Dyson's Brownian motions, intertwining and interlacing

A family of reflected Brownian motions is used to construct Dyson's process of non-colliding Brownian motions. A number of explicit formulae are given, including one for the distribution of a family of coalescing Brownian motions.


Introduction
The ordered eigenvalues Y 1 (t) ≤ Y 2 (t) ≤ . . . ≤ Y n (t) of a Brownian motion in the space of n × n Hermitian matrices form a diffusion process which satisfies the stochastic differential equations, where β 1 , β 2 , . . . , β n are independent real Brownian motions. This is a result that goes back to Dyson [7] and we will refer to Y as a Dyson non-colliding Brownian motion. A number of important papers in recent years have developed a link between random matrices and certain combinatorial models, involving random permutations, last passage percolation, random tilings, random growth models and queuing systems, see Baik, Deift and Johansson, [1] and Johansson, [14], amongst many others. A recent survey is given by König [13]. At the heart of this connection lies the Robinson-Schensted-Knuth algorithm, a combinatorial procedure which has its origins in group representation theory, and using this the following remarkable formula, was observed by Gravner, Tracy and Widom, [10] and Baryshnikov [2], representing the largest eigenvalue Y n (t) ( assuming Y (0) = 0) in terms of independent, real-valued, Brownian motions B 1 , B 2 , . . . , B n , O'Connell and Yor, [15], give a proof of this identity by considering reversibility properties of a queuing system, which in a subsequent paper, O'Connell [16], is shown to be linked to the RSK algorithm also. Another proof, again involving RSK, is given by Doumerc, [6]. In this paper an a different proof of the identity (2) is given, based around the following construction. Let Y (t); t ≥ 0 be a Dyson process, with components Y 1 , Y 2 , . . . Y n solving (1). Let X(t); t ≥ 0 be a process with (n + 1) components which are interlaced with those of Y , meaning for all t ≥ 0, and which satisfies the equations Here γ(t); t ≥ 0 is a standard Brownian motion in R n+1 , independent of the Brownian motion β which drives Y . The processes (L + i (t); t ≥ 0) and (L − i (t); t ≥ 0) are continuous non-decreasing processes that increase only at times when X i (t) = Y i (t) and X i (t) = Y i−1 (t) respectively: they are twice the semimartingale local times at zero of X i −Y i and X i −Y i−1 . The two exceptional cases L − 1 (t) and L + n+1 (t) are defined to be identically zero. Conditionally on Y the particles corresponding to X evolve as independent Brownian motions except when collisions occur with particles corresponding to Y . Think of the particles corresponding to the components of Y as being "heavy" so that in collisions with the "light" particles corresponding to components of X their motion is unaffected.
On the other hand the light particles receive a singular drift from the collisions which maintains the interlacing. We will verify that is possible to start X and Y from the origin so that x i = y j = 0 for all i and j. Then, see Proposition 5, the process X is distributed as a Dyson non-colliding process with (n + 1) particles. Thus if we observe only the particles corresponding to the components of X, the singular drifts that these particles experience from collisions with the unseen particles corresponding to Y are somewhat magically transmuted into an electrostatic repulsion. This is a consequence of a relationship between the semigroup of the extended process (X, Y ) and the semigroup of X that is called an intertwining relation.

A duality between interlaced Brownian motions
Consider a continuous, adapted, R n+1 × R n -valued process X(t), Y (t); t ≥ 0 having components X 1 (t), X 2 (t), . . . X n+1 (t) and Y 1 (t), Y 2 (t), . . . , Y n (t) which is defined on filtered probability space Ω, F , F t t≥0 , Q n x,y satisfying, for all t ≥ 0, the interlacing condition and the equations where, τ is the stopping time given by = 0 for all t ≥ 0, otherwise the processes L + i and L − i are continuous, non-decreasing and increase only when X i = Y i and X i = Y i−1 respectively, The process just defined is called a stopped, semimartingale reflecting Brownian motion. For general results on such processes see, for example, Dai and Williams, [4]. In this case it is not difficult to give a pathwise construction starting from the Brownian motions β i , for i ∈ {1, 2, . . . , n}, and γ i for i ∈ {1, 2, . . . , n + 1}, together with the choice of initial co-ordinates We obtain Y i immediately. X i is constructed by alternately using the usual Skorokhod construction to push X i up from Y i−1 and down from Y i . For more details see Section 3 of [19], where a similar construction is used. In fact by the same argument as Lemma 6 in Soucaliuc, Toth and Werner, [19] pathwise uniqueness holds, and hence the law of X, Y is uniquely determined. This uniqueness implies, by standard methods, that the process is Markovian, and in fact we are able to give an explicit formula for its transition probabilities. We denote by φ t the centered Gaussian density with variance t. Φ t is the corresponding distribution function Define q n t (x, y), (x ′ , y ′ ) for (x, y), (x ′ , y ′ ) ∈ W n+1,n and t > 0 to be equal to determinant of the (2n + 1) × (2n + 1) matrix uniformly for all w = (x, y) ∈ W n+1,n .
Proposition 2. (q n t ; t > 0) are a family of transition densities for the process X, Y killed at the instant τ , that is to say, for t > 0 and (x, y), (x ′ , y ′ ) ∈ W n+1,n , Proof. For any choice of z ′ ∈ R, each of the functions (t, z) satisfies the heat equation on (0, ∞)×R. Thus, by differentiating the determinant, we find that, We need to identify certain boundary conditions. We treat w ′ = (x ′ , y ′ ) ∈ W n+1,n as fixed. First consider (x, y) ∈ ∂W n+1,n satisfying y i = y i+1 for some i ∈ {1, 2, . . . n − 1}. We see that the ith and (i + 1)th rows of both C t (y, x ′ ) and D t (y, y ′ ) are equal, and hence q n t (x, y), (x ′ , y ′ ) = 0. Next consider (x, y) ∈ ∂W n+1,n satisfying x i = y i for some i ∈ {1, 2, . . . n}. Calculate ∂ ∂xi q n t ((x, y), (x ′ , y ′ )) by differentiating the ith rows of A t (x, x ′ ) and B t (x, y ′ ). Notice that, under our assumption that . Thus we deduce that ∂ ∂xi q n t ((x, y), (x ′ , y ′ )) = 0. Finally consider (x, y) ∈ ∂W n+1,n satisfying x i+1 = y i for some i ∈ {1, 2, . . . n}. Similarly to the previous case we obtain ∂ ∂xi+1 q n t ((x, y), (x ′ , y ′ )) = 0. Let f : W n+1,n → R be a bounded and continuous, and zero in a neighbourhood of the boundary. Then define a smooth function F on (0, ∞) × W n+1,n via By virtue of the above observations regarding q n t , and differentiating through the integral, we find that with the boundary conditions Fix T, ǫ > 0. Applying Itô's formula, we find that the process F (T + ǫ − t, (X t , Y t ) ; t ∈ [0, T ] is a local martingale, which is easily seen to be bounded and hence is a true martingale. Thus Appealing to the previous lemma, we may let ǫ ↓ 0 and so obtain, Since the part of the distribution of (X T , Y T ) that charges the boundary of W n+1,n exactly corresponds to the event {T ≥ τ } this suffices to prove the proposition.
We now consider a second reflected semimartingale Brownian motion X ,Ŷ having componentŝ x,y satisfying, for all t ≥ 0, the interlacing condition and the equationsŶ where, τ is the stopping time given byτ = inf t ≥ 0 : the processes L + i and L − i are continuous, non-decreasing and increase only whenŶ i =X i+1 and Y i =X i respectively, Notice the difference between this process and X, Y is the reflection rule: hereŶ is pushed offX whereas it was X that was pushed off Y . Define a family q n t ; t > 0 via The following proposition is proved by arguments exactly parallel to those just given in proof of Proposition 2.
Proposition 3. (q n t ; t > 0) are a family of transition densities for the process X ,Ŷ killed at the instantτ , that is to say, for t > 0 and (x, y), (x ′ , y ′ ) ∈ W n+1,n , The duality, represented by (10), between the transition semigroups of X, Y and X ,Ŷ is not unexpected. It is consistent with general results, see for example DeBlassie [5], and Harrison and Williams [11], which show that, in a variety of contexts, the dual of a reflected Brownian motion is another reflected Brownian motion where the direction of reflection at the boundary is obtained by reflecting the original direction of reflection across the normal vector. This is precisely the relationship holding between X, Y and X ,Ŷ here.

An intertwining involving Dyson's Brownian motions
It is known that Dyson's non-colliding Brownian motions can be obtained by means of a Doob htransform. Let W n = {y ∈ R n : y 1 ≤ y 2 ≤ . . . ≤ y n }. Suppose that Y t ; t ≥ 0 is when governed by the probability measure P n y a standard Brownian motion in R n , relative to a filtration {F t ; t ≥ 0}, starting from a point y ∈ W n and stopped at the instant τ = inf{t ≥ 0 : Y i (t) = Y j (t) for some i = j}. The transition probabilities of Y killed at the time τ are given explicitly by the Karlin-McGregor formula, [12], for y, y ′ ∈ W n , where, with φ t again denoting the Gaussian kernel with variance t, If the initial co-ordinates y satisfy y 1 < y 2 < . . . < y n , then we may define a new probability measure by the absolute continuity relation (13) P n,+ for t > 0, where h n is the function given by Under P n,+ y the process Y evolves as a Dyson non-colliding Brownian motion, that is to say τ is almost surely infinite and the stochastic differential equations (1) hold. The transition probabilities are related to those for the killed process by an h-transform for y, y ′ ∈ W n \ ∂W n . Finally we recall, see O'Connell and Yor, [15], that we may describe P n,+ 0 , the measure under which the non-colliding Brownian motion issues from the origin by specifying that it is Markovian with transition densities p n,+ t ; t > 0 and with the entrance law with the normalizing constant being Z n = (2π) n/2 j<n j!. Now suppose that X, Y is governed by the probability measure Q n x,y defined in the previous section. Recall that q n t ; t > 0 are the transition densities of the process killed at the time τ = inf{t ≥ 0 : Y i (t) = Y j (t) for some i = j}. Suppose the initial co-ordinates y of Y satisfy y 1 < y 2 < . . . < y n , then we may define a new probability measure Q n,+ x,y by the absolute continuity relation for t > 0. It follows from the fact that under Q n x,y the process Y evolves as a Brownian motion stopped at the instant τ , that h n (Y t∧τ ) is a martingale, and that this definition is hence consistent as t varies. Under the measure Q n,+ x,y , the process Y now evolves as a non-colliding Brownian motion satisfying the stochastic differential equation (1), whilst the process X satisfies (4). The corresponding transition densities q n,+ t ; t > 0 are obtained from those for the killed process by the h-transform for (x, y), (x ′ , y ′ ) ∈ W n+1,n with the components of y all distinct.
Lemma 4. The family of probability measures with densities given by ν n t ; t > 0 on W n+1,n , given by form an entrance law for q n,+ t ; t > 0 , that is to say, for s, t > 0 Accordingly we may define a probability measure Q n,+ 0,0 , under which the process X, Y is Markovian with transition densities q n,+ t ; t > 0 and with the entrance law It is easy to see that under this measure X, Y satisfies the equations (1) and (4), starting from the origin x = 0, y = 0. Presumably any solution to (1) and (4) starting from the origin has the same law, but we do not prove this.
We may now state the main result of this section.
Proposition 5. Suppose the process X t , Y t ; t ≥ 0 is governed by Q n,+ 0,0 then the process X t ; t ≥ 0 is distributed as under P n+1,+ 0 , that is as a Dyson non-colliding Brownian motion in W n+1 starting from the origin. This result is proved by means of a criterion described by Rogers and Pitman [18] for a function of a Markov process to be Markovian, see Carmona, Petit and Yor, [3], for futher examples of intertwinings. For x ∈ W n+1 let W n (x) = {y ∈ R n : x 1 ≤ y 1 ≤ . . . ≤ y n ≤ x n+1 , and define for x ∈ W n+1 \ ∂W n+1 and y ∈ W n (x). The normalizing constant being chosen so that λ n (x, ·) is the density of a probability measure on W n (x). This follows from the equality which is easily verified by writing h n (y) = det y j−1 i ; 1 ≤ i, j ≤ n . The proof of Proposition 5 depends on the following intertwining relation between q n,+ t ; t > 0 and p n+1,+ t ; t > 0 , for all t > 0, x ∈ W n+1 \ ∂W n+1 , and (x ′ , y ′ ) ∈ W n+1,n , This by be verified directly using the explicit formula for q n t given in the previous section. Alternatively the following derivation is enlightening. Recall that if X t ,Ŷ t ; t ≥ 0 is governed byQ n x,y then the process X t ; t ≥ 0 is a Brownian motion stopped at the instantτ = inf t ≥ 0;X i = X j for some i = j . Consequently the transition probabilities of the killed process satisfy Now using the duality between q n t andq n t and the symmetry of p n+1 t we may re-write this as Finally to obtain (24) we swop the roles of (x, y) and (x ′ , y ′ ) and use the expressions for q n,+ t and p n+1,+ t as h-transforms. As a first application of the intertwining we have the following.
Proof of Lemma 4. Notice that ν n t (x, y) = µ n+1 t (x)λ n (x, y). Hence, by virtue of the intertwining and the fact that µ n+1 t ; t > 0 is an entrance law for p n+1,+ t ; t > 0 we have, A similar argument, following [18] proves the proposition.
Notice that in the above proof, if we integrate y n over some smaller set than W n (x n ) we find that (27) Q n,+ 0,0 Y tn ∈ A|X t1 , X t2 , . . . , X tn = A∩W n (Xt n ) λ n (X tn , y dy. This may be interpreted as the following filtering property: the conditional distribution of Y t given X s ; s ≤ t is given by the density λ n (X t , ·) on W n (X t ).

Brownian motion in the Gelfand-Tsetlin cone
Proposition 5 lends itself to an iterative procedure. Let K be the cone of points x = x 1 , x 2 , . . . x n with x k = x k 1 , x k 2 , . . . , x k k ∈ R k satisfying the inequalities K is sometimes called the Gelfand-Tsetlin cone, and arises in representation theory. We will consider a process X(t) = X 1 (t), X 2 (t), . . . X n (t) taking values in K so that where γ k i (t); t ≥ 0 for 1 ≤ k ≤ n, 1 ≤ i ≤ k are independent Brownian motions, and L k,+ i (t); t ≥ 0 and L k,1 i (t); t ≥ 0 are continuous, increasing processes growing only when X k i (t) = X k−1 i (t) and X k i (t) = X k−1 i−1 (t) respectively, the exceptional cases L k,+ k (t) and L k,− 1 (t) being identically zero for all k. For initial co-ordinates satisfying x k i < x k i+1 for all k and i, we may give a pathwise construction, as in Section 2, based on alternately using the Skorokhod construction to reflect X k i downwards from X k−1 i and upwards from X k−1 i−1 . The potential difficulty that X k−1 i meets X k−1 i−1 does not arise. In order to construct X starting from the origin we use a different method. First we note that if the pair of processes X, Y , governed by the measure Q n,+ 0,0 satisfies equations (1) and (4), then, By repeated application of Proposition 5, there exists a process X(t); t ≥ 0 , starting from the origin, such that the process X k (t); t ≥ 0 is distributed as P k,+ 0 , for k = 1, 2, . . . , n, the pair of processes X k+1 (t), X k (t); t ≥ 0 are distributed as under Q k,+ 0,0 , for k = 1, . . . , n − 1, for k = 2, . . . , n−1 the process X k+1 (t); t ≥ 0 is conditionally independent of X 1 (t), . . . , X k−1 (t); t ≥ 0 given X k (t); t ≥ 0 .
By its very construction the process X satisfies the equations (29), for some Brownian motions γ k i , which by the observation (30) are independent. Even starting from the origin, pathwise uniqueness, and hence uniqueness in law hold for X. Consequently we may state the following proposition. Proposition 6. The process X(t); t ≥ 0 , satisfying (29), if started from the origin, satisfies for each k = 1, 2, . . . , n, X (k) (t); t ≥ 0 is distributed as under P k,+ 0 . The conditional distribution of X k (t); t ≥ 0 given X k−1 (t); t ≥ 0 factorizes in a Markovian fashion into the product of the conditional distribution of X k (s); 0 ≤ s ≤ t given X k−1 (s); 0 ≤ s ≤ t , and the conditional distribution of X k (u); t ≤ u given X k−1 (u); t ≤ u and X k (t). From this factorization we deduce that, for any t > 0, and k = 2, . . . , n − 1 the process X k+1 (s); 0 ≤ s ≤ t is conditionally independent of X 1 (s), . . . X k−1 (s); 0 ≤ s ≤ t given X k (s); 0 ≤ s ≤ t . For any x k ∈ W k we will denote by K(x k ) the set of all (x 1 , x 2 , . . . x k−1 ) such that for all i and j, Recall from (27) that the conditional distribution of X k (t) given X k+1 (s); 0 ≤ s ≤ t has the density λ k (X k+1 (t), ·) on W k (X k+1 (t)). Combining this with the conditional independence property we deduce that the conditional distribution of X 1 (t), X 2 (t) . . . X k (t) given X k+1 (s); 0 ≤ s ≤ t is uniform on K(X k+1 (t)). Finally using the fact that the distribution of X n (t) is given by the density µ n t on W n we deduce that the distribution of X(t) has the density with respect to Lebesgue measure on K. Baryshnikov, [2], studies this distribution in some detail. Let H(t); t ≥ 0 be a Brownian motion in the space of n × n Hermitian matrices, and consider the process H 1 (t), H 2 (t), . . . H n (t); t ≥ 0 where H k (t) is the k × k minor of H(t) = H n (t) obtain by deleting the last n − k rows and columns. It is a classical result that the eigenvalues of H k−1 (t) are interlaced with those of H k (t). Baryshnikov shows that, at any fixed instant t > 0, the distribution of the eigenvalues of H 1 (t), H 2 (t), . . . H n (t) is given by the density (31). However it is not the case that the eigenvalue process is distributed as the process X(t); t ≥ 0 . O'Connell, [16], describes another process Γ(t); t ≥ 0 taking values in K which is constructed via certain explicit path transformations. This process arises as the scaling limit of the RSK correspondence. The process X described above has several features in common with Γ. For each k, the subprocess Γ k (t); t ≥ 0 evolves as Dyson k-tuple starting from zero. Additionally but remarkably all other components Γ k l with l < k are given by explicit deterministic transformations applied to the processes Γ 1 1 , Γ 2 2 , . . . Γ n n . A feature that X certainly does not share. Notice that, for k ≥ 2, where L k,− k (t) grows only when X k k (t) = X k−1 k−1 (t). On applying the Skorokhod lemma, see Chapter VI of [17], we find that Iterating this relation we obtain which in the light of Proposition 6 proves the identity (2). This is essentially the same argument for (2) as given by O'Connell and Yor, [15], with Proposition 6 replacing the corresponding statement about Γ. We close this section by noticing that X 1 1 (t), X 2 2 (t), . . . , X n n (t); t ≥ 0 is Markovian and giving an explicit formula for it transition probabilities. For n ≥ 1 let Φ (n) t denote the nth order iterated integral of the Gaussian density φ t , Lemma 7. For any f : W n → R which is bounded and continuous and zero in a neighbourhood of the boundary of W n , uniformly for all x ∈ W n . Proposition 8. The process X 1 1 (t), X 2 2 (t), . . . , X n n (t); t ≥ 0 satisfying (33) is Markovian with transition densities given by r t (x, x ′ ).
Proof. For a fixed x ′ ∈ R, and any n, the function (t, x) → Φ (n) t (x ′ − x) solves the heat equation on (0, ∞) × R. From this we easily see that for a fixed x ′ ∈ R n , that the function (t, x) → r t (x ′ − x) solves the heat equation on (0, ∞) × R n . Moreover if x i = x i−1 for any i = 2, 3, . . . , n then the ith and (i − 1)th rows of the determinant defining ∂ ∂xi r t (x, x ′ ) are equal and hence this quantity is zero. Let f : W n → R be a bounded, continuous and are in a neighbourhood of the boundary of W n . Then define a smooth function F on (0, ∞) × W n via By virtue of the above observations regarding r t , and differentiating through the integral, we find that with the boundary conditions ∂F ∂x i (t, x) = 0 whenever x i = x i−1 for some i = 2, 3, . . . , n.
Let X denote a process governed by a probability R x , with components X 1 (t) ≤ X 2 (t) ≤ . . . ≤ X n (t) satisfying the equations X k (t) = x k + γ k (t) + L k (t), where γ k are independent Brownian motions and L k is an increasing process growing only when X k (t) = X k−1 (t), with L 1 being identically zero. Fix T, ǫ > 0. Applying Itô's formula, we find that the process F T + ǫ − t, X t ; t ∈ [0, T ] is a local martingale, which being bounded is a true martingale. Thus Appealing to the previous lemma, we may let ǫ ↓ 0 and so obtain, which, since it is clear the distribution of X(T ) does not charge the boundary of W n , proves the proposition.
In view of Proposition 6, we obtain from r t by a simple integration the following expression for the distribution function of the largest eigenvalue of H(t): Possibly this can be checked directly using the Heine identity.

Coalescing Brownian motions
In this section we consider the joint distribution of a family of coalescing Brownian motions. Fix z 1 ≤ z 2 ≤ . . . ≤ z n and consider the process of n coalescing Brownian Motions, where each process Z t (z i ); t ≥ 0 is a Brownian motion (relative to some common filtration) starting from Z 0 (z i ) = z i , with for each distinct pair i = j the process being a standard Brownian motion on the half-line [0, ∞) with an absorbing barrier at 0. Thus informally Z t (z i ); t ≥ 0 and Z t (z j ); t ≥ 0 evolve independently until they first meet, after which they coalesce and move together. Such families of coalescing Brownian motions have been wellstudied, for some recent works concerning them see [9] and [8].
For a fixed t > 0, the distribution of Z t (z) is supported on W n . That part of the distribution supported on the boundary of W n corresponds to the event that coalescence has occurred. Whereas the restriction of the distribution to the interior W n (corresponding to no coalescence) is given by Karlin-McGregor formula : In fact we can bootstrap from this result to a complete determination of the law of Z t (z), which can be expressed in the following neat way.
Proposition 9. For z, z ′ ∈ W n , the probability is given by the determinant of an n × n matrix with (i, j)th element given by Proof. First we note that by integrating the Karlin-McGregor formula we obtain We are going to obtain the desired result by showing how the indicator function of the event of interest can be expanded in terms of the indicator functions of the events of the form for increasing subsequences of indices i(1), i(2), . . . , i(s) and j(1), j(2), . . . , j(s). To this end I claim firstly that, whenever z, z ′ ∈ W n , I claim secondly that To prove the first claim take the matrix M = 1(z i ≤ z ′ j ) , and subtract from each column (other than the first) the values of the preceding column. The diagonal elements of this new matrix are Thus the product of these diagonal elements gives the desired result. We have to check that in the expansion of the determinant this is the only contribution. Suppose that ρ is a permutation, not the identity. Then we can find i < j with ρ(i) > i and ρ(j) ≤ i. Consider the product of the (i, ρ(i))th and (j, ρ(j))th elements of the matrix (after the column operations). We obtain This can only be non-zero if both z ′ ρ(i)−1 < z i and z j ≤ z ′ ρ(j) ; but z i ≤ z j so this would imply z ′ ρ(i)−1 < z ′ ρ(j) . In view of the fact ρ(i) − 1 ≥ ρ(j) this is impossible. Consider the matrix N appearing in the second claim. The product of its diagonal elements gives the desired result. To show that this is the only contribution to the determinant, take ρ a permutation, not equal to the identity and i < j with ρ(i) > i and ρ(j) ≤ i, as before. Then the product of the (i, ρ(i))th and (j, ρ(j))th elements of the matrix is Since z i ≤ z j for this to be non-zero we would have to have z ′ ρ(i) < z ′ ρ(j) , which is impossible for ρ(i) > ρ(j).
Let for appropriate signs s(i, j). Evaluating det(N ) via the second claim, and the minors M [i, j] via (general versions of ) the first claim we have obtained an expansion of 1(z 1 ≤ z ′ 1 , z 2 ≤ z ′ 2 , . . . , z n ≤ z ′ n ) as a linear combination of of terms of the form 1(z i(1) ≤ z ′ j(1) < z i(2) ≤ z ′ j(2) < . . . ≤ z ′ j(s) ). To complete the proof replace, in the above expansion, z i by Z t (z i ) and take expectations. On the lefthandside we obtain On the righthandside we have a linear combination of probabilities: The expression just obtained for the distribution of coalescing Brownian motions is closely related to the formula for the transition density of the interlaced Brownian motions given by Proposition 2. In fact it is easily verified that (43) q n t (x, y), (x ′ , y ′ ) = (−1) n ∂ n ∂y 1 . . . ∂y n This represents a duality between the the interlaced Brownian motions and coalescing Brownian motions which generalizes the well-known duality between Brownian motion on the half-line [0, ∞) with a reflecting Barrier at zero, and Brownian motion on the half-line with an absorbing barrier at zero. There is interesting alternative way of expressing the equality (43). The Arratia flow or Brownian web is a infinite family of coalescing Brownian motions, with a path starting from every point in space-time. Let t ∈ [s, ∞) → Z s,t (x) denote the path starting from (s, x). It is possible to define on the same probability space a dual flow with paths running backwards in time: s ∈ (−∞, t] →Ẑ s,t (x) being the path beginning at (t, x). For the details of this construction see [21] and [9]. The flow Z and its dualẐ are such that for any s, t, x and y, the two events Z s,t (x) ≤ y andẐ s,t (y) ≥ x differ by a set of zero probability. Using this we may rewrite (43) as (44) q n t (x, y), (x ′ , y ′ ) dx ′ dy = P Z 0,t (x i ) ∈ dx ′ i ,Ẑ 0,t (y ′ j ) ∈ dy j for all i, j .
This seems to fit with the fact that the paths ofẐ are "reflected off" those of Z, see [19] and [20].

Proofs of two lemmas
Proof of Lemma 7. The contribution to the determinant defining r t (x, x ′ ) coming from the principal diagonal is equal to the standard heat kernel in R n . The lemma will follow if we can show all other contributions to the determinant are uniformly negligible as t tends down to 0. Choose ǫ > 0 so that the function f is zero in an 2ǫ-neighbourhood of the boundary of W n . Then consider a contribution to the determinant corresponding to some permutation ρ which is not the identity. There exist i < j with ρ(i) > i and ρ(j) ≤ i, and the contribution corresponding to ρ consequently contains factors of Φ Noting that j − ρ(j) > 0 and i − ρ(i) < 0 we see that on the set {x ′ ρ(i) − x i > ǫ} ∪ {x ′ ρ(j) − x j < −ǫ} at least one of these factors, and indeed the entire contribution, tends to zero uniformly as t tends down to zero. But on the complement of this set we have x ′ ρ(i) ≤ x i + ǫ ≤ x j + ǫ ≤ x ′ ρ(j) + 2ǫ, and ρ(j) ≤ ρ(i) implies that x ′ ρ(j) ≤ x ′ ρ(i) , so we see that x ′ is within the 2ǫ-neighbourhood of the boundary of W n , and does not belong to the support of f . This proves the lemma.
Proof of Lemma 1. It is convenient to write z 1 = x 1 , z 3 = x 2 , . . . z 2n+1 = x n , and z 2 = y 1 , z 4 = y 2 , . . . , z 2n = y n , with a corresponding change of notation for x ′ i and y ′ i also. Now reorder the columns and rows of the determinant defining q n t so that the (i, j)th entry is a function of the difference z ′ j −z i . We may now argue in the same way as in the preceding proof. Choose ǫ > 0 so that the function f is zero in an 2ǫ-neighbourhood of the boundary of W n+1,n . Consider a contribution to the determinant corresponding to some permutation ρ which is not the identity. There exist i < j with ρ(i) > i and ρ(j) ≤ i, and the contribution corresponding to ρ consequently contains factors which are functions of z ′ ρ(i) − z i and z ′ ρ(j) − z j . Noting that j − ρ(j) > 0 and i − ρ(i) < 0, and checking the entries of the determinant above and below the diagonal we see that on the set {z ′ ρ(i) − z i > ǫ} ∪ {z ′ ρ(j) − z j < −ǫ} at least one of these factors, and indeed the entire contribution, tends to zero uniformly as t tends down to zero. As above, this proves the lemma.