# Elementary symmetric polynomials and martingales for Heckman-Opdam processes

Margit Rösler  Institut für Mathematik, Universität Paderborn, Warburger Str.
100, D-33102 Paderborn, Germany<EMAIL_ADDRESS>and  Michael Voit
Fakultät Mathematik, Technische Universität Dortmund, Vogelpothsweg 87,
D-44221 Dortmund, Germany<EMAIL_ADDRESS>
###### Abstract.

We consider the generators $L_{k}$ of Heckman-Opdam diffusion processes in the
compact and non-compact case in $N$ dimensions for root systems of type $A$
and $B$, with a multiplicity function of the form $k=\kappa k_{0}$ with some
fixed value $k_{0}$ and a varying constant $\kappa\in\,[0,\infty[$. Using
elementary symmetric functions, we present polynomials which are simultaneous
eigenfunctions of the $L_{k}$ for all $\kappa\in\,]0,\infty[$. This leads to
martingales associated with the Heckman-Opdam diffusions
$(X_{t,1},\ldots,X_{t,N})_{t\geq 0}$. As our results extend to the freezing
case $\kappa=\infty$ with a deterministic limit after some renormalization, we
find formulas for the expectations $\mathbb{E}(\prod_{j=1}^{N}(y-X_{t,j})),$
$y\in\mathbb{C}$.

###### 2010 Mathematics Subject Classification:

60J60, 33C67, 82C23, 60B20

##  1\. Introduction

In the theory of classical random matrix ensembles there exist several
formulas regarding determinants as follows: Let $X$ be a random variable with
values in some space of $N\times N$-matrices over
$\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H}$. Then the expectations
$\mathbb{E}(\det(X-yI_{N}))$ are classical orthogonal polynomials of degree
$N$ in $y\in\mathbb{C}$. Such results for the Hermite, Laguerre, and Jacobi
ensembles can for example be found in [DG, FG, A], where the expectations
above can be expressed via classical Hermite, Laguerre and Jacobi polynomials
of degree $N$. These results were extended to the spectra of $\beta$-ensembles
associated with $N$-dimensional time-homogeneous diffusion processes in [KVW,
V] by using some martingales constructed from these diffusions via elementary
symmetric polynomials. The generators of the diffusions were Dunkl-Bessel
Laplacians and (symmetric) Heckman-Opdam Laplacians in the compact $BC$ case.

In the present paper we study corresponding results for Heckman-Opdam
Laplacians in the non-compact $BC$ case as well Heckman-Opdam Laplacians of
type $A$ in the compact and noncompact setting. Together with [KVW, V], the
present paper covers the most important examples related with Calogero-Moser-
Sutherland particle models. The basic ideas here are similar to those in [KVW,
V]; however, while the approach in [KVW, V] is mainly based on Ito calculus,
we now focus on an algebraic point of view.

The idea is as follows. Let $(X_{t})_{t\geq 0}$ be a time-homogeneous
diffusion on a suitable closed set $C\subset\mathbb{R}^{N}$ (such as a Weyl
chamber or a fundamental alcove), where the paths are reflected at the
boundary $\partial C$. Then the generator $L$ of the associated transition
semigroup is an elliptic partial differential operator whose domain is
contained in the space of functions on $\mathbb{R}^{N}$ which admit
corresponding symmetries on $\partial C$. We are now interested in functions
$f:[0,\infty[\times\mathbb{R}^{N}\to\mathbb{C}$ for which $(f(t,X_{t}))_{t\geq
0}$ is a martingale (w.r.t. the canonical filtration), which essentially means
that $f$ is $L$-space-time-harmonic, i.e., $(\frac{\partial}{\partial
t}+L)f=0$; see Section III.10 of [RW]. Examples of such harmonic functions can
be given in terms of eigenfunctions of $L$. For a general background in
stochastic analysis we also recommend [P].

In the framework of Heckman-Opdam theory (see [HS], [HO]), we fix some
crystallographic root system $R$ (with associated Weyl group $W$) in
$\mathbb{R}^{N}$ and choose $C$ as an associated Weyl chamber or fundamental
alcove. We consider a non-negative multiplicity function $k\geq 0$ on $R$ of
the form $k=\kappa\cdot k_{0}$ with some fixed multiplicity $k_{0}$ and a
constant $\kappa>0$. We now study the $W$-invariant Heckman-Opdam Laplace
operators $L_{\kappa}:=L_{\kappa k_{0}}$ as generators of diffusions (see
[Sch1, Sch2, RR1]), where the parameter $\kappa$ is varying. We also study the
renormalized generators $\widetilde{L}_{\kappa}:=\frac{1}{\kappa}L_{\kappa}$
of the renormalized diffusions $(\widetilde{X}_{t}:=X_{t/\kappa})_{t\geq 0}$.
In suitable coordinates, these renormalized generators then have the form

$\widetilde{L}_{\kappa}f=\frac{1}{\kappa}\Delta f+Hf$

with some second-order differential operator $\Delta$ (often a classical
Laplacian) and some first-order drift operator $H$, where both operators are
independent of $\kappa$. This also works for $\kappa=\infty$ where
$(\widetilde{X}_{t})_{t\geq 0}$ is the deterministic solution of some ODE
associated with $H$. We now use elementary symmetric polynomials to construct
simultanous eigenfunctions $f$ of $\widetilde{L}_{\kappa}$ for all
$\kappa\in]0,\infty]$. In the next step we use martingales associated with
these functions $f$ together with information in the deterministic case
$\kappa=\infty$ in order to derive formulas for

(1.1)
$\mathbb{E}(\prod_{j=1}^{N}(\widetilde{X}_{t,i}-y))\quad\quad(y\in\mathbb{C}).$

For some values of $\kappa$, the operators $L_{\kappa}$ admit an
interpretation as Laplace-Beltrami operators on symmetric spaces. In these
cases, (1.1) leads to determinantal formulas for Brownian motions on the
symmetric spaces which are closely related, for instance, to some identities
in [R]. Considering $t\to\infty$ in the compact cases, this also leads to
determinantal formulas for the uniform distributions on compact symmetric
spaces.

The paper is organized as follows: In Section 2 we first recapitulate some
well-known facts on Heckman-Opdam hypergeometric functions and polynomials,
and the associated Laplacians in the compact and non-compact setting. We there
also prove that in the non-compact crystallographic case, the Heckman-Opdam
processes admit arbitrary exponential moments for arbitrary deterministic
starting conditions. In the main part of the paper, we then concentrate on
specific root systems. Section 3 is devoted to the compact case of type $A$,
which is related to Calogero-Moser-Sutherland particle models on the torus.
Here, for instance, our results lead to determinantal formulas for Brownian
motions and the uniform distribution on the unitary groups $U(N)$ and $SU(N)$.
In Section 4 we study the non-compact case of type $A$. Section 5 then
contains the non-compact case of type $BC$.

##  2\. Heckman-Opdam theory

Here we briefly collect some facts from Heckman-Opdam theory; see [HS, HO] for
a general background.

Let $(\mathfrak{a},\langle\,.\,,\,.\,\rangle)$ be a Euclidean space of
dimension $N$ with norm $|x|=\sqrt{\langle x,x\rangle}.$ We identify
$\mathfrak{a}$ with its dual space via the given scalar product. Let $R$ be a
crystallographic, possibly not reduced root system in $\mathfrak{a}$ with
associated finite reflection group $W$. Thus in particular, $R$ spans
$\mathfrak{a}.$ We fix a positive subsystem $R_{+}\subset R$ and a
$W$-invariant multiplicity $k:R\to[0,\infty[$. The Cherednik operators
associated with $R_{+}$ and $k$ are defined as

(2.1) $D_{\xi}(k)f(x)=\partial_{\xi}f(x)+\sum_{\alpha\in
R_{+}}\frac{k_{\alpha}\langle\alpha,\xi\rangle}{1-e^{-\langle\alpha,x\rangle}}(f(x)-f(\sigma_{\alpha}(x))-\langle\rho(k),\xi\rangle
f(x)$

for $\xi\in\mathfrak{a},$ with the (weighted) Weyl vector
$\rho(k):=\frac{1}{2}\sum_{\alpha\in R_{+}}k_{\alpha}\alpha$. The operators
$D_{\xi}(k),\,\xi\in\mathfrak{a}\,$ commute, and there is a $W$-invariant
tubular neighbourhood $U$ of $\mathfrak{a}$ in
$\mathfrak{a}_{\mathbb{C}}=\mathfrak{a}+i\mathfrak{a}$ and a unique analytic
function $(\lambda,z)\mapsto G(\lambda,k;z)$ on
$\mathfrak{a}_{\mathbb{C}}\times U$, the so called Opdam-Cherednik kernel,
which satisfies

(2.2) $G(\lambda,k;0)=1\quad\text{ and}\quad
D_{\xi}(k)G(\lambda,k;\,.\,)=\langle\lambda,\xi\rangle\,G(\lambda,k;\,.\,)\quad\text{
for all }\,\xi\in\mathfrak{a}.$

The hypergeometric function associated with $R$ is defined by

(2.3) $F(\lambda,k;z)=\frac{1}{|W|}\sum_{w\in W}G(\lambda,k;w^{-1}z).$

It is $W$-invariant in $\lambda$ and $z.$ To introduce the associated Heckman-
Opdam polynomials, we need the weight lattice and the set of dominant weights,

$P=\\{\lambda\in\mathfrak{a}:\langle\lambda,\alpha^{\vee}\rangle\in\mathbb{Z}\>\>\forall\alpha\in
R\,\\},\quad P_{+}=\\{\lambda\in P:\langle\lambda,\alpha^{\vee}\rangle\geq
0\,\,\forall\alpha\in R_{+}\,\\}\supset R_{+},$

where $\alpha^{\vee}=\frac{2\alpha}{\langle\alpha,\alpha\rangle}.$ $P_{+}$
carries the usual dominance order. Let

$\mathcal{T}:=\text{span}_{\mathbb{C}}\\{e^{i\lambda},\,\lambda\in P\\}$

be the space of trigonometric polynomials associated with $R$. The orbit sums

$M_{\lambda}=\sum_{\mu\in W\\!\lambda}e^{i\mu}\,,\quad\lambda\in P_{+}$

form a basis of the subspace $\mathcal{T}^{W}$ of $W$-invariant polynomials in
$\mathcal{T}$. For
$Q^{\vee}:=\text{span}_{\mathbb{Z}}\\{\alpha^{\vee},\,\alpha\in R\\}$,
consider the compact torus $\,T=\mathfrak{a}/2\pi Q^{\vee}$ with the weight
function

(2.4) $\delta_{k}(x):=\prod_{\alpha\in
R_{+}}\Bigl{|}\sin\Bigr{(}\frac{\langle\alpha,x\rangle}{2}\Bigr{)}\Bigr{|}^{2k_{\alpha}}.$

The Heckman-Opdam polynomials associated with $R_{+}$ and $k$ are given by

$P_{\lambda}(k;z)=M_{\lambda}(z)+\sum_{\nu<\lambda}c_{\lambda\nu}(k)M_{\nu}(z)\quad(\lambda\in
P_{+}\,,z\in\mathfrak{a}_{\mathbb{C}})$

where the coefficients $c_{\lambda\nu}(k)\in\mathbb{R}$ are uniquely
determined by the condition that $P_{\lambda}(k;\,.\,)$ is orthogonal to
$M_{\nu}$ in $L^{2}(T,\delta_{k})$ for all $\nu\in P_{+}$ with $\nu<\lambda$.
It is known that $\\{P_{\lambda}(k,\,.\,),\lambda\in P_{+}\,\\}$ is an
orthonormal basis of $L^{2}(T,\delta_{k})^{W}$ of all $W$-invariant functions
from $L^{2}(T,\delta_{k})$. According to [HS], the normalized polynomials

$R_{\lambda}(k,z):=P_{\lambda}(k;z)/P_{\lambda}(k;0)$

can be expressed in terms of the hypergeometic function as

(2.5) $R_{\lambda}(k,z)=F(\lambda+\rho(k),k;iz).$

Note that our notion slightly differs from [HS, HO], where the polynomials
$P_{\lambda}$ are defined as exponential polynomials on the torus
$i\mathfrak{a}/2\pi iQ^{\vee}$. We next introduce the Heckman-Opdam Laplacian

$\Delta_{k}:=\sum_{j=1}^{N}D_{\xi_{j}}(k)^{2}\,-\,|\rho(k)|^{2}$

with some orthonormal basis $\xi_{1},\ldots,\xi_{N}$ of $\mathfrak{a}$. The
operator $\Delta_{k}$ is independent of the choice of this basis. Denote by
$L_{k}$ the restriction of $\Delta_{k}$ to $W$-invariant functions. According
to [Sch2],

(2.6) $L_{k}f(x)=\Delta f(x)+\sum_{\alpha\in
R_{+}}k_{\alpha}\coth\Bigl{(}\frac{\langle\alpha,x\rangle}{2}\Bigr{)}\cdot\partial_{\alpha}f(x).$

We notice that by construction, for all $\lambda\in\mathfrak{a}_{\mathbb{C}}$,
the hypergeometric functions

$F_{\lambda}:=F(\lambda,k;\,.\,)$

are eigenfunctions of $L_{k}$ with eigenvalues

$\sum_{j=1}^{N}\langle\lambda,\xi_{j}\rangle^{2}-|\rho(k)|^{2}.$

The operator $L_{k}$ is independent of the choice of $R_{+}$ and generates a
Feller diffusion on the closed Weyl chamber
$\overline{\mathfrak{a}_{+}}\subset\mathfrak{a}$ associated with $R_{+}$
(called a radial Heckman-Opdam process), where the paths are reflected at the
boundaries, see [Sch1, Sch2]. The transition probabilities of this process,
with starting point $y\in\overline{\mathfrak{a}_{+}}$, are given by

$p_{t}^{W}\\!(x,y)d\mu(x)$

with the $W$-invariant heat kernel

$p_{t}^{W}\\!(x,y)=\int_{i\mathfrak{a}}e^{-\frac{1}{2}(|\lambda|^{2}+|\rho|^{2})}F_{\lambda}(x)F_{\lambda}(-y)d\nu^{\prime}(\lambda),$

where $\rho=\rho(k),$ $\nu^{\prime}$ is the symmetric Plancherel measure as in
[Sch2], and

$d\mu(x)=d\mu_{k}(x)=\prod_{\alpha\in
R_{+}}\big{|}2\sinh\langle\frac{\alpha}{2},x\rangle\big{|}^{2k_{\alpha}}\,dx.$

In the compact case we take the factor $i$ in (2.5) into account as in [RR1]
and consider the operator

(2.7) $\widehat{L}_{k}f(x):=\Delta f(x)+\sum_{\alpha\in
R_{+}}k_{\alpha}\cot\Bigl{(}\frac{\langle\alpha,x\rangle}{2}\Bigr{)}\cdot\partial_{\alpha}f(x).$

$\widehat{L}_{k}$ generates a Feller diffusion on a compact fundamental alcove
of the affine Weyl group $W_{\\!\mathit{aff}}=W\ltimes 2\pi Q^{\vee}$ in
$\mathfrak{a}$, again with reflecting boundaries. The trigonometric
polynomials $P_{\lambda}$ ($\lambda\in P_{+}$) are eigenfunctions of
$\widehat{L}_{k}$ with eigenvalues

$-\langle\lambda,\lambda+2\rho(k)\rangle\leq 0.$

They were used in [RR1] to construct the transition densities of the
diffusions with generators $\widehat{L}_{k}$. For root systems of type $BC$,
the $P_{\lambda}$ are multivariate Jacobi polynomials, and the associated
Feller diffusions are multivariate Jacobi processes which were studied in
[Dem].

Before turning to details for root systems of type $A$ and $BC$, we conclude
this section with an integrability result which assures the existence of
exponential moments for the radial Heckman-Opdam processes in the non-compact,
crystallographic case. This will be needed in Sections 4 and 5.

######  Lemma 2.1.

For each $y\in\mathfrak{a}$ and $\,\beta\in\mathfrak{a},$

(2.8)
$\int_{\mathfrak{a}}e^{\langle\beta,x\rangle}\,p_{t}^{W}\\!(x,y)d\mu(x)\,<\infty.$

.

###### Proof.

For $y=0$ this is obvious from Theorem 5.2 of [Sch2]. For general $y$, we
employ the $L^{p}$-theory for the hypergeometric transform developed in [NPP].
For suitable functions on $\mathfrak{a}$ and on $i\mathfrak{a}$ respectively,
the hypergeometric transform and the inverse hypergeometric transform are
given by

$\mathcal{H}f(\lambda)=\int_{\mathfrak{a}}f(x)F_{\lambda}(-x)d\mu(x),\quad\mathcal{I}(g)(x)=\int_{i\mathfrak{a}}g(\lambda)F_{\lambda}(x)d\nu^{\prime}(\lambda).$

We denote by $\mathbb{C}[\mathfrak{a}_{\mathbb{C}}]$ the space of polynomial
functions on $\mathfrak{a}_{\mathbb{C}}$ and by $\partial(q)$ the constant
coefficient differential operator associated with
$q\in\mathbb{C}[\mathfrak{a}_{\mathbb{C}}].$ Moreover, for $x\in\mathfrak{a}$
we denote by $C(x):=co(W.x)$ the convex hull of the $W$-orbit of $x$ in
$\mathfrak{a}.$ For an exponent $0<p\leq 2$, the $W$-invariant
$L^{p}$-Schwartz space is given by

$\mathcal{C}^{p}(\mathfrak{a})^{W}=\Big{\\{}f\in
C^{\infty}(\mathfrak{a})^{W}:\,\sup_{x\in\mathfrak{a}}\,\frac{(1+|x|)^{n}}{F_{0}(x)^{2/p}}\,\big{|}\partial(q)f(x)\big{|}\,<\infty\,\,\forall\,n\in\mathbb{N}_{0},\,q\in\mathbb{C}[\mathfrak{a}_{\mathbb{C}}]\Big{\\}}.$

Moreover, we consider the $W$-invariant Schwartz space
$\mathcal{S}(\mathfrak{a}_{\epsilon_{p}})^{W}$ with
$\epsilon_{p}=\frac{2}{p}-1,$ which consists of all $W$-invariant continuous
functions on the closed tube
$\mathfrak{a}_{\epsilon_{p}}=C(\epsilon_{p}\rho)+i\mathfrak{a}\subset\mathfrak{a}_{\mathbb{C}}$
which are holomorphic in its interior and satisfy

(2.9)
$\sup_{\lambda\in\mathfrak{a}_{\epsilon_{p}}}\,(1+|\lambda|)^{n}\big{|}\partial(q)g(\lambda)\big{|}\,<\infty\quad\text{for
all }n\in\mathbb{N}_{0},\,q\in\mathbb{C}[\mathfrak{a}_{\mathbb{C}}].$

By Theorem 5.6 of [NPP], the hypergeometric transform $\mathcal{H}$ is a
topological isomorphism from $\mathcal{C}^{p}(\mathfrak{a})^{W}$ onto
$\mathcal{S}(\mathfrak{a}_{\epsilon_{p}})^{W}$ with inverse $\mathcal{I}$.

We claim that for fixed $y\in\mathfrak{a},$ the function

$g(\lambda):=e^{\frac{1}{2}\langle\lambda,\lambda\rangle}F_{\lambda}(-y)$

belongs to $\mathcal{S}(\mathfrak{a}_{\epsilon_{p}})$ for each $p\in]0,2];$
here $\langle\,.\,,\,.\,\rangle$ denotes the bilinear extension of the given
scalar product to $\mathfrak{a}_{\mathbb{C}}.$ As soon as this is proven, it
will follow that
$\,p_{t}^{W}(\,.\,,y)=e^{-\frac{1}{2}|\rho|^{2}}\,\mathcal{I}(g)\,$ belongs to
$\mathcal{C}^{p}(\mathfrak{a})^{W}$ for each $p\in]0,2].$ In order to check
that $g\in\mathcal{S}(\mathfrak{a}_{\epsilon_{p}}),$ we only have to verify
the growth condition (2.9). Let $q\in\mathbb{C}[\mathfrak{a}_{\mathbb{C}}].$
Then in view of Theorem 3.4 (and Remark 3.2) of [Sch2],

$|\partial_{\lambda}(q)F_{\lambda}(-y)|\,\leq C(1+|y|)^{\deg q}F_{0}(-y)\cdot
e^{\max_{w\in W}\text{Re}\langle
w\lambda,-y\rangle}\quad(\lambda\in\mathfrak{a}_{\mathbb{C}}).$

This shows that $\partial_{\lambda}(q)F_{\lambda}(-y)$ is bounded as a
function of $\lambda$ on $\mathfrak{a}_{\epsilon_{p}}$. Moreover,
$\partial(q)e^{\frac{1}{2}\langle\lambda,\lambda\rangle}=\widetilde{q}(\lambda)e^{\frac{1}{2}\langle\lambda,\lambda\rangle}$
with some polynomial $\widetilde{q}$, and

$\big{|}e^{\frac{1}{2}\langle\lambda,\lambda\rangle}\big{|}\,\asymp\,e^{-\frac{1}{2}|\lambda|^{2}}\quad\text{on
}\mathfrak{a}_{\epsilon_{p}}.$

Therefore $\partial(q)g(\lambda)$ decays exponentially as $|\lambda|\to\infty$
within $\mathfrak{a}_{\epsilon_{p}}.$ It follows that
$g\in\mathcal{S}(\mathfrak{a}_{\epsilon_{p}})$ and thus
$\,p_{t}^{W}\\!(\,.\,,y)\in\mathcal{C}^{p}(\mathfrak{a})^{W}.$ In particular,
for each $p\in]0,2]$ there exists a constant $C_{p}>0$ such that

$p_{t}^{W}\\!(x,y)\leq C_{p}F_{0}(x)^{2/p}\quad\text{ for all
}x\in\mathfrak{a}.$

From [Sch2] we know that in the closed chamber
$\overline{\mathfrak{a}_{+}}\,,$ $\,F_{0}(x)\asymp
q_{0}(x)e^{-\langle\rho,x\rangle}\,$ with a certain positive polynomial
$q_{0}$. Hence there exists a nonnegative polynoimal $q_{p}$ (depending on
$p$), such that

$p_{t}^{W}\\!(x,y)\leq
q_{p}(x)e^{-\frac{2}{p}\langle\rho,x\rangle}\quad\text{for all
}x\in\overline{\mathfrak{a}_{+}}.$

Now fix $\beta\in\mathfrak{a}$ and note that $\rho$ is contained in the open
chamber $\mathfrak{a}_{+}.$ Choosing $p>0$ small enough, we therefore obtain

$\int_{\overline{\mathfrak{a}_{+}}}e^{\langle\beta,x\rangle}p_{t}^{W}\\!(x,y)d\mu(x)\,\leq\int_{\overline{\mathfrak{a}_{+}}}q_{p}(x)e^{\langle\beta,x\rangle-\frac{2}{p}\langle\rho,x\rangle+2\langle\rho,x\rangle}dx\,<\infty.$

This yields the assertion. ∎

##  3\. The compact case of type $A_{N-1}$

In this section we study Heckman-Opdam processes of type $A_{N-1}$ in the
compact setting. The generators of these processes are the Hamiltonians of
interacting particle models of Calogero-Sutherland type with $N$ particles on
the torus $\mathbb{T}:=\\{z\in\mathbb{C}:\>|z|=1\\}$; see [LV] for the
background. These processes are diffusions on some fundamental domain of
$W=S_{N}$ in $\mathbb{T}^{N}$. It will however be convenient to consider also
associated diffusions on $\mathbb{R}^{N}$ with $2\pi$-periodicity such that
the diffusions on $\mathbb{T}^{N}$ appear as images under $x\mapsto e^{ix}.$
To introduce the processes on $\mathbb{R}^{N}$, we consider the root system
$R=A_{N-1}=\\{\pm(e_{i}-e_{j}):1\leq i<j\leq N\\}$ in $\mathbb{R}^{N}$ with
positive subsystem $R_{+}=\\{e_{i}-e_{j}:i<j\\}$ and fix a multiplicity
parameter $k\in\,]0,\infty[$. Let

$\omega:=(1,\ldots,1)^{T}\in\mathbb{R}^{N}.$

Then $Q^{\vee}=\mathbb{Z}^{N}\cap(\mathbb{R}\omega)^{\perp},$ and a
fundamental domain for the action of $W_{\\!\mathit{aff}}=W\ltimes 2\pi
Q^{\vee}$ in $\mathbb{R}^{N}$ is given by

$\displaystyle C_{N}$
$\displaystyle=\\{x\in\mathbb{R}^{N}:0\leq\langle\alpha,x\rangle\leq
2\pi\,\,\forall\alpha\in R_{+}\\}$
$\displaystyle=\,\\{x\in\mathbb{R}^{N}:\,x_{1}\leq x_{2}\leq\ldots\leq
x_{N}\leq x_{1}+2\pi\\}.$

We consider the $W$-invariant Heckman-Opdam Laplacian

(3.1) $\widehat{L}_{k}f(x)=\Delta f(x)+k\sum_{j=1}^{N}\sum_{l\neq
j}\cot\Bigl{(}\frac{x_{j}-x_{l}}{2}\Bigr{)}\frac{\partial}{\partial
x_{j}}f(x)$

with reflecting boundaries, i.e. with domain

$D(\widehat{L}_{k})=\\{f|_{C^{N}}:f\in C^{2}(\mathbb{R}^{N})\text{ invariant
under }W_{\\!\mathit{aff}}\\}.$

$\widehat{L}_{k}$ is the generator of a Feller semigroup of transition
operators on $C_{N}$, c.f. [RR1]. Associated Feller diffusions
$(X_{t,k})_{t\geq 0}$ with continuous paths (which are reflected at the
boundary of $C_{N}$) are called Heckman-Opdam processes of type $A_{N-1}$ on
$C_{N}$. We also consider the renormalized generators

$\widetilde{L}_{k}:=\frac{1}{k}\widehat{L}_{k}=\frac{1}{k}\Delta+\sum_{j=1}^{N}\sum_{l\neq
j}\cot\Bigl{(}\frac{x_{j}-x_{l}}{2}\Bigr{)}\frac{\partial}{\partial x_{j}}$

which degenerate for $k\to\infty$ into

$\widetilde{L}_{\infty}=\sum_{j=1}^{N}\sum_{l\neq
j}\cot\Bigl{(}\frac{x_{j}-x_{l}}{2}\Bigr{)}\frac{\partial}{\partial x_{j}}.$

For $k\in]0,\infty[$, the process
$\widetilde{X}_{k}:=(\widetilde{X}_{t,k})_{t\geq 0}$ with
$\widetilde{X}_{t,k}:=X_{t/k,k}$ is a Feller diffusion associated with
$\widetilde{L}_{k}$. It can be also described as solution of the SDE

(3.2) $d\widetilde{X}_{t,k,j}=\frac{\sqrt{2}}{\sqrt{k}}dB_{t,j}+\sum_{l\neq
j}\cot\Bigl{(}\frac{\widetilde{X}_{t,k,j}-\widetilde{X}_{t,k,l}}{2}\Bigr{)}dt\quad\quad(j=1,\ldots,N)$

with some $N$-dimensional Brownian motion $(B_{t,1},\ldots,B_{t,N})_{t\geq
0}$. Moreover, for deterministic starting conditions, $\widetilde{L}_{\infty}$
is the generator of a deterministic process whose paths are the solution of
some initial value problem for the ODE

(3.3) $\frac{dx_{j}}{dt}(t)=\sum_{l\neq
j}\cot\Bigl{(}\frac{x_{j}(t)-x_{l}(t)}{2}\Bigr{)}\quad\quad(j=1,\ldots,N).$

As for Dunkl processes in [AV], one can show that for initial data in the
interior of $C_{N}$, the solution $(\widetilde{X}_{t,\infty})_{t\geq 0}$ of
this ODE exists for all $t\geq 0$ in the interior of $C_{N}$. We also point
out that in the Dunkl setting, the ODE analogous to (3.2) has unique solutions
for all $t\geq 0$ even for starting points at the boundary of the chamber, see
[VW]. We expect that such a result is also true in the present setting.

We need the following stationary solutions of (3.3):

######  Lemma 3.1.

For each $x_{1}\in\mathbb{R}$,

$\Bigl{(}x_{1},x_{1}+\frac{1}{N}2\pi,x_{1}+\frac{2}{N}2\pi,\ldots,x_{1}+\frac{N-1}{N}2\pi\Bigr{)}\in
C_{N}$

is a stationary solution of (3.3).

###### Proof.

Assume first that $N$ is odd. As the cotangent is odd and $\pi$-periodic, we
have for $j=1,\ldots,N$ that

$\sum_{l\neq
j}\cot\Bigl{(}\frac{(j-l)\pi}{N}\Bigr{)}=\sum_{l=1}^{N-1}\cot\Bigl{(}\frac{l\pi}{N}\Bigr{)}=\sum_{l=1}^{(N-1)/2}\Bigl{(}\cot\Bigl{(}\frac{l\pi}{N}\Bigr{)}+\cot\Bigl{(}\frac{(N-l)\pi}{N}\Bigr{)}\Bigr{)}=0.$

If $N$ is even, then our computation leads to the additional term
$\cot((N\pi/2)/N)=0$ in the last sum and thus to the same result. This yields
the claim. ∎

The generator $\widetilde{L}_{k}$ and the associated diffusion
$\widetilde{X}_{k}$ on $\mathbb{R}^{N}$ can be decomposed into two independent
parts, namely the center of gravity and the process of the distances of
neighboring particles. This reflects the fact that the usual representation of
the symmetric group $S_{N}$ on $\mathbb{R}^{N}$ decomposes into two
irreducible components. More precisely, consider the center-of-gravity process
$\widetilde{X}_{k}^{\mathit{cg}}:=(\widetilde{X}_{t,k}^{\mathit{cg}})_{t\geq
0}$ with

$\widetilde{X}_{t,k}^{cg}:=\frac{1}{N}(\widetilde{X}_{t,k,1}+\ldots+\widetilde{X}_{t,k,N})\cdot\omega$

which is the orthogonal projection of $\widetilde{X}_{t,k}$ onto
$\mathbb{R}\omega$. Then the diffusion

$\widetilde{X}_{k}^{\mathit{diff}}:=\widetilde{X}_{k}-\widetilde{X}_{k}^{\mathit{cg}}$

lives on the orthogonal complement
$(\mathbb{R}\omega)^{\perp}\subset\mathbb{R}^{N}$.

######  Lemma 3.2.

Let $k\in\,]0,\infty[$. If $\widetilde{X}_{k}$ starts in some deterministic
point, then the processes $\widetilde{X}_{k}^{\mathit{diff}}$ and
$\widetilde{X}_{k}^{\mathit{cg}}$ are stochastically independent.

###### Proof.

By the SDE (3.2) we have

(3.4)
$d\widetilde{X}_{t,k,j}^{\mathit{cg}}=\frac{\sqrt{2}}{N\sqrt{k}}d\Bigl{(}\sum_{l=1}^{N}B_{t,l}\Bigr{)}=:\frac{\sqrt{2}}{\sqrt{Nk}}\,dB_{t}$

with some one-dimensional Brownian motion $(B_{t})_{t\geq 0}$ while
$\widetilde{X}_{k}^{\mathit{diff}}$ satisfies

$d\widetilde{X}_{t,k,j}^{\mathit{diff}}=\frac{\sqrt{2}}{\sqrt{k}}\,d\Bigl{(}B_{t,j}-\frac{1}{N}\sum_{l=1}^{N}B_{t,l}\Bigr{)}+F_{j}(\widetilde{X}_{t,k}^{\mathit{diff}})\,dt\quad(j=1,\ldots,N)$

where the processes $(B_{t,j}-\frac{1}{N}\sum_{l=1}^{N}B_{t,l})_{t\geq 0}\,$
are stochastically independent of $(B_{t})_{t\geq 0}$, and the $F_{j}$ are
concrete continuous functions. This implies the claim by the very definition
of solutions of SDEs. ∎

We next transfer all data from $\mathbb{R}^{N}$ to the torus $\mathbb{T}^{N}$
via $z_{j}:=e^{ix_{j}}$ for $j=1,\ldots,N$, or for short,
$z:=e^{ix}\in\mathbb{T}^{N}$. A short computation shows that in
$z$-coordinates, the operator $L_{k}$ is given by the diffusion operator

(3.5) $\displaystyle H_{k}=$
$\displaystyle-\sum_{j=1}^{N}\Bigl{(}z_{j}\frac{\partial}{\partial
z_{j}}\Bigr{)}^{2}-k\sum_{j=1}^{N}\sum_{l\neq
j}\frac{z_{j}+z_{l}}{z_{j}-z_{l}}\cdot z_{j}\frac{\partial}{\partial z_{j}}$
$\displaystyle=$
$\displaystyle-\sum_{j=1}^{N}z_{j}^{2}\frac{\partial^{2}}{\partial
z_{j}^{2}}-(1-k(N-1))\sum_{j=1}^{N}z_{j}\frac{\partial}{\partial
z_{j}}-2k\sum_{j=1}^{N}\sum_{l\neq
j}\frac{z_{j}^{2}}{z_{j}-z_{l}}\frac{\partial}{\partial z_{j}}\,,$

acting on permutation invariant functions from $C^{2}(\mathbb{T}^{N})$. The
operator $H_{k}$ appears in a prominent way in the particle models of
Calogero-Sutherland type on $\mathbb{T}$; see Section 2 of [LV]. It is
obtained from the Calogero-Sutherland Hamiltonian by conjugation with the
ground state. The operator $H_{k}$ is the generator of the Feller diffusions
$Z_{k}:=(Z_{t,k}:=e^{iX_{t,k}})_{t\geq 0}$ on the alcove

$\mathbb{A}_{N}:=\\{e^{ix}:\>x\in C_{N}\\}\subset\mathbb{T}^{N}.$

Further, the operators $\widetilde{H}_{k}:=\frac{1}{k}H_{k}$ generate the
diffusions
$\widetilde{Z}_{k}:=(\widetilde{Z}_{t,k}:=e^{i\widetilde{X}_{t,k}})_{t\geq 0}$
for $k\in]0,\infty[$. Clearly, this also works for $k=\infty$ where
$\widetilde{Z}_{\infty}$ is deterministic for deterministic initial data.

######  Remark 3.3.

Besides the generators in (3.5), also the operators

(3.6) $D_{k}:=\sum_{j=1}^{N}z_{j}^{2}\frac{\partial^{2}}{\partial
z_{j}^{2}}+2k\sum_{j=1}^{N}\sum_{l\neq
j}\frac{z_{j}^{2}}{z_{j}-z_{l}}\frac{\partial}{\partial z_{j}}$

appear in the literature; see e.g. [F, St, OO]. Here $-H_{k}=D_{k}+E_{k}\,$
with the Euler operator

(3.7) $E_{k}:=(1-k(N-1))\sum_{j=1}^{N}z_{j}\frac{\partial}{\partial z_{j}}$

which commutes with $D_{k}$. The $-D_{k}$ are generators of diffusions with
additional drift on $\mathbb{T}$ which rotates the complete system at some
fixed speed. Clearly, the subsequent results can be easily translated to
$-D_{k}$. From a stochastic point of view, the diffusions associated with the
operators $H_{k}$ seem to be the most natural ones.

We recall that the (symmetric) eigenfunctions of $H_{k}$ are Jack polynomials.
To become precise, we introduce the following notations. We write

$\Lambda_{N}^{+}=\\{\lambda\in\mathbb{Z}_{+}^{N}:\lambda_{1}\geq\cdots\geq\lambda_{N}\\}$

for the set of partitions of length at most $N$. Denote by
$C_{\lambda}^{\alpha}\,,\,\lambda\in\Lambda_{N}^{+},$ the Jack polynomials of
index $\alpha>0$ in $N$ variables with the normalization

(3.8)
$(z_{1}+\cdots+z_{N})^{m}=\sum_{|\lambda|=m}C_{\lambda}^{\alpha}(z)\quad(m\in\mathbb{Z}_{+});$

see [St, BF]. The $C_{\lambda}^{\alpha}$ are symmetric and homogeneous of
degree $|\lambda|:=\lambda_{1}+\cdots+\lambda_{N}$. Moreover, by [St, BF], the
$C_{\lambda}^{\alpha}$ with index $\alpha=1/k$ are eigenfunctions of $D_{k}$
with eigenvalues

$d_{\lambda}(k)=\sum_{j=1}^{N}\lambda_{j}(\lambda_{j}-1+2k(N-j)).$

In addition, as $C_{\lambda}^{\alpha}$ is homogeneous of degree $|\lambda|$,

$E_{k}C_{\lambda}^{\alpha}=(1-k(N-1))|\lambda|\,C_{\lambda}^{\alpha}.$

In summary we obtain:

######  Lemma 3.4.

For $\lambda\in\Lambda_{n}^{+}$, $k\in]0,\infty[$ and $\alpha=1/k$,
$C_{\lambda}^{\alpha}$ is an eigenfunction of
$\widetilde{H}_{k}=\frac{1}{k}H_{k}$ with eigenvalue

$-\sum_{j=1}^{N}\lambda_{j}\bigl{(}\frac{\lambda_{j}}{k}+N+1-2j\bigr{)}\,\leq
0.$

It is well-known (c.f. [HO]) that the polynomials
$C_{\lambda}^{1/k},\,\lambda\in\Lambda_{n}^{+}\,$ form a complete orthogonal
system of $L^{2}(\mathbb{T}^{N},\mu_{k})^{W}$ with the probability measure

(3.9)
$d\mu_{k}(z)=\phi_{k}(z)dz,\,\,\,\phi_{k}(z):=\,c_{k}\cdot\\!\prod_{j,l:l\neq
j}|z_{j}-z_{l}|^{k}\cdot{\bf 1}_{\mathbb{A}_{N}}(z),$

where $c_{k}>0$ is a normalization constant and $dz$ denotes the Haar measure
on $\mathbb{T}^{N}$. Notice that these measures appear also in the context of
circular $\beta$-ensembles in random matrix theory.

We also notice that the elementary symmetric polynomials $e_{l}$
($l=0,\ldots,N$) in $N$ variables, which are determined by

$\prod_{j=1}^{N}(y-x_{j})\,=\sum_{l=0}^{N}(-1)^{l}e_{l}(x)\,y^{N-l}\quad(y\in\mathbb{C}),$

are Jack polynomials for all $k>0$ up to normalization. More precisely, by
(3.8),

(3.10)
$e_{l}(x)=\frac{1}{l!}C_{\lambda}^{\alpha}(x)\>\>\text{with}\>\>\lambda=1^{l}=(\underbrace{1,\ldots,1}_{l\text{
times}},0,\ldots,0)\quad(l=0,\ldots,N).$

Thus in view of Lemma 3.4, the polynomials $e_{l}$ are eigenfunctions of
$\widetilde{H}_{k}$ with eigenvalues $-l\bigl{(}\frac{1}{k}+N-l\bigr{)}$.
Moreover, by a continuity argument or by direct computation, the $e_{l}$ are
also eigenfunctions of

$\widetilde{H}_{\infty}=(N-1)\sum_{j=1}^{N}z_{j}\frac{\partial}{\partial
z_{j}}-2\sum_{j=1}^{N}\sum_{l\neq
j}\frac{z_{j}^{2}}{z_{j}-z_{l}}\frac{\partial}{\partial z_{j}}$

with eigenvalues $-l(N-l).$

We now use these properties to construct martingales from our processes
$\widetilde{Z}_{k}$ for $k\in]0,\infty[$. We recall that by Dynkin’s formula
(see e.g. Section III.10 of [RW]), the following holds for the generator $L$
of a Feller semigroup and an arbitrary associated Feller process
$(X_{t})_{t\geq 0}$: if $f$ is a bounded eigenfunction of $L$ with eigenvalue
$r\in\mathbb{R}$, then the process $(e^{-rt}f(X_{t}))_{t\geq 0}$ is a
martingale w.r.t. the canonical filtration. We thus have:

######  Corollary 3.5.

For $k\in]0,\infty[$ and $z\in\mathbb{A}_{N}$ consider the diffusion
$(\widetilde{Z}_{t,k})_{t\geq 0}$ on $\mathbb{A}_{N}$ with start in $z$. Then,
for $l=0,1,\ldots,N$, the process
$(e^{l(1/k+N-l)t}e_{l}(\widetilde{Z}_{t,k}))_{t\geq 0}$ is a martingale. In
particular, for $t\geq 0$,

$\mathbb{E}(e_{l}(\widetilde{Z}_{t,k}))=e^{-l(1/k+N-l)t}\,e_{l}(z).$

This statement also holds in the deterministic case $k=\infty$, where we also
obtain additional information for $t\to\infty$:

######  Corollary 3.6.

For each starting point $z$ in the interior of $\mathbb{A}_{N}$, the
deterministic process $(\widetilde{Z}_{t,\infty})_{t\geq 0}$ satisfies

(3.11)
$e_{l}(\widetilde{Z}_{t,\infty})=e^{-l(N-l)t}e_{l}(z)\quad\quad(l=0,1,\ldots,N).$

Moreover, the limit
$Z:=\lim_{t\to\infty}\widetilde{Z}_{t,\infty}\in\mathbb{T}^{N}$ exists and is
given by

(3.12) $Z=(Z_{1},Z_{1}e^{2\pi i/N},\,\ldots,Z_{1}e^{2\pi i(N-1)/N})$

where $Z_{1}\in\mathbb{T}$ is as follows: If
$z=(e^{ix_{1}},\ldots,e^{ix_{N}})$ with $(x_{1},\ldots,x_{N})\in C_{N}$, then

(3.13) $Z_{1}=e^{ix_{0}}\quad\text{with}\quad
x_{0}=\frac{x_{1}+\ldots+x_{N}-\pi(N-1)}{N}.$

###### Proof.

Eq. (3.11) is obvious. As each $\zeta\in\mathbb{A}_{N}$ is uniquely determined
by the elementary symmetric functions $e_{l}(\zeta)$, (3.11) and a continuity
argument imply that
$Z=(Z_{1},\ldots,Z_{N}):=\lim_{t\to\infty}\widetilde{Z}_{t,\infty}\in\mathbb{A}_{N}$
exists, and that

$\prod_{j=1}^{N}(y-Z_{j})=y^{N}-e_{N}(Z)=y^{N}-z_{1}z_{2}\cdots z_{N}\,.$

This shows that $Z$ has the form as stated in (3.12) for some
$Z_{1}\in\mathbb{T}$ with $Z_{1}^{N}=z_{1}\cdots z_{n}\,.$ To identify
$Z_{1}$, we write the initial condition as $z=(e^{ix_{1}},\ldots,e^{ix_{N}})$
with $(x_{1},\ldots,x_{N})\in C_{N}$. In the $x$-coordinates, our process
$(\widetilde{X}_{t,\infty}=(x_{t,1},\ldots,x_{t,N}))_{t\geq 0}$ satisfies the
ODE (3.3). This ODE yields that $x_{t,1}+\ldots+x_{t,N}$ is independent of
$t\in[0,\infty]$. Therefore, the form of
$Z=(e^{ix_{\infty,1}},\ldots,e^{ix_{\infty,N}})$ in (3.12) yields

$x_{1}+\ldots+x_{N}=x_{\infty,1}+\ldots+x_{\infty,N}=\sum_{j=0}^{N-1}(x_{\infty,1}+j\cdot
2\pi/N)=Nx_{\infty,1}+(N-1)\pi.$

This implies (3.13). ∎

In the next step we use the decomposition of the processes
$\widetilde{X}_{k}=\widetilde{X}_{k}^{\mathit{diff}}+\widetilde{X}_{k}^{\mathit{cg}}.$
Eq. (3.4) and the expectations of the geometric Brownian motion imply

(3.14)
$\mathbb{E}(e^{-il\widetilde{X}_{t,k,j}^{\mathit{cg}}})=\mathbb{E}\left(e^{-\frac{il\sqrt{2}}{\sqrt{Nk}}B_{t}-il\widetilde{X}_{0,k,j}^{\mathit{cg}}}\right)=e^{-\frac{tl^{2}}{Nk}-il\widetilde{X}_{0,k,j}^{\mathit{cg}}}\quad(j,l=1,\ldots,N).$

This yields:

######  Corollary 3.7.

Let $k\in\,]0,\infty[$ and $x\in C_{N}$ with $x_{1}+\ldots+x_{N}=0.$ Let
$z=e^{ix}\in\mathbb{A}_{N}$, and consider the diffusion
$(\widetilde{Z}_{t,k})_{t\geq 0}$ on $\mathbb{A}_{N}$ with start in $z$. Then,
for $t\geq 0$ and $l=1,\ldots,N$,

$\mathbb{E}(e_{l}(e^{i\widetilde{X}_{t,k}^{\mathit{diff}}}))=e^{-l(N-l+(N+l)/(Nk))t}e_{l}(z).$

###### Proof.

By our initial conditions and Eq. (3.4) we have
$\widetilde{X}_{t,k,j}^{\mathit{cg}}=\widetilde{X}_{t,k,1}^{\mathit{cg}}$ for
all $t,k$ and $j=1,\ldots,N$. Hence, the stochastic independence of
$\widetilde{X}_{k}^{\mathit{diff}},\widetilde{X}_{k}^{\mathit{cg}}$, Corollary
3.5, (3.14), and the initial condition imply

$\displaystyle\mathbb{E}(e_{l}(e^{i\widetilde{X}_{t,k}^{\mathit{diff}}}))$
$\displaystyle=\mathbb{E}(e_{l}(e^{i\widetilde{X}_{t,k}-i\widetilde{X}_{t,k}^{\mathit{cg}}}))=\mathbb{E}(e_{l}(\widetilde{Z}_{t,k})\cdot
e^{-il\widetilde{X}_{t,k,1}^{\mathit{cg}}})$
$\displaystyle=\mathbb{E}(e_{l}(\widetilde{Z}_{t,k}))\cdot\mathbb{E}(e^{-il\widetilde{X}_{t,k,1}^{\mathit{cg}}})=e^{-l(1/k+N-l)t}\,e_{l}(z)\cdot
e^{-\frac{tl^{2}}{Nk}}$

as claimed. ∎

######  Example 3.8.

For the starting configuration $x=(0,2\pi/N,\ldots,(N-1)2\pi/N)$ and
$z=e^{x}$, we have $e_{1}(z)=\ldots=e_{N-1}(z)=0$ and $e_{N}(z)=(-1)^{N-1}$.
Hence, by Corollary 3.7,

$\mathbb{E}\bigl{(}e_{l}(e^{i\widetilde{X}_{t,k}^{\mathit{diff}}})\bigr{)}=0\quad(l=1,\ldots,N-1),\quad\text{and}\quad\mathbb{E}\bigl{(}e_{N}(e^{i\widetilde{X}_{t,k}^{\mathit{\mathit{diff}}}})\bigr{)}=(-1)^{N-1}e^{-2Nt/k}$

for all $t\geq 0$. As $e_{0}=1$, we conclude that in this case for all
$y\in\mathbb{C}$ and $t\geq 0$,

(3.15)
$\mathbb{E}\bigl{(}\prod_{j=1}^{N}(y-e^{i\widetilde{X}_{t,k,j}^{\mathit{diff}}})\bigr{)}=\mathbb{E}\bigl{(}\sum_{l=0}^{N}y^{N-l}(-1)^{l}e_{l}(e^{i\widetilde{X}_{t,k}^{\mathit{diff}}})\bigl{)}=y^{N}-e^{-2Nt/k}.$

We point out that this result differs from the case $k=\infty$ where $x$ is a
stationary solution of the associated ODE by Lemma 3.1, and thus
$\mathbb{E}\bigl{(}\prod_{j=1}^{N}(y-e^{i\widetilde{X}_{t,\infty,j}^{\mathit{diff}}})\bigr{)}=y^{N}-1$
is independent from $t$.

We next study the limit $t\to\infty$ for $k\in]0,\infty[$ similar to the limit
results for $k=\infty$ in Corollary 3.6. We need the following well-known
result, which follows easily for instance from the explicit formulas for the
densities of our diffusions in [RR1]:

######  Lemma 3.9.

Let $k\in\,]0,\infty[$. Then for each starting point in $\mathbb{A}_{N}$, the
process $(\widetilde{Z}_{t,k})_{t\geq 0}$ converges in distribution for
$t\to\infty$ to the probability measure $\mu_{k}$ on $\mathbb{A}_{N}$ from
(3.9).

This observation and Corollary 3.5 for $t\to\infty$ imply:

######  Corollary 3.10.

Let $Z=(Z_{1},\ldots,Z_{N})$ be an $\mathbb{A}_{N}$-valued random variable
with the distribution $\mu_{k}$ of a circular $\beta$-ensemble. Then for each
$y\in\mathbb{C}$,

$\mathbb{E}\bigl{(}\prod_{j=1}^{N}(y-Z_{j})\bigr{)}=y^{N}.$

###### Proof.

$\mathbb{E}\bigl{(}\prod_{j=1}^{N}(y-Z_{j})\bigr{)}=\mathbb{E}\bigl{(}\sum_{l=0}^{N}y^{N-l}(-1)^{l}e_{l}(Z)\bigl{)}=y^{N}.$

∎

A corresponding result can be also stated under the condition that $Z$ only
takes values in the alcove

$\mathbb{A}_{N}^{1}:=\\{z\in\mathbb{A}_{N}:z_{1}\cdots z_{N}=1\\}.$

For this consider the process $\widetilde{X}_{k}^{\mathit{diff}}$ in the
decomposition
$\widetilde{X}_{k}=\widetilde{X}_{k}^{\mathit{diff}}+\widetilde{X}_{k}^{\mathit{cg}}$
above. Then $e^{i\widetilde{X}_{k}^{\mathit{diff}}}$ is a diffusion on
$\mathbb{A}_{N}^{1}$ which converges for $t\to\infty$ in distribution to the
conditional probability measure $\mu_{k}^{1}\in M^{1}(\mathbb{A}_{N}^{1})$ of
$\mu_{k}$ under the condition $\mathbb{A}_{N}^{1}$. (This is, up to
normalization, the measure with the density (2.4)). Eq. (3.15) now leads to:

######  Corollary 3.11.

Let $Z$ be an $\mathbb{A}_{N}^{1}$-valued random variable with distribution
$\mu_{k}^{1}$. Then for each $y\in\mathbb{C}$,

$\mathbb{E}\bigl{(}\prod_{j=1}^{N}(y-Z_{j})\bigr{)}=y^{N}.$

######  Remark 3.12.

It can be shown and is well known that the measures $\mu_{k}$ tend for
$k\to\infty$ weakly to the probability measure $\mu_{\infty}$ which appears as
image of the uniform distribution on $\mathbb{T}$ under the mapping

$\mathbb{T}\to\mathbb{A}_{N},\quad z\to(z,z\cdot e^{2\pi i/N},\ldots,z\cdot
e^{2\pi i(N-1)/N}).$

Corollary 3.10 and continuity show that a random variable
$Z=(Z_{1},\ldots,Z_{N})$ on $\mathbb{A}_{N}$ with distribution $\mu_{\infty}$
also satisfy $\mathbb{E}\bigl{(}\prod_{j=1}^{N}(y-Z_{j})\bigr{)}=y^{N}$. A
corresponding result holds also in the situation of Corollary 3.11.

Please notice that this differs from the situation in the end of Example 3.8
for $k=\infty$ where we have a purely deterministic situation and also a
slightly different result.

Corollaries 3.5, 3.7, 3.10, and 3.11 have applications to Brownian motions and
uniform probabilities on compact symmetric spaces of type $A.$

For a first example, consider the space $C(U(N))$ of all conjugacy classes of
$U(N)$, which can be identified with $\mathbb{A}_{N}$ up to the cyclic group
$\mathbb{Z}_{N}$, i.e., $C(U(N))\sim\mathbb{A}_{N}/\mathbb{Z}_{N}$. In fact,
the conjugacy classes are characterized via the ordered spectra of matrices
from $U(N)$ where, say, the eigenvalue with the smallest nonnegative argument
has the first position. On the other hand, elements in $\mathbb{A}_{N}$
describe configurations of ordered points on $\mathbb{T}$ where the position
of the first entry is arbitrary. It is also well known that the pushforward of
the uniform distribution (i.e., the normalized Haar measure) of $U(N)$ under
the natural projection $U(N)\mapsto C(U(N))\sim\mathbb{A}_{N}/\mathbb{Z}_{N}$
agrees with the pushforward of $\mu_{k}$ with $k=1$ under the canonical
mapping $\mathbb{A}_{N}\mapsto\mathbb{A}_{N}/\mathbb{Z}_{N}$. Corollary 3.10
thus leads to the following.

######  Corollary 3.13.

Let $Z$ be a uniformly distributed $U(N)$-valued random variable. Then for
each $y\in\mathbb{C}$, $\,\mathbb{E}\bigl{(}\det(yI_{N}-Z)\bigr{)}=y^{N}.$

The same procedure also works for $SU(N),$ where the space $C(SU(N))$ of
conjugacy classes corresponds to $\mathbb{A}_{N}^{1}/\mathbb{Z}_{N}$, and the
pushforward of the uniform distribution corresponds to $\mu_{k}^{1}\in
M^{1}(\mathbb{A}_{N}^{1})$ with $k=1$. Hence, by Corollary 3.11:

######  Corollary 3.14.

Let $Z$ be an $SU(N)$-valued random variable which is uniformly distributed.
Then for $y\in\mathbb{C}$, $\mathbb{E}\bigl{(}\det(yI_{N}-Z)\bigr{)}=y^{N}$.

Corollaries 3.13 and 3.14 are special cases of well-known general formulas for
integrals of polynomials on unitary groups w.r.t. uniform distributions in
[CS]. On the other hand, our approach leads to generalizations of these
formulas for Brownian motions on $U(N)$ and $SU(N)$.

For $k=1/2$ or $k=2$, our results above are related to the compact symmetric
spaces $U(n)/O(n)$ and $U(2n)/Sp(n)$ associated with the root system
$A_{N-1}$.

##  4\. The non-compact case of type $A_{N-1}$

In this section we start with the $W$-invariant Heckman-Opdam Laplacians

(4.1) $L_{k}f(x)=\Delta f(x)+k\sum_{j=1}^{N}\sum_{l\neq
j}\coth\Bigl{(}\frac{x_{j}-x_{l}}{2}\Bigr{)}\frac{\partial}{\partial
x_{j}}f(x)$

for $k\in[0,\infty[$ on the Weyl chamber

$C_{N}^{A}:=\\{x\in\mathbb{R}^{N}:\>x_{1}\geq x_{2}\geq\ldots\geq x_{N}\\}$

of type $A_{N-1}$. As in Section 3, $L_{k}$ is the generator of a Feller
diffusion $(X_{t,k})_{t\geq 0}$ on $C_{N}^{A}$ with reflecting boundaries.
Again we study the renormalized generators
$\widetilde{L}_{k}:=\frac{1}{k}L_{k}$ which degenerate for $k\to\infty$ into

$\widetilde{L}_{\infty}=\sum_{j=1}^{N}\sum_{l\neq
j}\coth\Bigl{(}\frac{x_{j}-x_{l}}{2}\Bigr{)}\frac{\partial}{\partial x_{j}}.$

The process $\widetilde{X}_{k}:=(\widetilde{X}_{t,k}:=X_{t/k,k})_{t\geq 0}$
for $k\in]0,\infty[$ then solves the SDE

(4.2) $d\widetilde{X}_{t,k,j}=\frac{\sqrt{2}}{\sqrt{k}}dB_{t,j}+\sum_{l\neq
j}\coth\Bigl{(}\frac{\widetilde{X}_{t,k,j}-\widetilde{X}_{t,k,l}}{2}\Bigr{)}dt\quad\quad(j=1,\ldots,N)$

which degenerates for $k=\infty$ into the ODE

(4.3) $\frac{dx_{j}}{dt}(t)=\sum_{l\neq
j}\coth\Bigl{(}\frac{x_{j}(t)-x_{l}(t)}{2}\Bigr{)}\quad\quad(j=1,\ldots,N).$

Again, for initial data in the interior of the chamber $C_{N}^{A}$, the
solution $(\widetilde{X}_{t,\infty})_{t\geq 0}$ of these differential
equations exists for all $t\geq 0$ in the interior of $C_{N}^{A}$.

As in the preceding section, we next decompose the diffusions
$\widetilde{X}_{k}$ into the center-of-gravity process
$\widetilde{X}_{k}^{\mathit{cg}}:=(\widetilde{X}_{t,k}^{\mathit{cg}})_{t\geq
0}$ with

$\widetilde{X}_{t,k}^{\mathit{cg}}:=\frac{1}{N}(\widetilde{X}_{t,k,1}+\ldots+\widetilde{X}_{t,k,N})\cdot\omega$

and the process
$\widetilde{X}_{k}^{\mathit{diff}}:=\widetilde{X}_{k}-\widetilde{X}_{k}^{\mathit{cg}}\,$
on the Weyl chamber

$C_{N,0}^{A}:=\\{x\in C_{N}^{A}:x_{1}+\ldots+x_{N}=0\\}$

which describes the distances of the particles.

As in the proof of Lemma 3.2, the processes
$\widetilde{X}_{k}^{\mathit{diff}}$ and $\widetilde{X}_{k}^{\mathit{cg}}$ are
stochastically independent.

In the next step we observe that the processes $\widetilde{X}$,
$\widetilde{X}_{k}^{\mathit{diff}}$, $\widetilde{X}_{k}^{\mathit{cg}}$ admit
arbitrary exponential moments for arbitrary deterministic starting points. In
fact, for $\widetilde{X}_{k}^{\mathit{diff}}$ this follows from Lemma 2.1.
Moreover, $\widetilde{X}_{k}^{\mathit{cg}}$ is a classical one-dimensional
Brownian motion up to scaling and has therefore arbitrary exponential moments.
Finally, the independence of $\widetilde{X}_{k}^{\mathit{diff}}$ and
$\widetilde{X}_{k}^{\mathit{cg}}$ ensures that
$\widetilde{X}_{k}=\widetilde{X}_{k}^{\mathit{diff}}+\widetilde{X}_{k}^{\mathit{cg}}\,$
has this property as well.

With the existence of exponential moments in mind, we now follow Section 3 and
observe that the trigonometric elementary symmetric polynomials

$\widetilde{e}_{l}(x):=e_{l}(e^{x})\quad(l=0,\ldots,N,\,\,x\in C_{N}^{A})$

are eigenfunctions of $\widetilde{L}_{k}$ for all $k$ with eigenvalues
$l\bigl{(}\frac{1}{k}+N-l\bigr{)}\,\geq 0$. This leads to martingales for the
diffusions $(\widetilde{X}_{t,k})_{t\geq 0}$ on $C_{N}^{A},$ similar to
Corollaries 3.5 and 3.6 as follows:

######  Corollary 4.1.

For $k\in\,]0,\infty[$ and $x\in C_{N}^{A}$ consider the diffusion
$(\widetilde{X}_{t,k})_{t\geq 0}$ on $C_{N}^{A}$ with start in $x$. Then, for
$l=0,1,\ldots,N$, the process
$\bigl{(}e^{-l(1/k+N-l)t}\,\widetilde{e}_{l}(\widetilde{X}_{t,k})\bigr{)}_{t\geq
0}$ is a martingale. In particular, for $t\geq 0$,

$\mathbb{E}\bigl{(}\widetilde{e}_{l}(\widetilde{X}_{t,k})\bigr{)}=e^{l(1/k+N-l)t}\,\widetilde{e}_{l}(x).$

This result also holds for $k=\infty$. More precisely, the solution
$(\widetilde{X}_{t,\infty})_{t\geq 0}$ of the ODE (4.3) with start $x$ in the
interior of $C_{N}^{A}$ satisfies

(4.4)
$\widetilde{e}_{l}(\widetilde{X}_{t,\infty})=e^{l(N-l)t}\,\widetilde{e}_{l}(x)\quad\quad(l=0,1,\ldots,N).$

Using the independence of the processes $\widetilde{X}_{k}^{\mathit{diff}}$
and $\widetilde{X}_{k}^{\mathit{cg}}$, we also obtain the following analog of
Corollary 3.7.

######  Corollary 4.2.

For $k\in\,]0,\infty[$ and $x\in C_{N,0}^{A}$ consider the diffusion
$\widetilde{X}_{k}^{\mathit{diff}}$ on the chamber $C_{N,0}^{A}$ with start in
$x$. Then, for $t\geq 0$ and $l=1,\ldots,N$,

$\mathbb{E}\bigl{(}\widetilde{e}_{l}(\widetilde{X}_{t,k}^{\mathit{diff}})\bigr{)}=e^{l(N-l+(N+l)/(Nk))t}\,\widetilde{e}_{l}(x).$

This holds also for $k=\infty$ where the expectations can be omitted.

Corollary 4.2 can be restated as:

######  Corollary 4.3.

For $k\in\,]0,\infty[$ and $x\in C_{N,0}^{A}$ consider the diffusion
$\widetilde{X}_{k}^{\mathit{diff}}$ on $C_{N,0}^{A}$ with start in $x$. Then,
for $t\geq 0$ and $y\in\mathbb{C}$,

$\mathbb{E}\bigl{(}\prod_{j=1}^{N}(y-e^{\widetilde{X}_{k,t,j}^{\mathit{diff}}})\bigr{)}=P_{t,N,k,x}(y)$

with the polynomial

$P_{t,N,k,x}(y):=\sum_{l=0}^{N}y^{N-l}(-1)^{l}e^{l(N-l+(N+l)/(Nk))t}\,\widetilde{e}_{l}(x).$

For general starting points $x\in C_{N,0}^{A}$, these polynomials do not seem
to have particularly nice properties. For the starting configuration $x=0\in
C_{N,0}^{A}$, which is of particular interest here, we get

$P_{t,N,k,0}(y)=\sum_{l=0}^{N}{N\choose
l}(-1)^{l}y^{N-l}e^{l(N-l+(N+l)/(Nk))t}.$

We do not have much information about these polynomials. This is a contrast to
the Bessel processes of type $A$ where in a corresponding formula classical
Hermite polynomials appear; see [KVW].

######  Example 4.4.

  1. (1)

Let $N=2$. We try to solve the ODE (4.3) with the singular starting point
$x=0\in C_{2}^{A}.$ In fact, (4.3) yields that $x_{1}(t)=-x_{2}(t)$ and that
$y(t):=x_{1}(t)-x_{2}(t)$ satisfies $\frac{dy}{dt}(t)=2\coth(y(t)/2)$. On the
other hand, (4.4) for $N=2$, $l=1$ suggests that

$e^{x_{1}(t)}+e^{-x_{1}(t)}=2e^{t}$

and thus $x_{1}(t)={\rm arcosh}(e^{t})$. It is now easily checked that in
fact,

$t\mapsto({\rm arcosh}(e^{t}),-{\rm arcosh}(e^{t}))$

is continuous on $[0,\infty)$ and solves (4.3) for $t>0.$ Moreover, for
$k=\infty$ we have $x^{\mathit{diff}}\\!(t)=x(t)$, and

$P_{t,2,\infty,0}(y)=y^{2}-2e^{t}\,y+1.$

These polynomials have the zeros $e^{\pm x_{1}(t)}$ as claimed.

  2. (2)

Let $N=3.$ We again try to solve the ODE (4.3) with $x=0\in\partial
C_{3}^{A}$. Here symmetry arguments imply that the solution of the ODE (4.3)
must have the form $x(t)=(x_{1}(t),0,-x_{1}(t))$. On the other hand, formula
(4.4) with $N=2$, $l=1,2$ suggests that

$e^{x_{1}(t)}+e^{-x_{1}(t)}+1=3e^{2t}$

and thus $\,x_{1}(t)={\rm arcosh}\bigl{(}3(e^{2t}-1)/2\bigr{)}.\,$ This indeed
gives a solution as in example (1). Moreover, the polynomial

$P_{t,3,\infty,0}(y)=y^{3}-3e^{2t}y^{2}+3e^{2t}y-1$

has the zeros $1$ and $e^{\pm x_{1}(t)}$ as claimed.

Unfortunately, we have no closed formulas for $x(t)$ for general dimension $N$
even for the starting point $x=0\in C_{N}^{A}$.

We finally mention that Corollaries 4.2 and 4.3 for $k=1/2,1,2$ have
applications to Brownian motions on the noncompact symmetric spaces of type A,
i.e. on $GL(N,\mathbb{F})/U(N,\mathbb{F})$ for
$\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H}$ similar to the results in the
end of Section 3.

##  5\. The non-compact case of type $BC_{N}$

We here start with the nonreduced root system

$R=BC_{N}=\\{\pm e_{i},\pm 2e_{i},\pm(e_{i}\pm e_{j});\>\>1\leq i<j\leq
N\\}\subset\mathbb{R}^{N}$

for $N\geq 2$ with the multiplicity $k=(k_{1},k_{2},k_{3}),$ where
$k_{1},k_{2}\geq 0$, $k_{3}>0$ are the values on the roots
$e_{i},2e_{i},e_{i}\pm e_{j}$. As indicated in the introduction, we now
reparametrize the multiplicity $k$. This will be not natural at a first
glance, but it will turn out to be useful in the end. We here mainly follow
the notations in [Dem, V] and define

(5.1) $\kappa=k_{3},\quad q=N-1+\frac{1+2k_{1}+2k_{2}}{2k_{3}},\quad
p=N-1+\frac{1+2k_{2}}{2k_{3}}.$

Then $\kappa>0$, $q\geq p\geq N-1+1/{2\kappa}$, and

(5.2)
$k=(k_{1},k_{2},k_{3})=\kappa\cdot\Bigl{(}q-p,k_{0,2},1\Bigr{)}\quad\text{with}\quad
k_{0,2}:=p-(N-1)-\frac{1}{2\kappa}.$

We now regard $p,q$ as fixed parameters and $\kappa>0$ as a varying parameter,
and denote the multiplicity $k$ in (5.2) by $k_{\kappa}$. The associated
$W$-invariant Heckman-Opdam Laplacian (2.6) is

$\displaystyle L_{\kappa}=\Delta+\sum_{i=1}^{N}\kappa\Bigl{(}($ $\displaystyle
q-p)\coth(\frac{x_{i}}{2})+2k_{0,2}\coth(x_{i})\,$ (5.3)
$\displaystyle+\sum_{j:j\neq
i}\bigl{(}\coth(\frac{x_{i}-x_{j}}{2})+\coth(\frac{x_{i}+x_{j}}{2})\bigr{)}\Bigr{)}\partial_{i}$

on the Weyl chamber

$C_{N}^{B}:=\\{x\in\mathbb{R}^{N}:\>x_{1}\geq x_{2}\geq\ldots\geq x_{N}\geq
0\\}$

of type $B_{N}$, associated with $R_{+}=\\{e_{i},\,2e_{i},\,e_{i}-e_{j}:1\leq
i<j\leq N\\}.$ As in the preceding sections, the operators $L_{\kappa}$ are
the generators of diffusions $(X_{t,\kappa})_{t\geq 0}$ on $C_{N}^{B}$ where
the paths are reflected at $\partial C_{N}^{B}$. The renormalized operators

$\widetilde{L}_{\kappa}:=\frac{1}{\kappa}L_{\kappa}\,$

then are the generators of the diffusions
$(\widetilde{X}_{t,\kappa}:=X_{t/\kappa,\kappa})_{t\geq 0}$ which may be
regarded as solutions of the SDE

$\displaystyle d\widetilde{X}_{t,\kappa,j}=$
$\displaystyle\frac{\sqrt{2}}{\sqrt{\kappa}}dB_{t,j}+\sum_{l\neq
j}\Bigl{(}\coth\Bigl{(}\frac{\widetilde{X}_{t,\kappa,j}-\widetilde{X}_{t,\kappa,l}}{2}\Bigr{)}+\coth\Bigl{(}\frac{\widetilde{X}_{t,\kappa,j}+\widetilde{X}_{t,\kappa,l}}{2}\Bigr{)}\Bigr{)}dt$
(5.4)
$\displaystyle+\Bigl{(}(q-p)\coth\bigl{(}\frac{\widetilde{X}_{t,\kappa,j}}{2}\bigr{)}+2k_{0,2}\coth(\widetilde{X}_{t,\kappa,j})\Bigr{)}dt\quad\quad(j=1,\ldots,N).$

For $\kappa\to\infty$, the generator degenerates into

$\displaystyle\widetilde{L}_{\infty}=$
$\displaystyle\sum_{i=1}^{N}\Bigl{(}(q-p)\coth(\frac{x_{i}}{2})+2(p-(N-1))\coth(x_{i})$
(5.5) $\displaystyle+\sum_{j:j\neq
i}\Bigl{(}\coth(\frac{x_{i}-x_{j}}{2})+\coth(\frac{x_{i}+x_{j}}{2})\Bigr{)}\Bigr{)}\partial_{i},$

and (5) becomes the ODE

(5.6) $\displaystyle\frac{dx_{j}}{dt}(t)=$ $\displaystyle\sum_{l\neq
j}\Bigl{(}\coth\Bigl{(}\frac{x_{j}(t)-x_{l}(t)}{2}\Bigr{)}+\coth\Bigl{(}\frac{x_{j}(t)+x_{l}(t)}{2}\Bigr{)}\Bigr{)}$
$\displaystyle+(q-p)\coth\bigl{(}\frac{x_{j}(t)}{2}\bigr{)}+2\bigl{(}p-(N-1)\bigr{)}\coth
x_{j}(t)\quad\quad(j=1,\ldots,N).$

Again, for initial data in the interior of the chamber $C_{N}^{B}$, the
solution $(\widetilde{X}_{t,\infty})_{t\geq 0}$ of these differential
equations exists for $t\geq 0$ in the interior of $C_{N}^{B}$.

We now turn to eigenfunctions of the operators $\widetilde{L}_{\kappa}$ and
consider the Heckman-Opdam hypergeometric functions
$F_{BC}(\lambda,k_{\kappa};x)$ associated with $BC_{N}$ in the variable
$x\in\mathbb{R}^{N},$ with spectral parameter $\lambda\in\mathbb{C}^{N}$ and
multiplicity $k_{\kappa}$. By Section 2, for $\kappa\in\,]0,\infty[$, the
functions $x\mapsto F_{BC}(\lambda,k_{\kappa};x)$ are eigenfunctions of the
renormalized Laplacian $\widetilde{L}_{\kappa}$ with the eigenvalues

$r_{\lambda,\kappa}:=\frac{1}{\kappa}\Bigl{(}\sum_{j=1}^{N}\lambda_{j}^{2}-|\rho(\kappa)|^{2}\Bigr{)},$

where

(5.7) $\rho(\kappa)=\frac{1}{2}\sum_{\alpha\in
R_{+}}k_{\kappa}(\alpha)\alpha\quad\text{with}\quad\rho(\kappa)_{j}=\frac{\kappa}{2}\bigl{(}(q-p)+2k_{0,2}+2(N-j)\bigr{)}.$

We further consider the associated (normalized) Heckman-Opdam polynomials
$R_{\lambda}(k,\,.\,)=R_{\lambda}^{BC}(k,\,.\,),$ as introduced in (2.5),
which are indexed by $P_{+}=\Lambda_{N}^{+}$ (the set of partitions of length
at most $N$). These are multivariate generalizations of the classical Jacobi
polynomials which are well-studied in the literature, see for instance [BO, L,
RR2]. The polynomials $\widetilde{R}_{\lambda}$ defined by

$\widetilde{R}_{\lambda}(\cos x):=R_{\lambda}(k;x)$

form an orthogonal basis of $L^{2}(\mathbb{A}_{N},w_{k})$ on

$\mathbb{A}_{N}:=\\{x\in\mathbb{R}^{N}|\>-1\leq y_{1}\leq...\leq y_{N}\leq
1\\}$

with the weight function

(5.8)
$w_{k}(y):=\prod_{i=1}^{N}(1-y_{i})^{k_{1}+k_{2}-1/2}(1+y_{i})^{k_{2}-1/2}\cdot\prod_{i<j}|y_{i}-y_{j}|^{2k_{3}}.$

Here we are mainly interested in the fact that for $k=k_{\kappa}$ and
$\lambda\in\Lambda_{N}^{+}$, the exponential polynomials

$H_{\lambda}(x):=\widetilde{R}_{\lambda}(\cosh x)=F_{BC}(\lambda+\rho(k),k;x)$

are eigenfunctions of $\widetilde{L}_{\kappa}$ with the eigenvalues

$\displaystyle r_{\lambda}$
$\displaystyle=\frac{1}{\kappa}\langle\lambda,\lambda+2\rho(k)\rangle$
$\displaystyle=\frac{1}{\kappa}\sum_{j=1}^{N}\lambda_{j}(\lambda_{j}+k_{1}+2k_{2}+2k_{3}(N-j))$
(5.9)
$\displaystyle=\sum_{j=1}^{N}\lambda_{j}\Bigl{(}\frac{\lambda_{j}-1}{\kappa}+p+q+2-2j\Bigr{)}.$

We now consider the partitions $\lambda(n):=1^{n}\in\Lambda_{+}^{N}$ for
$n=0,\ldots,N.$ It is known (see Section 5 of [V] and in particular Lemma 5.1
there) that the Jacobi polynomials $\widetilde{R}_{\lambda(n)}$ are of the
form

(5.10) $\widetilde{R}_{\lambda(n)}=\sum_{l=0}^{n}c_{n,l}(p,q)\cdot
e_{l}\quad\text{with }\>c_{n,n}(p,q)\neq 0,$

where the $e_{l}$ are again the elementary symmetric polynomials in $N$
variables and the coefficients $c_{n,l}(p,q)\in\mathbb{R}$ depend on $p,q$
only and not on $\kappa$. This observation will be crucial in the following
and is the reason for our parametrization of $k$ by $p,q,\kappa$ above. For
more details on the $c_{n,l}(p,q)$ we refer to [V]. In summary, the functions
$H_{\lambda(n)}$ with $n=0,\ldots,N$ are independent of $\kappa$ and
simultanous eigenfunctions of the operators $\widetilde{L}_{\kappa}$ for all
$\kappa\in\,]0,\infty[$ with the eigenvalues

$r_{n}=n(p+q-n+1).$

Clearly, this observation also holds for $\kappa=\infty$. This implies:

######  Lemma 5.1.

For each starting point $x\in C_{N}^{B}$ of the processes
$(\widetilde{X}_{t,\kappa})_{t\geq 0}$ with $\kappa\in\,]0,\infty]$, the
processes

(5.11) $\Bigl{(}e^{-r_{n}t}\cdot
H_{\lambda(n)}\bigl{(}\widetilde{X}_{t,\kappa}\bigr{)}\Bigr{)}_{t\geq 0}$

are martingales for $n=0,\ldots,N$, where the numbers $r_{n}$ and the
functions $H_{\lambda(n)}$ do not depend on $\kappa$.

In particular, for $x$ in the interior of $C_{N}^{B}$, the solution
$(\widetilde{X}_{t,\infty})_{t\geq 0}$ of the ODE (5.6) with
$\widetilde{X}_{0,\infty}=x$ satisfies

$H_{\lambda(n)}(\widetilde{X}_{t,\infty})=e^{r_{n}t}H_{\lambda(n)}(x)\quad\text{for}\quad
t\geq 0,\>n=0,\ldots,N.$

###### Proof.

This follows from our preceding considerations and the fact that the random
variables $H_{\lambda(n)}(\widetilde{X}_{t,\kappa})$ are integrable for
$\kappa<\infty$ and $t>0$ by Lemma 2.1. ∎

We may invert (5.10) and write the elementary symmetric polynomials $e_{l}$ as
linear combinations of the $\widetilde{R}_{\lambda(n)}$ for $l,n=0,\ldots,N$
with coefficients independent of $\kappa$. Lemma 5.1 thus implies:

######  Corollary 5.2.

Fix some deterministic starting point $x\in C_{N}^{B}$ as well as the
parameters $p,q$. Consider the associated diffusions
$(\widetilde{X}_{t,\kappa})_{t\geq 0}$ for $\kappa\in\,]0,\infty[$. Then there
are coefficients $a_{n,l}\in\mathbb{R}$ for $0\leq l\leq n\leq N$ such that

$\mathbb{E}\bigl{(}e_{n}(\cosh(\widetilde{X}_{t,\kappa}))\bigr{)}=\sum_{l=0}^{n}a_{n,l}\,e^{r_{l}t}$

with $r_{0}=0$ where the coefficients $a_{n,l}$ and the exponents $r_{l}$
depend on $p,q$ and $x$ only and not on $\kappa$. The same holds for
$\kappa=\infty$ and starting points $x$ in the interior of $C_{N}^{B}.$

######  Example 5.3.

For $x=0\in C_{N}^{B}$, we have $H_{\lambda(n)}(0)=1\,$ and thus

$\mathbb{E}\bigl{(}H_{\lambda(n)}(\widetilde{X}_{t,\kappa})\bigr{)}=e^{r_{n}t}\quad\quad(n=0,\ldots,N,\>t\geq
0,\,\kappa\in\,]0,\infty[).$

We finally turn to an application concerning a determinantal formula. Again we
fix some starting point $x\in C_{N}^{B}$ as well as $p,q$ and consider the
associated diffusions $(\widetilde{X}_{t,\kappa})_{t\geq 0}$ for
$\kappa\in\,]0,\infty[$. Then by Corollary 5.2, for all $t\geq 0$ and
$y\in\mathbb{C}$,

(5.12)
$\displaystyle\mathbb{E}\Bigl{(}\prod_{j=1}^{N}\bigl{(}y-\cosh(\widetilde{X}_{t,\kappa,j})\bigr{)}\Bigr{)}$
$\displaystyle=\sum_{n=0}^{N}(-1)^{n}\,\mathbb{E}\bigl{(}e_{n}(\cosh(\widetilde{X}_{t,\kappa})\bigr{)}\cdot
y^{N-n}$
$\displaystyle=\sum_{n=0}^{N}(-1)^{n}\,e_{n}\bigl{(}\cosh(\widetilde{X}_{t,\infty})\bigr{)}\cdot
y^{N-n}$
$\displaystyle=\prod_{j=1}^{N}\,\bigl{(}y-\cosh(\widetilde{X}_{t,\infty,j})\bigr{)}=:D_{t,x}(y).$

It is an interesting task to find particularly nice starting points $x\in
C_{N}^{B}$ for which $D_{t,x}(y)$ can be determined explicitly.

For instance, in the setting of multivariate Bessel processes of types
$A_{N-1}$ or $B_{N}$ and start in the origin, $D_{t,0}(y)$ is a classical one-
dimensional Hermite or Laguerre polynomial of degree $N$ in $y$, which is
scaled by a factor $\sqrt{t}$; for the details see [KVW]. Moreover, in the
setting of Heckman-Opdam Jacobi processes of type $BC$ on the compact alcove
$\mathbb{A}_{N}$, and with the same paramatrization of the multiplicity $k$ by
$p,q,\kappa$ as here, there is a (unique) stationary solution $x_{0}$ of the
ODE in the interior of $\mathbb{A}_{N}$, whose coordinates are the ordered
zeroes of some classical one-dimensional Jacobi polynomial of degree $N$ in
$y$. The indices of this Jacobi polynomial are determined by $p,q$. This means
that for this particular starting point $x_{0}$, the function $D_{t,x_{0}}(y)$
is just this specific Jacobi polynomial and is independent of $t\geq 0$ (due
to stationarity). We refer to [V] for further details.

We expect that in our non-compact $BC$ setting, $D_{t,x}(y)$ should be of
particular interest when the associated ODE (5.6) starts in $x=0\in\partial
C_{N}^{B},$ where we expect that the corresponding initial value problem is
uniquely solvable, as in the Dunkl setting in [VW]. It seems that the explicit
solution of this initial value problem is more involved than in the cases
considered in [KVW, V].

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