is a function of φ s+δ,s (x 1 ) and φ s+δ,s (x 2 ). Proposition 8.5 says that d T V ((φ s+δ,s (x 1 ), φ s+δ,s (x 2 )), (φ s+δ,s (x ′ 1 ), φ s+δ,s (x ′ 2 ))) → 0 so we can deduce that Combining this with the fact that E φ (M δ st (x 1 , x 2 , b, a, φ)|F s+δ ) is uniformly integrable for (x 1 , x 2 ) in each compact set we are done.
By applying the triangle inequality the following 3 claims will now suffice to complete the proof, firstly and thirdly E φ n E xM ǫ (φ n ) → E φ n E x M(φ n ) as ǫ → 0 uniformly in n.
We first prove the second claim. T η monotonically increases to T 0 as η → 0 and thusM ǫ is monotonically increasing to M as ǫ → 0. Thus the second claim holds by the Monotone Convergence Theorem.
Using the strong Markov property at time T ǫ we can see that Putting this together and averaging over x we have proved the third claim. Finally, we will show that E xM ǫ is a continuous function of φ from which our first claim immediately follows due to weak convergence and we will be done.
Combining this with the corresponding lower bound whose derivation is similar we find Thus E x 1 b(t, φ(x 1 )) and the second term of E xM ǫ are continuous in φ. Similarly we can conclude that E x 1 a(t, φ(x 1 )) is continuous wrt φ and further as the products of intervals generate the Borel σ-algebra on R 2 that a(t, φ(x 1 ))dµ(x) is a continuous function of φ for each measure µ that is bounded, compactly supported and absolutely continuous wrt Lebesgue measure on R 2 . This will be useful after we rewrite the third term of E xM ǫ as To show this is continuous it suffices to show that E x a(t, φ(x 1 )) ǫ 0 ½ {t>Tη } ǫ dη is continuous and uniformly bounded ∀t > s. The boundedness is immediate. The continuity is not immediate from (17) being continuous because T η depends on φ, however it can be shown as follows. Let T 0 η be the T η corresponding to φ 0 and define T δ η similarly. Let µ 0 be the measure on R 2 with Radon-Nikodym derivative ǫ 0 ½ {t>Tη } ǫ dη with respect to the uniform probability measure on I 1 × I 2 and define µ δ similarly. Then The second of these terms is small due to the continuity of (17), the first term is bounded by The contribution to this integral when |x 1 − x 2 | < 2δ is clearly small, we will show that the contribution when x 1 ≥ x 2 + 2δ is small and as the case for x 1 ≤ x 2 − 2δ is similar we will then be done. Conditional on x 1 ≥ x 2 + 2δ we have T 0 η+2δ (x 1 − δ, x 2 + δ) ≤ T δ η (x 1 , x 2 ) ≤ T 0 η−2δ (x 1 + δ, x 2 − δ) and thus our integrand is zero unless By changing variables in our integral to an orthonormal basis of R 3 that includes 2η−x 1 +x 2 √ 6 as the variable for the inner integral, we find that that inner integral is bounded by 2 √ 6δ and the endpoints for the two outer integrals are bounded independently of δ. So our integral is small.