Noncolliding Squared Bessel Processes

0.28 MB·English

by  Makoto Katori

Noncolliding Squared Bessel Processes
Makoto Katori and Hideki Tanemura
∗ †
16 January 2011
1
1
0
Abstract
2
n We consider a particle system of the squared Bessel processes with index ν > 1
a conditioned never to collide with each other, in which if 1 < ν < 0 the origin−is
J −
assumed to be reflecting. When the number of particles is finite, we prove for any
6
1 fixed initial configuration that this noncolliding diffusion process is determinantal in
the sensethat any multitime correlation function is given by adeterminant with a con-

]
R tinuous kernel called the correlation kernel. When the number of particles is infinite,
P we give sufficient conditions for initial configurations so that the system is well de-
. fined. There the process with an infinite number of particles is determinantal and the
h
t correlation kernel is expressed using an entire function represented by the Weierstrass
a
m canonical product, whose zeros on the positive part of the real axis are given by the
[ particle-positions in the initial configuration. From the class of infinite-particle initial

configurations satisfying our conditions, we report one example in detail, which is a
3
v fixed configuration such that every point of the square of positive zero of the Bessel
4 function J is occupied by one particle. The process starting from this initial configu-
ν
4
ration shows a relaxation phenomenon converging to the stationary process, which is
1
0 determinantal with the extended Bessel kernel, in the long-term limit.
. Keywords Noncolliding diffusion process, Squared Bessel process, Fredholm determi-
8
0 nants, Entire functions, Weierstrass canonical products, Infinite particle systems
0
1
:
v 1 Introduction
i
X
r Let M be the space of nonnegative integer-valued Radon measures on R, which is a Polish
a
space with the vague topology. We say ξ M,n N 1,2,... converges to ξ M
n
∈ ∈ ≡ { } ∈
weakly in the vague topology, if lim ϕ(x)ξ (dx) = ϕ(x)ξ(dx) for any ϕ C (R),
n R n R 0
where C (R) is the set of all continu→o∞us real-valued functions with compact sup∈ports in
0
R R
R. Any element of M can be represented by δ ( ) with a sequence of points in R,
i Λ xi ·
x = (x ) , satisfying ♯ x : x I < for a∈ny compact subset I R, and with a
i i Λ i i
∈ { ∈ } ∞ P ⊂
∗DepartmentofPhysics,FacultyofScienceandEngineering,ChuoUniversity,Kasuga,Bunkyo-ku,Tokyo
112-8551,Japan; e-mail: [email protected]
†Department of Mathematics and Informatics, Faculty of Science, Chiba University, 1-33 Yayoi-cho,
Inage-ku, Chiba 263-8522,Japan; e-mail: [email protected]
1
countable set (an index set) Λ. We call an element ξ of M an unlabeled configuration,
and a sequence of points x a labeled configuration. For A R, we write the restriction
⊂
of ξ M on A as (ξ A)( ) = δ ( ). Let R = x R : x 0 and define
M+ =∈ (ξ R )( ) : ξ(∩) M· . In tih∈eΛ:pxir∈eAsenxti p·aper we c+onsid{er a∈one-par≥ame}ter family of
+
{ ∩ · · ∈ } P
M+-valued processes with a parameter ν > 1,
−
Ξ(ν)(t, ) = δ ( ), t [0, ), (1.1)
· X(ν)(t) · ∈ ∞
i
i
X
describing a particle system of squared Bessel processes with index ν > 1 (BESQ(ν))
−
(ν)
interacting with each other by long-ranged repulsive forces, such that X (t)’s satisfy the
i
SDEs
(ν) (ν)
dX (t) = 2 X (t)dB (t)+2(ν +1)dt
i i i
q 1
(ν)
+ 4X (t) dt, i = 1,2, , t [0, ) (1.2)
i X(ν)(t) X(ν)(t) ··· ∈ ∞
jX:j6=i i − j
with a collection of independent standard Brownian motions (BMs), B (t),i N , and,
i
{ ∈ }
if 1 < ν < 0, with a reflection wall at the origin. Note that for the BM in Rd, B(t) =
−
(B (t),...,B (t)),d N, the square of its distance from the origin, X(t) B(t) 2 =
1 d
∈ ≡ | |
d B (t)2, solves the SDE, dX(t) = 2 X(t)dB(t)+2(ν +1)dt with e
i=1 i
e e e
P p d
e
ν = 1, (1.3)
2 −
where B(t) is a standard BM which is different from B (t),1 i d [24, 4]. We give the
i
≤ ≤
initial configuration of the process ξ( ) = Ξ(ν)(0, ) = δ ( ) and the process is denoted by
· · i xi ·
(Ξ(ν)(t),Pξ). e
Whenνthe number of particles is finite, ξ (R ) =PN < , the process (Ξ(ν)(t),PξN) is
N + ∞ ν
realized in the following systems.
(i) When ν N N 0 , i.e., when the corresponding dimension d given by (1.3) is a
0
∈ ≡ ∪{ }
positive even integer, (Ξ(ν)(t),PξN) is realized as the eigenvalue process of the Laguerre
ν
process [20]. Let M(t) be an (N + ν) N matrix, whose entries are independent
×
complex BMs having the real and imaginary parts given by independent standard
BMs, and set L(t) = M(t) M(t). The N N matrix-valued process L = (L(t))
∗ t [0, )
× ∈ ∞
is called the Laguerre process. The matrix L(t) is Hermitian and positive definite,
and its N eigenvalues satisfy (1.2) with i = 1,2,...,N. When the entries of M(t)
are independent standard real BMs, the matrix-valued process (M(t)TM(t)) is
t [0, )
∈ ∞
called the Wishart process [5], and thus L is also called the complex Wishart process.
The eigenvalue processes of the real and complex Wishart processes are related with
the random matrix theory [23, 9] for the chiral Gaussian ensembles studied in the high
energy physics (see [15] and references therein).
2
(ii) Let (N)bethespaceofN N Hermitianmatrices. Andletsp(2N,C)andso(2N,C)
H ×
be the symplectic Lie algebra and the orthogonal Lie algebra, having 2N 2N-matrix
×
representations, respectively. Ifthe2N 2N matrixisinthespace (2N) (2N)
C
× H ≡ H ∩
sp(2N,C) or in (2N) (2N) so(2N,C), its eigenvalues are given by N pairs of
D
H ≡ H ∩
positive and negative ones with the same absolute value, (λ , λ ) : λ 0,1 i
i i i
{ − ≥ ≤ ≤
N . Consider the (2N)-valued and the (2N)-valued Brownian motions. The
C D
} H H
dynamics of positive eigenvalues of them are described by (1.2) with ν = 1/2(d = 3)
for the former case and with ν = 1/2(d = 1) for the latter case, respectively [15].
−
The pairing of positive and negative eigenvalues simulates the particle-hole symmetry
in the energy space of the Bogoliubov-de Gennes formalism of superconductivity and
these processes are related with the random matrix theory studied in the solid-state
physics [1].
(iii) Let p(ν)(t,y x),y R , be the transition probability density for BESQ(ν), ν > 1,
+
| ∈ −
1 y ν/2 x+y √xy
exp I , t > 0,x > 0,
ν
2t x − 2t t
 (cid:16) (cid:17)yν (cid:18) (cid:19) (cid:18) (cid:19)
p(ν)(t,y|x) =  (2t)ν+1Γ(ν +1)e−y/2t, t > 0,x = 0, (1.4)


δ(y x), t = 0,x R ,
 +
 − ∈




if 1 < ν < 0, theorigin is assumed to be reflecting [24, 4], where I (x) is the modified
ν
−
Bessel function of the first kind defined by
∞ 1 x 2n+ν
I (x) = (1.5)
ν
Γ(n+1)Γ(n+1+ν) 2
Xn=0 (cid:16) (cid:17)
with the Gamma function Γ(z) = 0∞e−uuz−1du, ℜu > 0. First we consider the
following Karlin-McGregor determinant [14],
R
f(ν)(t,y x) = det p(ν)(t,y x ) , t 0, x = (x )N ,y = (y )N W+, (1.6)
N | 1 i,j N i| j ≥ i i=1 i i=1 ∈ N
≤ ≤
h i
where W+ = x = (x ,...,x ) RN : 0 x < < x . The transition
N { 1 N ∈ ≤ 1 ··· N}
probability density of the N-particle system of BESQ(ν) conditioned never to collide
with each other, which we call the noncolliding BESQ(ν), is given by the h-transform
of (1.6),
1
p(ν)(t,y x) = h (y)f(ν)(t,y x) (1.7)
N | N N | h (x)
N
with the harmonic function given by the Vandermonde determinant [11, 20, 15]
hN(x) = 1 di,ejtN[xij−1] = (xj −xi). (1.8)
≤ ≤ 1 i<j N
≤Y≤
3
It is easy to confirm that p(ν)(t, x) satisfies the following backward Kolmogorov equa-
N ·|
tion
∂ N ∂2 N ∂
u(t,x) = 2 x u(t,x)+2(ν +1) u(t,x)
∂t i∂x2 ∂x
i=1 i i=1 i
X X
N N
x ∂
+ 4 i u(t,x), t 0, x W+,
x x ∂x ≥ ∈ N
i j i
i=1 j=1 −
XX
j=i
6
and it implies that the process (Ξ(ν)(t),PξN),ν > 1, is realized as the noncolliding
ν −
BESQ(ν) [15]. We put
M+ = ξ M+ : ξ( x ) 1 for any x R . (1.9)
0 { ∈ { } ≤ ∈ }
We see that Ξ(ν)(t) M+, t > 0.
∈ 0 ∀
In the present paper, we call the process (Ξ(ν)(t),Pξ) the noncolliding BESQ(ν). See
ν
[28, 16, 21] for related noncolliding diffusion processes.
Assume that ξ M+ with ξ (R ) = N N. For any M N and any time sequence
N N +
∈ ∈ ∈
0 < t < < t < , the formula (1.7) and the Markov property of the system give the
1 M
··· ∞
multitime probability density of (Ξ(ν),PξN) as [11, 17]
ν
M−1 f(ν)(t ,x(1) x)
pξN(t ,ξ(1);...;t ,ξ(M)) = h (x(M)) f(ν)(t t ;x(m+1) x(m)) N 1 | (1.10)
ν 1 N M N N N m+1− m | h (x)
N
m=1
Y
with ξ ( ) = N δ ( ) M+, 0 x x x for the initial configuration and
N · i=1 xi · ∈ ≤ 1 ≤ 2 ≤ ··· ≤ N
ξ(m)( ) = N δ ( ) M+, x(m) = (x(m),...,x(m)) W+ for the configurations at times
N · i=P1 xi(m) · ∈ 0 1 N ∈ N
t ,1 m M. In (1.10), if some of x ’s in x coincide, the factor f(ν)(t ,x(1) x)/h (x) is
m ≤ P≤ i N 1 | N
interpreted using l’Hoˆpital’s rule.
For x(m) = (x(m),...,x(m)) W+ with ξ(m)( ) = N δ ( ) and N 1,2,...,N ,
1 N ∈ N N · i=1 x(m) · ′ ∈ { }
i
we put xN(m′) = (x1(m),...,xN(m′)) ∈ W+N′, 1 ≤ m ≤ M.PFor a sequence (Nm)Mm=1 of positive
integers less than or equal to N, we define the (N ,...,N )-multitime correlation function
1 M
by
ρξN t ,x(1);...;t ,x(M)
ν 1 N1 M NM
(cid:16) M N (cid:17) M
1
= dx(m)pξN t ,ξ(1);...;t ,ξ(M) , (1.11)
i ν 1 N M N (N N )!
ZQMm=1R+N−Nm mY=1i=YNm+1 (cid:16) (cid:17)mY=1 − m
which is symmetric in the sense that ρξN(...;t ,σ(x(m));...) = ρξN(...;t ,x(m);...) with
ν m Nm ν m Nm
σ(x(m)) (x(m),...,x(m) ) for any permutation σ ,1 m M. For any M N,
Nm ≡ σ(1) σ(Nm) ∈ SNm ≤ ∀ ≤ ∈
f C (R ),θ R,1 m M, 0 < t < < t < , the Laplace transform of (1.10)
m 0 + m 1 M
∈ ∈ ≤ ≤ ··· ∞
4
is considered as a functional of χ(x) = (χ (x),...,χ (x)), where χ (x) eθmfm(x) 1,1
1 M m
≡ − ≤
m M,x R ;
+
≤ ∈
M M
ξN[χ] dx(m)pξN t ,ξ(1);...;t ,ξ(M) exp θ f (x)ξ(m)(dx)
Gν ≡ ν 1 N M N m m N
RNM ( R )
Z + mY=1 (cid:16) (cid:17) mX=1 Z
M
= EξN exp θ f (x)Ξ(ν)(t ,dx) . (1.12)
m m m
" ( R )#
m=1 Z
X
It is the generating function of multitime correlation functions, since if we expand it with
respect to χ (x(m))’s, ρξN’s appear as coefficients in terms;
m i ν
N N M
1
ξN[χ] = dx(m)
Gν ··· Nm! RNm Nm
NX1=0 NXM=0mY=1 Z +
M Nm
χ (x(m))ρξN t ,x(1);...;t ,x(M) . (1.13)
× m i ν 1 N1 M NM
mY=1Yi=1 (cid:16) (cid:17)
In the present paper, first we prove that, for any fixed initial configuration ξ M+
N
∈
with ξ (R ) = N N, there is a function KξN(s,x;t,y), which is continuous with respect
N + ∈ ν
to (x,y) (0, )2 for any fixed (s,t) [0, )2, and that the function (1.12) is given by the
∈ ∞ ∈ ∞
Fredholm determinant in the form
ξN[χ] = Det δ δ(x y)+KξN(t ,x;t ,y)χ (y) . (1.14)
Gν 1 m,n M " mn − ν m n n #
(x,≤y) (0≤, )2
∈ ∞
By definition of Fredholm determinant (see Eq.(4.8) in Section 4.1), (1.14) means that any
multitime correlation function is given by a determinant
ρξN t ,x(1);...;t ,x(M) = det KξN(t ,x(m);t ,x(n)) . (1.15)
ν 1 N1 M NM 1 i Nm,1 j Nn" ν m i n j #
(cid:16) (cid:17) ≤≤1 m,n≤M≤
≤ ≤
The function KξN is called the correlation kernel and it determines the finite dimensional
ν
distributions of the process (Ξ(ν)(t),PξN) through (1.15). It is an extension of determinantal
ν
(Fermion) point process of distributions studied by Soshnikov [26] and Shirai and Takahashi
[25] to the cases on T R with T = t ,...,t , M N,0 < t < < t < . See [12]
+ 1 M 1 M
× { } ∈ ··· ∞
for variety of examples of determinantal point processes. We express this result by simply
saying that the noncolliding BESQ(ν) is determinantal with a correlation kernel KξN for any
ν
ξ M+ with ξ (R ) = N N (Theorem 2.1).
N N +
∈ ∈
Next we consider the infinite-particle limits. For ξ M+ with ξ(R ) = , when
+
∈ ∞
Kνξ∩[0,L] converges to a continuous function as L → ∞, the limit is written as Kξν. If
sup Kξ [0,L](s,x;t,y) < , L > 0 for any (s,t) (0, )2 and any compact interval
x,y∈I | ∩ | ∞ ∀ ∈ ∞
I (0, ), we can obtain the convergence of generating functions of multitime correlation
⊂ ∞
5
functions, Gνξ∩[0,L][χ] → Gνξ[χ], as L → ∞. It implies Pνξ∩[0,L] → ∃Pξν as L → ∞ in the sense
of finite dimensional distributions weakly in the vague topology. In this case, we say that
the noncolliding BESQ(ν) (Ξ(ν)(t),Pξ) with an infinite number of particles ξ(R ) = is
ν + ∞
well defined with the correlation kernel Kξ [19]. We will give sufficient conditions so that
the process (Ξ(ν)(t),Pξ) is well defined, in which the correlation kernel is generally expressed
ν
using a double integral of an entire function represented by the Weierstrass canonical prod-
uct having zeros on supp ξ, where supp ξ = x R : ξ( x ) > 0 (Theorem 2.2). As an
{ ∈ { } }
application of this theorem, we will study the following example of infinite particle system,
which is a non-equilibrium dynamics exhibiting a relaxation phenomenon.
Consider the Bessel function
∞ ( 1)n z 2n+ν
J (z) = − . (1.16)
ν
Γ(n+1)Γ(n+1+ν) 2
Xn=0 (cid:16) (cid:17)
It is an analytic function of z in a cut plane. The function J (z)/zν is an entire function. As
ν
usual we define zν to be exp(νlogz), where the argument of z is given its principal value;
zν = exp ν log z +√ 1arg(z) , π < arg(z) π. (1.17)
| | − − ≤
h n oi
The function J (z) is analytically continued outside this range of arg(z) so that the relation
ν
J (emπ√ 1z) = eνmπ√ 1J (z) (1.18)
ν − − ν
holds [29]. If ν > 1, J (z) has an infinite number of pairs of positive and negative zeros
ν
−
with the same absolute value, which are all simple. We write the positive zeros of J (z)
ν
arranged in ascending order of the absolute value as
0 < j < j < j < .
ν,1 ν,2 ν,3
···
Explicitly J (z) is expressed using the infinite product of the Weierstrass primary factors of
ν
genus zero as (see Chapter XV of [29]),
(z/2)ν ∞ z2
J (z) = 1 . (1.19)
ν Γ(ν +1) − j2
i=1(cid:18) ν,i(cid:19)
Y
The configuration in which every point of the square of positive zero of J (z) is occupied by
ν
one particle, denoted by
∞
2
ξJhνi(·) = δjν2,i(·), (1.20)
i=1
X
satisfies the conditions of Theorem 2.2. We will determine the correlation kernel of the
noncolliding BESQ(ν) starting from ξh2i explicitly (Theorem 2.3 (i)) and prove that the
Jν
process shows a relaxation phenomenon to a stationary process,
(Ξ(ν)(t+θ),PξJh2νi) (Ξ(ν)(t),P ) as θ
ν → Jν → ∞
6
weakly in the sense of finite dimensional distributions (Theorem 2.3 (ii)). Here (Ξ(ν)(t),P )
Jν
is the equilibrium dynamics, which is determinantal with the correlation kernel
1
due 2u(s t)J (2√ux)J (2√uy) if s < t
− − ν ν
 Z0


 J (2√x)√yJ (2√y) √xJ (2√x)J (2√y)
KJν(t−s,y|x) =  ν ν′ x− y ν′ ν if t = s (1.21)

 −
 ∞due 2u(s t)J (2√ux)J (2√uy) if s > t,
 −Z1 − − ν ν


(x,y) (0, )2, whereJ (z) = dJ(z)/dz. Note that this kernel is temporally homogeneous
′
∈ ∞
but spatially inhomogeneous. Let µ be the determinantal (Fermion) point process on R ,
Jν +
in which, for any N N, N-point correlation function is given by [9]
∈
ρ (x ) = det K (x x ) , x = (x ,...,x ) (0, )N,
ν N 1 i,j N Jν i| j N 1 N ∈ ∞
≤ ≤
h i
with the Bessel kernel
K (y x) K (0,y x)
Jν | ≡ Jν |
J (2√x)√yJ (2√y) √xJ (2√x)J (2√y)
= ν ν′ − ν′ ν
x y
−
√xJ (2√x)J (2√y) J (2√x)√yJ (2√y)
ν+1 ν ν ν+1
= − . (1.22)
x y
−
In particular, the density of particle at x (0, ) is given by
∈ ∞
ρ (x) = K (x x) limK (y x)
ν Jν | ≡ y x Jν |
→
ν2
= (J (2√x))2 + 1 (J (2√x))2.
ν′ − 4x ν
(cid:18) (cid:19)
= (J (2√x))2 J (2√x)J (2√x). (1.23)
ν ν+1 ν 1
− −
The probability measure µ is obtained in an N limit called the hard-edge scaling-
Jν → ∞
limit of the distribution of squares of eigenvalues of random matrices in the chiral Gaussian
unitary ensemble [23, 9]. The process (Ξ(ν)(t),P ) is a reversible process with respect to
Jν
µ . The correlation kernel (1.21) is called the extended Bessel kernel [10, 27].
Jν
In the random matrix theory, three kinds of determinantal point processes of infinite
particle systems have been well studied, the correlation kernels of which are given by (i) the
sine kernel, (ii) the Airy kernel, and (iii) the Bessel kernel [23, 9]. They are obtained by
taking (i) the bulk, (ii) the soft-edge, and (iii) the hard-edge scaling limits in the Gaussian
unitary ensemble (GUE) for (i) and (ii) and in the chiral GUE for (iii), respectively. These
three determinantal point processes have been extended to time-dependent versions so that
7
they describe equilibrium dynamics, which are reversible with respect to the determinantal
point processes [17]. Corresponding to these three stationary processes, the present authors
introduced three relaxation processes with infinite numbers of particles realized in (i) the
Dyson model (the noncolliding Brownian motion) starting from Z (i.e. the zeros of sin(πz))
in [19], (ii) the Dyson model with drift terms starting from the Airy zeros in [18], and (iii)
the noncolliding BESQ(ν) starting from the squares of positive zeros of J in the present
ν
paper. The scaling limits are performed by increasing the number of particles in the system
N , while in our setting determinantal processes with infinite numbers of particles
→ ∞
N = are well constructed based on the theory of entire functions. In our relaxation
∞
processes in non-equilibrium we only have to wait for sufficiently long time to observe the
three determinantal point processes. In order to give temporally inhomogeneous correlation
kernels explicitly, we have reported the relaxation processes with the special initial config-
urations. The universality of the three determinantal point processes in a wide variety of
fields of mathematics, physics, and others [23, 9] implies robustness of relaxation phenomena
with respect to initial configurations. Mathematical justification of this fact will be reported
in the future.
The present paper is organized as follows. In Section 2 preliminaries and main results are
given. In Section 3 the properties of special functions used in this paper are given. Section
4 is devoted to proofs of results.
2 Preliminaries and Main Results
We introduce the following operations; for ξ( ) = δ ( ) M,
· i Λ xi · ∈
∈
(shift) with u R, τ ξ( ) = δ ( ), P
∈ u · xi+u ·
i Λ
X∈
(dilatation) with c > 0, c ξ( ) = δ ( ),
◦ · cxi ·
i Λ
X∈
(square) ξ 2 ( ) = δ ( ),
h i · x2i ·
i Λ
X∈
and for ξ( ) = δ ( ) M+,
· i Λ xi · ∈
∈
P
(square root) ξ 1/2 ( ) = δ +δ ( ) .
h i · √xi −√xi ·
Xi∈Λ (cid:16) (cid:17)
Note that the notation (1.20) states that this configuration is obtained as the square of
the point-mass distribution on the positive zeros of Jν(z) denoted by ξJν(·) = ∞i=1δjν,i(·).
We use the convention such that
P
f(x) = exp ξ(dx)logf(x) = f(x)ξ( x )
{ }
R
x ξ (cid:26)Z (cid:27) x supp ξ
Y∈ ∈Y
8
for ξ M and a function f on R. For a multivariate symmetric function g we write g((x) )
x ξ
∈ ∈
for g((x ) ).
i i Λ
The tr∈ansition probability density of BESQ(ν), given by (1.4), satisfies the Chapman-
Kolmogorov equation
∞dyp(ν)(t s,z y)p(ν)(s,y x) = p(ν)(t,z x), (2.1)
− | | |
Z0
for 0 s t,x,z R . Here we define the modified Bessel function of the first kind on C
+
≤ ≤ ∈
as
e νπ√ 1/2J (eπ√ 1/2z), π < arg(z) π/2,
I (z) = − − ν − − ≤ (2.2)
ν e3νπ√ 1/2J (e 3π√ 1/2z), π/2 < arg(z) π,
(cid:26) − ν − − ≤
where J is defined by (1.16) so that (1.18) holds [2]. This definition is consistent with (1.5),
ν
associated with the relation
I (emπ√ 1z) = eνmπ√ 1I (z). (2.3)
ν − − ν
For t R, we define
∈
1 y ν/2 x+y √xy
exp I , t R 0 ,x C 0 ,
ν
2 t x − 2t t ∈ \{ } ∈ \{ }
 | | (cid:16) (cid:17)yν (cid:18) (cid:19) (cid:18) (cid:19)
p(ν)(t,y|x) =  (2 t )ν+1Γ(ν +1)e−y/2t, t ∈ R\{0},x = 0,
 | |
δ(y x), t = 0,x C,

 − ∈
 (2.4)


y C. Then by using the modified version of Weber’s integral of the Bessel functions
∈
(see Eqs.(3.1) and (3.2) in Section 3.1), (2.1) is extended to the following equations. For
0 s t,x,z C,
≤ ≤ ∈
∞dyp(ν)( t,z y)p(ν)(t s,y x) = p(ν)( s,z x), (2.5)
− | − | − |
Z0
∞dyp(ν)(t s,z y)p(ν)( t,y x) = p(ν)( s,z x). (2.6)
− | − | − |
Z0
We define
ν/2
x
p(ν)(t,y x), t R,x C 0 ,
y | ∈ ∈ \{ }
pJν(t,y|x) =  (cid:18) (cid:19) (2.7)

 y ν/2p(ν)(t,y 0), t R,x = 0,
−
| ∈
y C. When t 0,x,y R , it has the expression
+
∈ ≥ ∈
p (t,y x) = ∞duJ (2√ux)J (2√uy)e 2ut
Jν | ν ν −
Z0
= 2 ∞dwwJ (2w√x)J (2w√y)e 2w2t. (2.8)
ν ν −
Z0
9
The Chapman-Kolmogorov equation (2.1) and its extensions (2.5) and (2.6) are mapped to
∞
dyp (t s,z y)p (s,y x) = p (t,z x), (2.9)
Jν − | Jν | Jν |
Z0
∞
dyp ( t,z y)p (t s,y x) = p ( s,z x), (2.10)
Jν − | Jν − | Jν − |
Z0
∞
dyp (t s,z y)p ( t,y x) = p ( s,z x) (2.11)
Jν − | Jν − | Jν − |
Z0
for 0 s t,x,z C.
≤ ≤ ∈
For ξ M+ with ξ (R ) = N N, we define the functions of z C,
N N +
∈ ∈ ∈
z
Π (ξ ,z) = 1 , (2.12)
0 N
− x
x∈ξYN∩{0}c(cid:16) (cid:17)
Π(ν)(ξ ,z) = zν/2Π (ξ ,z). (2.13)
N 0 N
For a C we also define
∈
z a
Φ (ξ ,a,z) = Π (τ ξ ,z a) = 1 − , (2.14)
0 N 0 a N
− − − x a
x∈ξNY∩{a}c(cid:18) − (cid:19)
z ν/2
Φ (ξ ,a,z), if a = 0,
0 N
Φ(ν)(ξ ,a,z) = a 6 (2.15)
N  (cid:16) (cid:17)

 Π(ν)(ξ ,z), if a = 0.
N
Theorem 2.1 (i) Forany fixed configuration ξ M+ with ξ (R ) = N N, (Ξ(ν)(t),PξN)
N ∈ N + ∈ ν
is determinantal with the correlation kernel
1 ε 1
KξN(s,x;t,y) = lim − dy dzp(ν)(s,x z) Φ (ξ ,z,y )p(ν)( t,y y)
ν ε↓0 2π√−1 Z−∞ ′IΓy′(ξN) | y′ −z 0 N ′ − ′|
1(s > t)p(ν)(s t,x y), (s,t) [0, )2,(x,y) (0, )2, (2.16)
− − | ∈ ∞ ∈ ∞
where Γ (ξ ) denotes a counterclockwise contour on the complex plane C encircling the
y′ N
points in supp ξ on R but not the point y ( , ε], and 1(ω) is the indicator function
N + ′
∈ −∞ −
of condition ω.
(ii) If ξ M+ with ξ (R ) = N N, the correlation kernel is given by
N ∈ 0 N + ∈
0
KξN(s,x;t,y) = ∞ξ (dx) dy p(ν)(s,x x)Φ (ξ ,x,y )p(ν)( t,y y)
ν N ′ ′ | ′ 0 N ′ ′ − ′|
Z0 Z
−∞
1(s > t)p(ν)(s t,x y), (s,t) [0, )2,(x,y) (0, )2. (2.17)
− − | ∈ ∞ ∈ ∞
Without changing any finite dimensional distributions of the process, the correlation kernel
(2.17) can be replaced by
0
KξN(s,x;t,y) = ∞ξ (dx) dy p (s,x x)Φ(ν)(ξ ,x,y )p ( t,y y)
Jν N ′ ′ Jν | ′ N ′ ′ Jν − ′|
Z0 Z
−∞
1(s > t)p (s t,x y), (s,t) [0, )2,(x,y) (0, )2. (2.18)
− Jν − | ∈ ∞ ∈ ∞
10