Maximum of Dyson Brownian motion and non-colliding systems with a boundary

We prove an equality-in-law relating the maximum of GUE Dyson's Brownian motion and the non-colliding systems with a wall. This generalizes the well known relation between the maximum of a Brownian motion and a reflected Brownian motion.


Introduction and Results
Dyson's Brownian motion model of GUE (Gaussian unitary ensemble) is a stochastic process of positions of m particles, X(t) = (X 1 (t), . . . , X m (t)) described by the stochastic differential equation, where B i , 1 ≤ i ≤ m are independent one dimensional Brownian motions [5]. The process satisfies X 1 (t) < X 2 (t) < · · · < X m (t) for all t > 0. We remark that the process X can be started from the origin, i.e., one can take X i (0) = 0, 1 ≤ i ≤ m. See [8].
One can introduce similar non-colliding system of m particles with a wall at the origin [6,7,14]. The dynamics of the positions of the m particles X (C) = (X (C) 1 , . . . , X (C) m ) satisfying 0 < X 1 (t) < X 2 (t) < · · · < X m (t) for all t > 0 are described by the stochastic differential equation, This process is referred to as Dyson's Brownian motion of type C. It can be interpreted as a system of m Brownian particles conditioned to never collide with each other or the wall. One can also consider the case where the wall above is replaced by a reflecting wall [7]. The dynamics of the positions of the m particles X (D) = (X (D) 1 , . . . , X (D) m ) satisfying 0 ≤ X 1 (t) < X 2 (t) < · · · < X m (t) for all t > 0, is described by the stochastic differential equation, where L(t) denotes the local time of X (D) 1 at the origin. This process will be referred to as Dyson's Brownian motion of type D. Some authors consider a process defined by the s.d.e.s (1.3) without the local time term. In this case the first component of the process is not constrained to remain non-negative, and the process takes values in the Weyl chamber of type D, {|x 1 | < x 2 < x 3 . . . < x m }. The process we consider with a reflecting wall is obtained from this by replacing the first component with its absolute value, with the local time term appearing as a consequence of Tanaka's formula.
It is known the processes X (C,D) can be obtained using the Doob h-transform, see [6]. Let (P 0,(C,D) t ; t ≥ 0) be the transition semigroup for m independent Brownian motions killed on exiting {0 < x 1 < x 2 . . . < x m }, resp. the transition semigroup for m independent Brownian motions reflected at the origin killed on exiting {0 ≤ x 1 < x 2 . . . < x m }. From the Karlin-McGregor formula, the corresponding densities can be written as (1.6) For notational simplicity we suppress the index C, D for the semigroups and in h in the following. Then one can show that h(x) is invariant for the P 0 t semigroup and we may define a Markov semigroup by This is the semigroup of the Dyson non-colliding system of Brownian motions of type C and D. Similarly to the X process, the processes X (C) and X (D) can also be started from the origin (see [9] or use Lemma 4 in [7] and apply the same arguments as in [8]). In GUE Dyson's Brownian motion of n particles, let us take the initial conditions to be X i (0) = 0, 1 ≤ i ≤ n. The quantity we are interested in is the maximum of the position of the top particle for a finite duration of time, max 0≤s≤t X n (s). In the sequel we write sup instead of max to conform with common usage in the literature. Let m be the integer such that n = 2m when n is even and n = 2m − 1 when n is odd. Consider the non-colliding systems of X (C) , X (D) of m particles starting from the origin, X Our main result of this note is Theorem 1. Let X and X (C) , X (D) start from the origin. Then for each fixed t ≥ 0, one has To prove the theorem we introduce two more processes Z j and Y j . In the Z process, Here the reflection means the Skorokhod construction to push Z j+1 up from Z j . More precisely, where B i , 1 ≤ i ≤ n are independent Brownian motions, each starting from 0. The process is the same as the process (X 1 1 (t), X 2 2 (t), . . . , X n n (t); t ≥ 0) studied in section 4 of [15]. The representation (1.9) was given earlier in [2]. In the Y process, 0 ≤ Y 1 ≤ Y 2 ≤ . . . ≤ Y n , the interactions among Y i 's are the same as in the Z process, i.e., Y j+1 is reflected by Y j , 1 ≤ j ≤ n−1, but Y 1 is now a Brownian motion reflected at the origin (again by Skorokhod construction). Similarly to (1.9), (1.10) From the results in [4,8,15], we know In this note we show Proposition 2. The following equalities in law hold between processes: The proof of this proposition is given in Section 2. The idea behind it is that the processes ) j≥1 could be realized on a common probability space consisting of Brownian motions satisfying certain interlacing conditions with a boundary [15,16]. Such a system is expected to appear as a scaling limit of the discrete processes considered in [3,16]. In this enlarged process, the processes Y n (t) and X (C,D) m (t) just represent two different ways of looking at the evolution of a specific particle and so the statement of Proposition 2 follows immediately. Justification of such an approach is however quite involved, and we prefer to give a simple independent proof. See also [4] for another representation of X (C,D) m in terms of independent Brownian motions.
Then to prove (1.8) it is enough to show (1.14) This is shown in Section 3. For n = 1 case, this is well known from the Skorokhod construction of reflected Brownian motion [10]. The n > 1 case can also be understood graphically by reversing time direction and the order of particles. This relation could also be established as a limiting case of the last passage percolation. In fact the identities in our theorem was first anticipated from the consideration of a diffusion scaling limit of the totally asymmetric exclusion process with 2 speeds [1].

Proof of proposition 2
In this section we prove the relation between X (C,D) and Y , (1.13). The following Lemma is a generalization of the Rogers-Pitman criterion [11] for a function of a Markov process to be Markovian.
is a Markov process with state space E, evolving according to a transition semigroup (P t ; t ≥ 0) and with initial distribution µ. Suppose that {Y (t) : t ≥ 0} is a Markov process with state space F , evolving according to a transition semigroup (Q t ; t ≥ 0) and with initial distribution ν. Suppose further that L is a Markov transition kernel from E to F , such that µL = ν and the intertwining P t L = LQ t holds. Now let f : E → G and g : F → G be maps into a third state space G, and suppose that Then we have in the sense of finite dimensional distributions.
We let (Y (t) : t ≥ 0) be the process Y of n reflected Brownian motions with a wall introduced in the previous section. It is clear from the construction (1.10) that the process Y is a time homogeneous Markov process. We denote its transition semigroup by Q t ; t ≥ 0). It turns out that there is an explicit formula for the corresponding densities. Recall φ t (z) = 1 √ 2πt e −z 2 /(2t) . Let us define φ  Proposition 5. The transition densities q t (y, y ′ ) from y = (y 1 , . . . , y n ) at t = 0 to y ′ = (y ′ 1 , . . . , y ′ n ) at t of the Y process can be written as where a i,j is given by The same type of formula was first obtained for the totally asymmetric simple exclusion process by Schütz [13]. The formula for the Z process was given as a Proposition 8 in [15], see also [12].
Proof of Proposition 5. For a fixed y ′ , define G t (y, t) to be (2.3) as a function of y and t.
For n = 2m, resp. n = 2m − 1 we take (X(t), t ≥ 0) to be Dyson Brownian motion of type C, resp. of type D. The transition semigroup P t ; t ≥ 0 of this process is given by  (1.7).
LQ t = P t L. (2.8) Now if we apply Lemma 4 with f (x) = x m , g(y) = y n and the initial conditions starting from the origin we obtain (1.13).
Proof of Proposition 6. The kernels P t (x, ·) and L(x, ·) are continuous in x. Thus we may consider x in the interior of E, and it is enough to prove From the definition of the kernel L 0 , this is equivalent to showing where q t and p 0 are densities corresponding to Q t and P 0 t . Integrations with respect to z are on the LHS with b(z) = x fixed and on the RHS with e(z) = y fixed.
Let us consider the case where n = 2m. Using the determinantal expressions for q t and p 0 t we show that both sides of (2.10) are equal to the determinant of size 2m whose (i, j) matrix element is a 2i,j (0, y j ) for 1 ≤ i ≤ m, 1 ≤ j ≤ 2m and a 2m,j (x i−m , y j ) for m + 1 ≤ i ≤ 2m, 1 ≤ j ≤ 2m.
The integrand of the LHS of (2.10) is We perform the integral with respect to z 1 , . . . , z 2m−1 in this order. After the integral up to z 2l−1 , 1 ≤ l ≤ m, we get the determinant of size 2m whose (i, j) matrix element is a 2i,j (0, y j ) for 1 ≤ i ≤ l, a 2l,j (z 2l i−l , y j ) for l + 1 ≤ i ≤ 2l and a i,j (e(z) i , y j ) for 2l + 1 ≤ i ≤ 2m. Here we use a property of a i,j , a i,j (y, y ′ ) = ∞ y a i−1,j (u, y ′ )du, (2.12) and do some row operations in the determinant. The case for l = m gives the desired expression.
The integrand of the RHS of (2.10) is with the condition e(z) = y.

Proof of proposition 3
Using (1.10) repeatedly, one has Y n (t) = sup with t n+1 = t. By renaming t − t n−i+1 by t i and changing the order of the summation, we have Y n (t) = sup