Elliptic Determinantal Processes and Elliptic Dyson Models

We introduce seven families of stochastic systems of interacting particles in one-dimension corresponding to the seven families of irreducible reduced affine root systems. We prove that they are determinantal in the sense that all spatio-temporal correlation functions are given by determinants controlled by a single function called the spatio-temporal correlation kernel. For the four families ${A}_{N-1}$, ${B}_N$, ${C}_N$ and ${D}_N$, we identify the systems of stochastic differential equations solved by these determinantal processes, which will be regarded as the elliptic extensions of the Dyson model. Here we use the notion of martingales in probability theory and the elliptic determinant evaluations of the Macdonald denominators of irreducible reduced affine root systems given by Rosengren and Schlosser.


Introduction and main results
Stochastic analysis on interacting particle systems is important to provide useful models describing equilibrium and non-equilibrium phenomena studied in statistical physics [15]. Determinantal process is a stochastic system of interacting particles which is integrable in the sense that all spatio-temporal correlation functions are given by determinants controlled by a single function called the spatio-temporal correlation kernel [4,20]. Since the generating functions of correlation functions are generally given by the Laplace transforms of probability densities, the stochastic integrability of determinantal processes is proved by showing that the Laplace transform of any multi-time joint probability density is expressed by the spatio-temporal Fredholm determinant associated with the correlation kernel. The purpose of this paper is to present new kinds of determinantal processes in which the interactions between particles are described by the logarithmic derivatives of Jacobi's theta functions. A classical example of determinantal processes is Dyson's Brownian motion model with parameter β = 2, which is a dynamical version of the eigenvalue statistics of random matrices in the Gaussian unitary ensemble (GUE), and we call it simply the Dyson model [6,15,38]. We will extend the Dyson model to the elliptic-function-level in this paper. We use the notion of martingales in probability theory [14,15] and the elliptic determinantal evaluations of the Macdonald denominators of seven families of irreducible reduced affine root systems given by Rosengren and Schlosser [34] (see also [26,42]).
Among the seven families of irreducible reduced affine root systems, R = A N −1 , B N , B ∨ N , C N , C ∨ N , BC N , and D N , we reported the results only for the system R = A N −1 in the previous papers This paper is a contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications. The full collection is available at https://www.emis.de/journals/SIGMA/EHF2017.html arXiv:1703.03914v3 [math.PR] 4 Oct 2017 [16,17] as follows. Assume 0 < t * < ∞, 0 < r < ∞, 0 < N < ∞, and let A 2πr N (t * − t, x) = 1 2πr ∂ ∂v log ϑ 1 (v; τ ) v=x/2πr, τ =iN (t * −t)/2πr 2 , t ∈ [0, t * ), (1.1) where θ 1 (v; τ ) denotes one of the Jacobi theta functions. See Appendix A for the Jacobi theta functions and related functions. For N ∈ {2, 3, . . . }, we define the Weyl chamber We introduced a one-parameter (β > 0) family of systems of stochastic differential equations (SDEs) for t ∈ [0, t * ) [17] (A N −1 ) X for u = (u 1 , . . . , u N ) ∈ W N , where W j (t), t ≥ 0, j = 1, . . . , N are independent one-dimensional standard Brownian motions, and We called this family of N -particle systems on R, X A N −1 (t) = X . This implies that the elliptic Dyson model of type A is realized as a system of interacting Brownian bridges (see, for instance, [5,Part I,Section IV.4.22]) pinned at the configuration v A N −1 at time t * [17]. When the system of SDEs is temporally homogeneous, it is well known that the corresponding Kolmogorov (Fokker-Planck) equation for transition probability density can be transformed into a Calogero-Moser-Sutherland quantum system (see, for instance, [8,Chapter 11]). The present system of interacting Brownian bridges is temporally inhomogeneous, however, and the Kolmogorov equation is mapped into a Schrödinger-type equation with time-dependent Hamiltonian and the time-dependent ground energy [17]. The obtained quantum system is elliptic, but different from the elliptic Calogero-Moser-Sutherland model extensively studied as a quantum integrable system [7,25,31,39,40]. We found at the same time that the interaction among particles vanishes when the parameter is chosen to be a special value (β = 2 in our case) as found in the usual Calogero-Moser-Sutherland models. We applied the determinantal-martingale-method [14] and proved that the elliptic Dyson model of type A with β = 2 is an integrable stochastic process in a sense that it is determinantal for a set of observables [16,17].
In the present paper, we report the results for other six systems, R = B N , B ∨ N , C N , C ∨ N , BC N , D N . Here we first construct the six families of determinantal processes (Theorem 1.1). Then for the three families B N , C N and D N , we clarify the systems of SDEs (with parameter β = 2) which are solved by our new determinantal processes (Theorem 1.2).
In the interval [0, πr], we consider the N -particle system of one-dimensional standard Brownian motions B(t) = (B 1 (t), . . . , B N (t)), t ≥ 0 started at u = (u 1 , . . . , u N ) ∈ W (0,πr) N with either an absorbing or reflecting boundary condition at the endpoints of the interval, 0 and πr. The transition probability density of each particle is generally denoted by p [0,πr] . The boundary conditions at 0 and πr are indicated by b and b , respectively, and the transition probability density with the specified boundary conditions b, b is written as p [0,π] b,b . In this paper the absorbing (resp. reflecting) boundary condition is abbreviated as 'a' (resp. 'r'). By the reflection principle of Brownian motion (see, for instance, [5, Appendixes 1.5 and 1.6]), if both boundaries are absorbing, the transition probability density is given by 11) and if both are reflecting, it is given by for x, y ∈ [0, πr], t ≥ 0, where p BM (t, y|x) denotes the transition probability density of the one-dimensional standard Brownian motion We write the probability law of such a system of boundary-conditioned Brownian motions in [0, πr] as P i.e., the first collision-time of the N -particle system of Brownian motions in the interval [0, πr]. Let 1(ω) be the indicator function of a condition ω; 1(ω) = 1 if ω is satisfied, and 1(ω) = 0 otherwise. Then we define where D R u are given by (1.5)-(1.7) and F t denotes the filtration associated with the Brownian motion (see Section 2.1). That is, the Radon-Nikodym derivative of P R u with respect to the Wiener measure P [0,πr] u of Brownian motion is given by 1(T collision > t)D R u (t, B(t)) at each time t ∈ [0, t * ). Therefore, a stochastic process with N particles governed by P R u is well-defined as a realization of non-intersecting paths which are absolutely continuous to the N -particle paths of independent Brownian motions in [0, πr].
For y ∈ R, δ y (·) denotes the delta measure such that δ y ({x}) = 1 if x = y and δ y ({x}) = 0 otherwise. The first theorem of this paper is the following. 15) gives a probability measure and defines a measure-valued stochastic process The process Ξ R (t) t∈[0,t * ) , P R u is determinantal with the spatio-temporal correlation kernel with the sets of entire functions (the elliptic Lagrange interpolation functions) The second theorem of this paper is the following. we put an absorbing boundary condition at 0 and a reflecting boundary condition at πr, for Ξ C N (t) t∈[0,t * ) , P C N u we put an absorbing boundary condition both at 0 and πr, and for Ξ D N (t) t∈[0,t * ) , P D N u we put a reflecting boundary condition both at 0 and πr, respectively. If we set Ξ R (t) = N j=1 δ X R j (t) , then X R (t) = X R 1 (t), . . . , X R N (t) , R = B N , C N , D N , solve the following systems of SDEs in [0, πr] with a reflecting boundary condition at πr, where W j (t), t ≥ 0, j = 1, 2, . . . , N are independent one-dimensional standard Brownian motions.
We call the systems (1.22)-(1.24) the elliptic Dyson models of types B, C, D, respectively. By (A.9) in Appendix A.2, we see that in The above implies that the elliptic Dyson models of types B, C, and D are realized as the systems of interacting Brownian bridges pinned at the configurations corresponding to the situation such that the absorbing boundary condition is imposed at 0 for B N , and at 0 and πr for C N , while the reflecting boundary condition is imposed at other endpoints of the interval [0, πr].
We see by (1.1) and (A.4). Hence in the limit t * → ∞, the systems of SDEs (1.2) with β = 2, and of (1.22)-(1.24) become the following temporally homogeneous systems of SDEs for t ∈ [0, ∞): ds, j = 1, 2, . . . , N, in [0, πr] with a reflecting boundary condition at πr, ds, j = 1, 2, . . . , N, in [0, πr], (1.27)   [2,18,19,23,24,28,41]. The correlation kernel for a determinantal process is in general a function of two points on the spatio-temporal plane, say, (s, x) and (t, y). In the present formula (1.17) in Theorem 1.1, the dependence of the correlation kernel K R u on one spatio-temporal point (s, x) is explicitly given by the transition probability density p [0,πr] of a single Brownian motion in an interval [0, πr] with given boundary conditions. The dependence of K R u on another spatio-temporal point (t, y) is described by the following two procedures. The information of interaction among particles and initial configuration is expressed by a set of entire functions Φ R u,u j (z) N j=1 , (1.19)-(1.21), which are static functions without time-variable t. Then evolution in time t is given by the integral transformation (1.18) specified by the transition probability density p BM with time duration t of a single Brownian motion. One of the benefits of such separation of static information and dynamics in the formula is that we can trace a relaxation process to equilibrium, which is a typical non-equilibrium phenomenon, for any initial configuration in W (0,πr) N [14]. In order to demonstrate this fact, in Section 5 we will take the temporally homogeneous limit t * → ∞ to make the systems have equilibrium processes and study the relaxation to equilibria for the trigonometric Dysom models of types C and D. Another possible benefit of the present formula for spatio-temporal correlation kernels is that the infinite particle systems will be studied if we can control the set of entire functions Φ R u,u j (z) N j=1 in the infinite-particle limit N → ∞. For the original Dyson model [22] and other related systems [21,23], we have applied the Hadamard theorem on the Weierstrass canonical product-formulas of entire functions [27] to analyze the infinite particle systems. In the present case with (1.19)-(1.21), we have to treat the rations of infinite products of the Jacobi theta functions in this limit N → ∞. Asymptotic analysis with N → ∞ will be an interesting future problem.
The paper is organized as follows. In Section 2 we explain the relationship between determinantal martingales and determinantal processes used in this paper, and determinantal equalities (Lemma 2.4) obtained from the elliptic determinant evaluations of the Macdonald denominators given by Rosengren and Schlosser [34]. There we explain how to derive the determinantal martingale-functions (1.5)-(1.7), which define the interacting systems of Brownian motions by (1.15). Proofs of Theorems 1.1 and 1.2 are given in Sections 3 and 4, respectively. We study the temporally homogeneous limit t * → ∞ in Section 5, and relaxation processes to equilibria are clarified for the trigonometric Dyson models of types C and D. Concluding remarks and open problems are given in Section 6. Notations and formulas of the Jacobi theta functions and related functions are listed in Appendix A.

Determinantal martingales and determinantal equalities 2.1 Notion of martingale
Martingales are the stochastic processes preserving their mean values and thus they represent fluctuations. A typical example of martingale is the one-dimensional standard Brownian motion as explained below. Let B(t), t ≥ 0 denote the position of the standard Brownian motion in R starting from the origin 0 at time t = 0. The transition probability density from the position x ∈ R to y ∈ R in time duration t ≥ 0 is given by (1.13). Let B(R) be the Borel set on R. Then for an arbitrary time sequence The collection of all paths is denoted by Ω and there is a subset Ω ⊂ Ω such that P[ Ω] = 1 and for any realization of path ω ∈ Ω, B(t) = B(t, ω), t ≥ 0 is a real continuous function of t. In other words, B(t), t ≥ 0 has a continuous path almost surely (a.s. for short). For each t ∈ [0, ∞), we write the smallest σ-field (completely additive class of events) generated by the Brownian motion up to time t as F t = σ(B(s) : 0 ≤ s ≤ t). We have a nondecreasing family {F t : t ≥ 0} such that F s ⊂ F t for 0 ≤ s < t < ∞, which we call a filtration, and put F = t≥0 F t . The triplet (Ω, F, P) is called the probability space for the one-dimensional standard Brownian motion, and P is especially called the Wiener measure. The expectation with respect to the probability law P is written as E. When we see p BM (t, y|x) as a function of y, it is nothing but the probability density of the normal distribution with mean x and variance t, and hence it is easy to verify that which means that B(t), t ≥ 0 is a martingale. We see, however, Now we assume that B(t), t ≥ 0 is a one-dimensional standard Brownian motion which is independent of B(t), t ≥ 0, and its probability space is denoted by Ω, F, P . Then we introduce a complex-valued martingale called the complex Brownian motion The probability space of Z(t), t ≥ 0 is given by the product space (Ω, F, P) ⊗ ( Ω, F, P) and we write the expectation as E = E ⊗ E. For the complex Brownian motion, by the independence of its real and imaginary parts, we see that which implies that for any n ∈ N 0 , Z(t) n , t ≥ 0 is a martingale. This observation will be generalized as the following stronger statement; if F is an entire and non-constant function, then F (Z(t)), t ≥ 0 is a time change of a complex Brownian motion (see, for instance, [33, If we take the expectation E with respect to Z(·) = B(·) of the both sides of the above equality, we have In this way we can obtain a martingale F (t, B(t)) ≡ E[F (Z(t))], t ≥ 0 with respect to the filtration F t of the one-dimensional Brownian motion. The present argument implies that if we have proper entire functions, then we will obtain useful martingales describing intrinsic fluctuations involved in interacting particle systems.

Basic equalities and determinantal martingales
Let f j , j ∈ I, be an infinite series of linearly independent entire functions, where the index set We then define a set of N distinct entire and non-constant functions of z ∈ C by By definition, it is easy to verify that It should be noted that given 3) can be regarded as the Lagrange interpolation functions [10,11,12]. We can prove the following lemmas.

4)
and the coefficients φ u,u j (k), j, k ∈ {1, 2, . . . , N } satisfy the relations In particular, if j is an element of {1, 2, . . . , N }, and write its determinant as |f u | = 0. The minor determinant |f u (k, j)| is defined as the deter- , which is obtained from f u by deleting the k-th row and the j-th column, k, j ∈ {1, 2, . . . , N }. Then the determinant in the numerator of (2.2) is expanded along the j-th column and we obtain (2.4) with which proves (2.5), for the summation in the r.h.s. is the expansion of det along the k-th row. It is immediate to conclude (2.6) from (2.5), if both of j and k are elements of {1, 2, . . . , N }.
Then we obtain the following equalities. holds.
Proof . (i) Multiply the both sides of (2.6) by f k (z) and take summation over k ∈ {1, 2, . . . , N }. We consider N pairs of independent copies (B k (t), B k (t)), k = 1, 2, . . . , N , of (B(t), B(t)), t ≥ 0, and define N independent complex Brownian motions each of which starts from u k ∈ R. The probability law and expectation of them are given by P u ⊗ P and E u ⊗ E, respectively. Then for each complex Brownian motion Z k (t), t ≥ 0, k = 1, 2, . . . , N , we have N distinct time-changes of complex Brownian motions, Φ u,u j (Z k (t)), t ≥ 0, j = 1, 2, . . . , N , started from the real values, 0 or 1; Φ u,u j (Z k (0)) = Φ u,u j (u k ) = δ jk , j, k = 1, 2, . . . , N . Therefore, we can conclude that, if we take expectation E, we will obtain N distinct martingales By this definition, f (0, x) = f (x), ∈ I. Then we define a multivariate function of t ≥ 0 and which gives a martingale as a functional of t ≥ 0 and By the multi-linearity of determinant and (2.9), we see We call D u (t, B(t)), t ≥ 0, the determinantal martingale [14].

Auxiliary measure and spatio-temporal Fredholm determinant
Now we introduce an auxiliary measure P u , which is complex-valued in general, but is absolutely continuous to the Wiener measure P u as (Note that this is generally different from P R u given by (1.15), since the condition 1(T collision > t) is omitted, and hence it is not the measure representing any noncolliding particles.) Here we consider the corresponding auxiliary system of N particles on R governed by P u starting from (2.1) and each particle of which has a continuous path a.s. We consider the unlabeled configuration of X(t) as Consider an arbitrary number M ∈ N and an arbitrary set of strictly increasing times t = be the set of all continuous real-valued functions with compact supports on R. For g = (g t 1 , g t 1 , . . . , g t M ) ∈ C c (R) M , we consider the following functional of g which is the Laplace transform of the multi-time distribution function P u on a set of times t with the functions g. If we put then (2.15) can be written as Explicit expression of (2.16) is given by the following multiple integrals The following was proved as Theorem 1.3 in [14].
s, t > 0, x, y ∈ R. Then the following equality holds for an arbitrary number M ∈ N, an arbitrary set of strictly increasing times t = {t 1 , t 2 , . . . , t M }, 0 ≡ t 0 < t 1 < · · · < t M < ∞, and where dx This proposition is general and it proves that the auxiliary system (2.14) is determinantal. The measure (2.13) which governs this particle system is, however, complex-valued in general, and hence the system is unphysical. The problem is to clarify the proper conditions which should be added to (2.13) to construct a non-negative-definite real measure, i.e., the probability measure, which defines a physical system of interacting particles. As a matter of course, this problem depends on the choice of an infinite set of linearly independent entire functions f j , j ∈ I.

Elliptic determinant evaluations of the Macdonald denominators
Here we report the results when we choose the entire functions f j , j ∈ Z as follows z ∈ C, j ∈ Z, with N ∈ N, 0 < r < ∞, and τ ∈ C with 0 < τ < ∞, where 19) and N R are given by (1.4). These functions (2.18) were used to express the determinant evaluations by Rosengren and Schlosser [34] for the Macdonald denominators W R (x) for seven families of irreducible reduced affine root systems where k 0 is a constant, k 1 is a single variable function, k 2 is an antisymmetric function of two variables, k 2 (u, v) = −k 2 (v, u), and k sym (u) is a symmetric function of u which cannot be factorized as N =1 k 1 (u ). For u = (u 1 , . . . , u j−1 , u j , u j+1 , . . . , u N ) ∈ W N , we replace the j-th component u j by a variable z and write the obtained vector as u (j) (z) = (u 1 , . . . , u j−1 , z, u j+1 , . . . , u N ). Under the assumption (2.20), Φ u,u j (z) defined by (2.2) is also factorized as Then the determinantal equality (2.9) becomes det 1≤j,k≤N

24)
holds with the following factors,

Determinantal martingale-functions
Since f R j (z; τ ), j ∈ Z given by (2.18) are entire and non-constant functions of z ∈ C, give the single-variable martingale-functions such that, if f R j (t, B(t); τ ), t ≥ 0, j ∈ Z are finite, then they give martingales with respect to the filtration F t of one-dimensional Brownian motion B(t), t ≥ 0, We have found that f R j (t, x; τ )'s are expressed using f R j (t, x; ·) by shifting with (1.4) and multiplying time-dependent factors. With 0 < t * < ∞, we put (1.3).
Lemma 2.5. For f R j , j ∈ Z given by (2.18), we obtain the following infinite series of linearly independent martingale-functions
Proof . By (2.18), it is enough to calculate with a constant α. By the definition (A.1) of ϑ 1 , this is equal to we can see that (2.29) is equal to Therefore, if we set τ = iN R t * /2πr 2 , (2.28) are obtained.
We call the multivariate function D u (t, x) of x ∈ R N defined by (2.11) the determinantal martingale-function, since if it is finite, it gives the determinantal martingale when we put the N -dimensional Brownian motion B(t), t ≥ 0 into x. The following is proved. Proposition 2.6. For f R j , j ∈ Z given by (2.18), the determinantal martingale-functions are factorized as follows Proof . By the second equality of (2.12) and Lemma 2.5, where the multi-linearity of determinant was used in the second equality. Then applying Lemma 2.4, we obtain (2.30) with . 3 Proof of Theorem 1.1

By (2.25) and the summation formulas
We also see that if The inequality |x 1 | < x 2 means that x 2 is greater than both of x 1 and its reflection at 0, and x N −1 < x N ∧ (2πr − x N ) means that x N −1 is smaller than both of x N and its reflection at πr. The above observation implies that if u ∈ W (0,πr) N , then Due to the Karlin-McGregor-Lindström-Gessel-Viennot (KMLGV) formula [9,13,29] for non-intersecting paths, the transition probability density of the N particle system in [0, πr] governed by the probability law P R u defined by (1.15) is given by Since we consider the measure-valued stochastic process (1.16), the configuration is unlabeled and hence all the observables at each time should be symmetric functions of particle positions.

(3.3)
By definition of determinant, the above is equal to this is equal to . Therefore, we obtain the equality that is, the noncolliding condition, T collision > t M , can be omitted in (3.3). The expression (3.4) is interpreted as the expectation of the product of symmetric functions M m=1 g tm (·) with respect to the signed measure which is a modification of (2.13) obtained by replacing P u by P    Proof . For given t ∈ (0, t * ) and y ∈ W (0,πr) N , put with the KMLGV determinant where w(s, x) is a C 1,2 -function which will be specified later, and p [0,πr] = p Therefore, the following equation holds the above equation is written as  . Therefore, if we can choose g R (s) so that the equation holds, then the r.h.s. of (4.8) vanishes and we can conclude that u(s, x) given in the form (4.5) with (4.9) and (4.10) solve the Kolmogorov equations (4.1)-(4.3). From (4.9) and (4.10) with (1.3), we find that whereθ 1 (v; τ ) = ∂ϑ 1 (v; τ )/∂τ . If we use the equation (A.2), then the above is written as where ϑ 1 (v; τ ) = ∂ 2 ϑ 1 (v; τ )/∂v 2 . From (4.11), we find that and hence By definition (1.1), where we have used the fact concluded from (A.7). Then (4.12) holds, if (4.14) Now we use the following expressions for A 2πr N R and its spatial derivative [16,17] A 2πr Here η 1 N R (t * − s) is given by (A.10) and we put where the Weierstrass ℘ function and zeta function ζ are defined by (A.14) in Appendix A.5. Applying (4.15) to (4.13) gives (4.17) Using (4.15) and (4.17), the l.h.s. of (4.14) can be written using ζ N R , ℘ N R and η 1 N R . Moreover, from the functional equation (A.15) given in Appendix A.5, we can derive the following equalities and 1≤j,k≤N, j =k (4.21) and 1≤j,k, ≤N, Using the above formulas and the values of N R given by (1.4), we can show that (4.14) is reduced to the following simple equations (4.23) They give the conditions for g R (s) so that (4.12) holds. Since (A.11) gives for R = B N , we find that with constants c R , R = B N , C N , D N . By (4.7), the conditions (4.4) determine the constants.
In conclusion, under the condition (4.4), solves (4.1), solves (4.2), and  [15,33]. Due to the behavior (A.8) of A 2πr N , we can show that [15,33] particles in [0, πr] following (1.22) do not arrive at 0, and those following (1. 23) do not arrive at 0 nor πr with probability one, and thus we do not need to impose any boundary condition at these endpoints of [0, πr] for these systems of SDEs. Hence the proof is completed.

Relaxation to equilibrium processes in trigonometric Dyson models of types C and D
As a corollary of Theorems 1.1 and 1.2, we obtain the following trigonometric determinantal processes by taking the temporally homogeneous limit t * → ∞.
Corollary 5.1. Assume that u ∈ W (0,πr) . Then the trigonometric Dyson models of types C and D given by (1.27) and (1.28), respectively, are determinantal with the spatio-temporal correlation kernelš We find that in the limit t * → ∞, the equality (2.24) with R = C N and D N gives det 1≤j,k≤N By Lemma 2.1, we see that (5.2) can be expanded aš We note that the transition probability densities (1.11) and (1.12) are written as (A.12) and (A.13), respectively. Hence we have Now we show relaxation processes to equilibria, which are typical non-equilibrium phenomena.
Proposition 5.2. For any initial configuration u ∈ W (0,πr) , the trigonometric Dyson models of types C and D exhibit relaxation to the equilibrium processes. The equilibrium processes are determinantal with the correlation kernelš Proof . Here we prove the convergence of correlation kernels. It implies the convergence of the generating functions of spatio-temporal correlation functions (i.e., the Laplace transformations of multi-time joint probability densities). Hence the convergence of process is concluded in the sense of finite-dimensional distributions [22]. Assume u ∈ W (0,πr) and let It is easy to verify that and thuš Then we have e n 2 (t−s)/2r 2 sin nx r sin ny r , and g D N eq (t − s, x, y) = 1 πr e n 2 (t−s)/2r 2 cos nx r cos ny r , in the same sense. It is easy to confirm that the limit kernels are represented by (5.7) and (5.8), if we use (5.6). The limit kernels (5.7) and (5.8) depend on time difference t−s, which implies that the determinantal processes defined by them are temporally homogeneous. The determinantal processes with the spatio-temporal correlation kernels (5.7) and (5.8) are equilibrium processes, which are reversible with respect to the determinantal point processes with the spatial correlation kernelš (x, y) ∈ [0, πr] 2 , respectively. The convergence of processes is irreversible. Thus all statements of the present proposition have been proved.
We note thať as required.

Concluding remarks and open problems
In the previous paper [16,17] and in this paper (Theorem 1.1), we have introduced seven families of interacting particle systems Ξ R (t) = N j=1 δ X R j (t) , t ∈ [0, t * ) governed by the probability laws P R u associated with the irreducible reduced affine root systems denoted by R = A N −1 , B N , B ∨ N , C N , C ∨ N , BC N , D N . When we proved that they are determinantal processes, we showed that without change of expectations of symmetric functions of X R j (·) N j=1 , P R u can be replaced by the signed . As the simplest corollary of this fact, we can conclude that at any time 0 < t < t * , This is nothing but a rather trivial statement such that the processes Ξ R (t) t∈[0,t * ) are wellnormalized, but it provides nontrivial multiple-integral equalities including parameters t ∈ [0, t * ) and u = (u 1 , . . . , u N ). For example, for R = D N , we will have , where η(τ ) is the Dedekind modular function (1.10) and In the previous papers [16,17] and in Theorem 1.2 we have identified the systems of SDEs which are solved by the four families of determinantal processes, Ξ A N −1 (t) t∈[0,t * ) , P A N −1 u , u ∈ W (0,2πr) N , and Ξ R (t) t∈[0,t * ) , P R u , u ∈ W (0,πr) N with R = B N , C N and D N . The systems of SDEs for other cases R = B ∨ N , C ∨ N , BC N are not yet clarified. The exceptional cases of reduced irreducible affine root systems and the non-reduced irreducible affine root systems [30] will be studied from the view point of the present stochastic analysis.
The classical Dyson models of type A in R given by (1.29) and of types C and D in [0, ∞) given by (1.30) and (1.31) are realized as the eigenvalue processes of Hermitian-matrix-valued Brownian motions with specified symmetry [6,18]. It will be an interesting problem to construct the matrix-valued Brownian motions such that the eigenvalue processes of them provide the elliptic Dyson models; (1.2) with β = 2 and (1.22)-(1.24).