Complex Brownian Motion Representation of the Dyson Model

Dyson's Brownian motion model with the parameter $\beta=2$, which we simply call the Dyson model in the present paper, is realized as an $h$-transform of the absorbing Brownian motion in a Weyl chamber of type A. Depending on initial configuration with a finite number of particles, we define a set of entire functions and introduce a martingale for a system of independent complex Brownian motions (CBMs), which is expressed by a determinant of a matrix with elements given by the conformal transformations of CBMs by the entire functions. We prove that the Dyson model can be represented by the system of independent CBMs weighted by this determinantal martingale. From this CBM representation, the Eynard-Mehta-type correlation kernel is derived and the Dyson model is shown to be determinantal. The CBM representation is a useful extension of $h$-transform, since it works also in infinite particle systems. Using this representation, we prove the tightness of a series of processes, which converges to the Dyson model with an infinite number of particles, and the noncolliding property of the limit process.


Introduction and Results
Dyson's Brownian motion model is a one-parameter family of the systems of one-dimensional Brownian motions with long-ranged repulsive interactions, whose strength is represented by a parameter β > 0. The system solves the following stochastic differential equations (SDEs), where B i (t)'s are independent one-dimensional standard Brownian motions [3,19].In the present paper we consider the case with β = 2, since in this special case the system is realized by the following three processes [9,10], (i) the process of eigenvalues of Hermitian matrix-valued diffusion process in the Gaussian unitary ensemble (GUE) [3,16,5], (ii) the system of one-dimensional Brownian motions conditioned never to collide with each other [6], (iii) the harmonic transform of the absorbing Brownian motion in a Weyl chamber of type A n−1 [6], , with a harmonic function given by the Vandermonde determinant ( In a sense, the case with β = 2 plays the role in the one-parameter family of interacting particle systems (1.1), which is similar to the role of the three-dimensional Bessel process playing in the family of d-dimensional Bessel processes with a parameter d [14].We call the case with β = 2 of Dyson's Brownian motion model simply the Dyson model in this paper.
Let M be the space of nonnegative integer-valued Radon measures on R, which is a Polish space with the vague topology.Any element ξ of M can be represented as ξ(•) = i∈I δ x i (•) with a countable index set I and a sequence of points in R, x = (x i ) i∈I satisfying ξ(K) = ♯{x i : x i ∈ K} < ∞ for any compact subset K ⊂ R. In this paper the cardinality of a finite set S is denoted by ♯S.We call an element ξ of M an unlabeled configuration, and a sequence x a labeled configuration.We write the restriction of configuration in A ⊂ R as (ξ ∩A)(•) = i∈I:x i ∈A δ x i (•), a shift of configuration by u ∈ R as τ u ξ(•) = i∈I δ x i +u (•), and a square of configuration as ξ 2 (•) = i∈I δ x 2 i (•), respectively.The set of M-valued continuous functions defined on [0, ∞) is denoted by C([0, ∞) → M), which has the topology of uniform convergence on any compact sets.For ξ ∈ M with ξ(R) ∈ N ≡ {1, 2, . . .}, we introduce a one-parameter family of entire functions of z ∈ C [15] parameterized by u ∈ C, {Φ u ξ (z) : u ∈ C}, in which with supp ξ = {x ∈ R : ξ({x}) > 0}.The zero set of the function (1.3) is supp ξ ∩ {u} c .
With the SDEs (1.1), we consider the diffusion process Ξ(t) = i∈I δ X i (t) in M and the process under the initial configuration ξ = i∈I δ x i ∈ M is denoted by (Ξ(t), P ξ ).We write the expectation with respect to P ξ as E ξ .We introduce a filtration {F (t)} t∈[0,∞) on the space C([0, ∞) → M) defined by F (t) = σ(Ξ(s), s ∈ [0, t]).Let C 0 (R) be the set of all continuous real-valued functions with compact supports.For any integer M ∈ N, a sequence of times t = (t 1 , t 2 , . . ., t M ) with 0 < t 1 < • • • < t M < T < ∞, and a sequence of functions f = (f t 1 , f t 2 , . . ., f t M ) ∈ C 0 (R) M , the moment generating function of multitime distribution of (Ξ(t), P ξ ) is defined by We put M 0 = {ξ ∈ M : ξ({x}) ≤ 1 for any x ∈ R}.Since any element ξ of M 0 is determined uniquely by its support, it is identified with a countable subset {x i } i∈I of R. For a finite index set I and v = (v i ) i∈I , v i ∈ R, let Z i (t), t ≥ 0, i ∈ I be a sequence of independent complex Brownian motions (CBMs) on a probability space (Ω, F , P v ) with Z i (0) = v i , i ∈ I.We write the expectation with respect to P v as E v .The real part and the imaginary part of Z i (t) are denoted by V i (t) = ReZ i (t) and W i (t) = ImZ i (t), respectively, i ∈ I, which are independent one-dimensional standard Brownian motions.For any sequences since Φ x ξ is an entire function.Each of them is a time change of a CBM [18].A key observation for the present study is that the equality det 1≤i,j≤ξ(R) R) .It implies that from a harmonic function h(z) given by (1.2), we have a martingale for a system of independent CBMs {Z i (•) : . Let 1(ω) be the indicator function of a condition ω; 1(ω) = 1 if ω is satisfied and 1(ω) = 0 otherwise, and I p = {1, 2, . . ., p} for p ∈ N. The main theorem of the present paper is the following.
In particular, the moment generating function (1.4) is given by where We call the above results the complex Brownian motion (CBM) representations of the Dyson model.In order to show the simplest application of this representation, we consider the density function at a single time for (Ξ(t), P ξ ) denoted by ρ ξ (t, x).It is defined as a continuous function of x ∈ R for 0 < t < T such that for any χ ∈ C 0 (R) (1.9) 3) and (1.5), the equality (1.7) gives the following expression for (1.9) where p s,t (x, y) = e −(y−x) 2 /2(t−s) / 2π(t − s), 0 ≤ s < t, x, y ∈ R, since V (0) = ReZ(0) = v ∈ supp ξ and W (0) = ImZ(0) = 0.Then, if we define the function for (x, y) ∈ R 2 , (s, t) ∈ (0, T ) 2 , we obtain the expression for the density function for any initial configuration ξ ∈ M 0 .
The above calculation will be fully generalized and the following formula can be derived from our CBM representations. (1.11) Then the moment generating function (1.8) for the multitime distribution is given by a Fredholm determinant δ st δ x (y) + K ξ (s, x; t, y)χ t (y) . (1.12) By definition of Fredholm determinant (see, for example, [5]) the moment generating function (1.12) can be expanded with respect to χ tm (•), 1 ≤ m ≤ M, as with where Nm denotes (x Nm ) and dx The functions ρ ξ 's are multitime correlation functions, and Ψ t ξ [f ] can be regarded as a generating function of them.The function K ξ given by (1.11) with (1.10) is thus called the correlation kernel [12].In general, when the moment generating function for the multitime distribution is given by a Fredholm determinant, the process is said to be determinantal [10,12].The results by Eynard and Mehta reported in [4] for a multi-layer matrix model can be regarded as the theorem that the Dyson model is determinantal for the special initial configuration ξ = ξ(R)δ 0 , i.e., all particles are put at the origin, for any ξ(R) ∈ N. The correlation kernel is expressed by using the Hermite orthogonal polynomials [17].The present authors proved that, for any fixed initial configuration ξ ∈ M with ξ(R) ∈ N, the Dyson model (Ξ(t), P ξ ) is determinantal, in which the correlation kernel is given by where Γ(ξ) is a closed contour on the complex plane C encircling the points in supp ξ on the real line R once in the positive direction (Proposition 2.1 in [12]).In order to derive (1.15), we used the integral representations of multiple Hermite polynomials given by Bleher and Kuijlaars [2].
In the present paper, we assume ξ ∈ M 0 preventing the initial configuration from having any multiple points.This restriction is only for simplicity of calculation.(Note that, if ξ ∈ M 0 , the Cauchy integrals in (1.15) can be readily performed and the expression (1.11) with (1.10) is obtained.)The fact that we would like to report here is that, although the theory of (bi-)orthogonal functions are very useful to analyze determinantal processes [12,11,13], it is not necessary to deriving the Eynard-Mehta-type determinantal expressions for multitime correlation functions.The essential point may be the extension of h-transform to the conformal martingale of CBMs in the determinantal form det 1≤i,j≤ξ(R) [Φ u i ξ (Z j (•))], which we have named the CBM representation.In other words, the proof of Corollary 1.2 will provide a probability-theoretical derivation of the Eynard-Mehta-type correlation kernel.
The CBM representation is indeed a non-trivial extension of h-transforms, since it is valid also for infinite particle systems.For example, if ξ ∈ M 0 is chosen so that and for any compact set then, also in the case ξ(R) = ∞, (1.7) holds for any F (t)-measurable bounded function F which is determined by the behavior of particles in some compact set, and (1.8) is valid.
Then we can obtain the convergence of moment generating functions We give useful sufficient conditions of ξ for (1.16), (1.17) and (1.18) which are exactly the same as the sufficient conditions for initial configuration introduced in [12] so that the Dyson model is well defined as a determinantal process even if ξ(R) = ∞ : For L > 0, α > 0 and ξ ∈ M we put It was shown that, if ξ ∈ M 0 satisfies the conditions (C.1) and (C.2), then (1.16) holds, and for some c, C > 0 and θ ∈ (max{α, (2 − β)}, 2), which are determined by the constants C 0 , C 1 , C 2 and the indices α, β in the conditions (Lemma 4.4 in [12]).We have noted that in the case that ξ ∈ M 0 satisfies the conditions (C.1) and (C.2) with constants C 0 , C 1 , C 2 and indices α and β, then ξ ∩ [−L, L], ∀L > 0 does as well.Hence, this estimation implies (1.17) and (1.18) .Moreover, even if ξ(R) = ∞, K ξ given by (1.11) with (1.10) is welldefined as a correlation kernel and dynamics of the Dyson model with an infinite number of particles (Ξ(t), P ξ ) exists as a determinantal process [12].In order to demonstrate the usefulness of the CBM representations to characterize infinite particle systems, we show that the following estimate is readily obtained from the expression (1.7).Let C ∞ 0 (R) be the set of all infinitely differentiable real functions with compact supports. (1.20) By a criterion of Kolmogorov (see, for example, [7,1]), Proposition 1.3 implies that the sequence of the process (Ξ(t), Then we can conclude the following.
Finally in the present paper, we show that the noncolliding property of the Dyson model with an infinitely many particles is obtained by using the CBM representations.Proposition 1.5 Suppose that the initial configuration ξ ∈ M 0 satisfies the conditions (C.1) and (C.2) with constants C 0 , C 1 , C 2 and indices α, β.Then P ξ Ξ(t) ∈ M 0 , t > 0 = 1.
In the following four sections, we give the proofs of Theorem 1.1, Corollary 1.2, Proposition 1.3, and Proposition 1.5, respectively.

Proof of Theorem 1.1
It is sufficient for the proof of Theorem 1.1 to consider the case that F is given as We give the proof for the case with M = 2, i.e., for ξ The generalization for M > 2 is straightforward by the Markov property of Brownian motion as implied in the following proof.
We use the fact that the Dyson model is obtained as the h-transform of the absorbing Brownian motion in the Weyl chamber W A ξ(R) [6].Put τ = inf{t > 0 : V(t) / ∈ W A ξ(R) }, then the LHS of (2.1) is given by (2.2) For a finite set S, we write the collection of all permutations of elements in S as S(S).In particular, we express S(I p ) simply by S p , p ∈ N. Then by the Karlin-McGregor formula [8] (2.2) is given by with σ(u) = (u σ(1) , . . ., u σ(ξ(R)) ) for each permutation σ ∈ S ξ(R) , in which all contributions from the paths with τ ≤ t 1 are canceled out by taking the signed sum, σ∈S ξ(R) sgn(σ) Here the Markov property of Brownian motion has been used.The Karlin-McGregor formula is again used to realize the condition 1(τ > t 2 − t 1 ) and the above is written as Note that the average of g 1 (V(t 1 ))g 2 (V(t 2 )) with positive weight |h(V(t 2 ))|/h(u) over the paths conditioned τ > t 2 in (2.2) is now replaced by that with signed weight h(V(t 2 ))/h(u) over all paths in (2.3).Then we use the equality (1.6) in (2.3).Note that V i (t), 1 ≤ i ≤ ξ(R) and W i (t), 1 ≤ i ≤ ξ(R) are independent.Then we can regard the probability space (Ω, F , P v ) as a product of two probability spaces (Ω 1 , F 1 , P 1 ) and (Ω 2 , F 2 , P 2 ), and V i (t), 1 ≤ i ≤ ξ(R) are F 1 -measurable and W i (t), 1 ≤ i ≤ ξ(R) are F 2 -measurable.We write E α for the expectation with P α , α = 1, 2. Then where we have used the independence of Z j (t), 1 ≤ j ≤ ξ(R), in the last equality.By the binomial expansion, E 2 [(Z j (t) Since G(x) is a monic polynomial with degree i − 1, Combining (1.6), (2.3), (2.4) and the fact (1.5), we have (2.1).
For the proof of (1.8) with M = 2, we first prove that for any Applying (2.1) with g m (x) = Jm⊂I ξ(R) ,♯Jm=Nm jm∈Jm χ tm (x jm ), m = 1, 2, we see that the LHS of (2.5) equals to where we have used the fact (1.5) with Φ v i ξ (Z j (0)) = δ ij .Then the RHS of the last equation coincides with the RHS of (2.5).By using relation exp If we take the summation of (3.1) over all 0 ≤ N m ≤ ξ(R), 1 ≤ m ≤ M, the LHS gives (1.13) with (1.14) and the RHS does (1.8).In this section we will prove (3.1).So in the following, we fix Then the integrand in the LHS of (3.1) is simply written as N i=1 χ τ i (x i ) det 1≤i,j≤N [K ξ (τ i , x i ; τ j , x j )], and the integral The determinant is defined using the notion of permutations and we note that any permutation σ ∈ S N can be decomposed into a product of exclusive cyclic permutations.Let the number of cycles in the decomposition be ℓ(σ) and express σ by For each 1 ≤ λ ≤ ℓ(σ), we write the set of entries {c λ (i)} q λ i=1 of c λ simply as {c λ }, in which the periodicity c λ (i + q λ ) = c λ (i), 1 ≤ i ≤ q λ is assumed.By definition, for each 1 ≤ λ ≤ ℓ(σ), c λ (i), 1 ≤ i ≤ q λ are distinct indices chosen from I N , and where the definition (1.11) of the correlation kernel K ξ is used.In order to express binomial expansions for (3.2), we introduce the following notations: For each cyclic permutation the LHS of (3.1) is expanded as From now on, we will explain how to rewrite G(c λ , M λ ) until (3.8).We note that if we set which is the integral over R M λ , then (3.3) is obtained by performing the integral of it over In (3.5), use the integral representation (1.10) for ) by putting the integral variables to be v = v c λ (i) and w = w c λ (i+1) .We obtain .
Using Fubini's theorem, (3.6) is given by For each 1 coincides with the conditional expectation of i j=i+1 Using only the entries of M λ , we can define a subcycle c λ of c λ uniquely as follows: Since c λ is a cyclic permutation, Moreover, we decompose the set M λ into M subsets, M λ = M m=1 J λ m , by letting Note that by definition J λ m ∩ J λ m ′ = ∅, m = m ′ in general, and Finally we arrive at the following expression of G(c λ , M λ ), where the martingale property (1.5) is used.
λ=1 ♯M c λ = ♯M, the LHS of (3.1), which is written as (3.4), becomes now (3.9) Note that ℓ( σ) = ℓ(σ).The obtained (J m ) M m=1 's form a collection of series of index sets satisfying the following conditions, which we write as J ({N m } M m=1 ): , which means that from the original index set I N = M m=1 I (m) with ♯I (m) = N m , 1 ≤ m ≤ M, we obtain a subset M by eliminating A m elements at each level 1 ≤ m ≤ M. By this reduction, we obtain σ ∈ S(M) from σ ∈ S N .It implies that, for all σ ∈ S(M), the number of σ's in S N which give the same σ and (J m ) M m=1 by this reduction is given by M m=1 A m !, where 0! ≡ 1.Then (3.9) is equal to Since the CBMs are i.i.d. in P v , the integral in (3.10) has the same value for all (J m ) M m=1 ∈ Λ 1 with M m=1 J m = M and it is also equal to In Λ 2 , on the other hand, N 1 elements in J 1 is chosen from I p , and then for each 2 ) is equal to the RHS of (3.1) and the proof is completed.

Proof of Proposition 1.3
Suppose that the initial configuration ξ ∈ M 0 satisfies the conditions (C.1) and (C.2) with constants C 0 , C 1 , C 2 and indices α, β.Since Theorem 1.1 can be generalized to the case with an infinite number of particles, we see that the LHS of (1.20) is given by ) and F 4 (x 4 ) = x 1 x 2 x 3 x 4 .Then Proposition 1.3 is concluded from the following estimate.ϕ(V i (t)) − ϕ(V i (s)) 1 L (V i (s), V i (t))  for some c 3 , c ′ 3 , which are also independent of s, t, and θ ∈ (max{α, (2−β)}, 2).Combining the above estimates with the fact that, for any c, c ′ > 0, R ξ(dv)e c|v| θ −c ′ |v| 2 < ∞, which is derived from the condition (C.2) (i) and the fact θ < 2, we obtain the lemma.