Functional central limit theorems for multivariate Bessel processes in the freezing regime

TL;DR

This paper establishes functional central limit theorems for multivariate Bessel processes in the freezing regime, revealing Gaussian limits with explicit matrix exponential representations, advancing understanding of their asymptotic behavior.

Contribution

It introduces functional CLTs for multivariate Bessel processes with specific initial conditions, extending previous fixed-time results to a locally uniform in time setting.

Findings

Gaussian limiting processes with explicit matrix exponential forms

Functional CLTs valid for initial conditions scaled by k

Extension of previous fixed-time limit theorems to a dynamic, uniform-in-time context

Abstract

Multivariate Bessel processes $(X_{t,k})_{t\ge0}$ describe interacting particle systems of Calogero-Moser-Sutherland type and are related with $\beta$-Hermite and $\beta$-Laguerre ensembles. They depend on a root system and a multiplicity $k$ which corresponds to the parameter $\beta$ in random matrix theory. In the recent years, several limit theorems were derived for $k\to\infty$ with fixed $t>0$ and fixed starting point. Only recently, Andraus and Voit used the stochastic differential equations of $(X_{t,k})_{t\ge0}$ to derive limit theorems for $k\to\infty$ with starting points of the form $\sqrt k\cdot x$ with $x$ in the interior of the corresponding Weyl chambers. Here we provide associated functional central limit theorems which are locally uniform in $t$. The Gaussian limiting processes admit explicit representations in terms of matrix exponentials and the solutions of the associated deterministic dynamical systems.

Equations

$$C_N^A:=\{x\in {\mathbb{R}}^N:x_1\ge x_2\ge \dots \ge x_N\},$$

$$Lf:=\frac{1}{2}\Delta f+k\sum_{i=1}^{N}\Bigl{(}\sum_{j\neq i}\frac{1}{x_{i}-x_{j}}\Bigr{)}\frac{\partial}{\partial x_{i}}f,$$

$$D(L):=\{f{|}_{C_N^A}:f\in C^{(2)}({\mathbb{R}}^N),f\text{ invariant under all coordinate permutations}\}.$$

$$C_N^B:=\{x\in {\mathbb{R}}^N:x_1\ge x_2\ge \dots \ge x_N\ge 0\},$$

$$Lf:=\frac{1}{2}\Delta f+k_{2}\sum_{i=1}^{N}\sum_{j\neq i}\Bigl{(}\frac{1}{x_{i}-x_{j}}+\frac{1}{x_{i}+x_{j}}\Bigr{)}\frac{\partial}{\partial x_{i}}f\quad+k_{1}\sum_{i=1}^{N}\frac{1}{x_{i}}\frac{\partial}{\partial x_{i}}f,$$

$$K_t(x,A)=c_k{\int }_A\frac{1}{t^{{\gamma }_k+N/2}}{e}^{-(\Vert x{\Vert }^2+\Vert y{\Vert }^2)/(2t)}{J}_k(\frac{x}{\sqrt{t}},\frac{y}{\sqrt{t}})\cdot w_k(y)dy$$

$$w_k^A(x):=\prod _{i<j}(x_i-x_j{)}^{2k},w_k^B(x):=\prod _{i<j}(x_i^2-x_j^2{)}^{2k_2}\cdot \prod _{i=1}^N{x}_i^{2k_1},$$

$$\frac{c_k}{t^{\gamma +N/2}}{e}^{-\Vert y{\Vert }^2/(2t)}\cdot w_k(y)dy$$

$$d{X}_{t,k}=d{B}_t+\frac{1}{2}(\nabla (\ln w_k))(X_{t,k})dt$$

$$\sqrt{k}(\frac{X_{t,k}}{\sqrt{k}}-\varphi (t,x))⟶W_t\text{for}k\to \infty$$

$$d{X}_{t,k}^i=d{B}_t^i+k\sum _{j\ne i}\frac{1}{X_{t,k}^i-X_{t,k}^j}dt(i=1,\dots ,N).$$

$$d{\tilde{X}}_{t,k}^i=\frac{1}{\sqrt{k}}d{B}_t^i+\sum _{j\ne i}\frac{1}{{\tilde{X}}_{t,k}^i-{\tilde{X}}_{t,k}^j}dt(i=1,\dots ,N).$$

$$\frac{dx}{dt}(t)=H(x(t))$$

$$H(x):=\Bigl{(}\sum_{j\neq 1}\frac{1}{x_{1}-x_{j}},\ldots,\sum_{j\neq N}\frac{1}{x_{N}-x_{j}}\Bigr{)}$$

$$\varphi (t,c\cdot z)=\sqrt{2t+c^2}\cdot z$$

$$z:=(z_1,\dots ,z_N)\in C_N^A$$

$$\frac{d}{dt}\Vert \varphi (t,x){\Vert }_2^2$$

$$=2\sum _{i,j=1,\dots ,N,i\ne j}\frac{{\varphi }_i(t,x)}{{\varphi }_i(t,x)-{\varphi }_j(t,x)}=N(N-1).$$

$$\Vert \varphi (t,x){\Vert }_2^2=N(N-1)t+\Vert x{\Vert }^2.$$

$$\underset{0\le s\le t,k\ge k_0}{\sup }\Vert X_{s,k}-\sqrt{k}\varphi (s,x)\Vert <\infty a.s..$$

$$d{W}_t^i=d{B}_t^i+\sum _{j\ne i}\frac{W_t^j-W_t^i}{({\varphi }_i(t,x)-{\varphi }_j(t,x){)}^2}dt(i=1,\dots ,N).$$

$$d{W}_t=d{B}_t+A(t,x)W_{t}dt$$

$$A(t,x{)}_{i,j}:=\frac{1}{({\varphi }_i(t,x)-{\varphi }_j(t,x){)}^2},A(t,x{)}_{i,i}:=-\sum _{j\ne i}\frac{1}{({\varphi }_i(t,x)-{\varphi }_j(t,x){)}^2}$$

$$W_t=e^{{\int }_0^{t}A(s,x)ds}{\int }_0^t{e}^{-{\int }_0^{s}A(u,x)du}d{B}_s(t\ge 0).$$

$$\underset{0\le s\le t,k\ge k_0}{\sup }\sqrt{k}\cdot \Vert X_{s,k}-\sqrt{k}\varphi (s,x)-W_s\Vert <\infty a.s.,$$

$$\displaystyle R_{t,k}^{i}=k\int_{0}^{t}\sum_{j\neq i}\Biggl{(}$$

$$\displaystyle-\frac{W_{s}^{j}-W_{s}^{i}}{\bigl{(}\sqrt{k}(\phi_{i}(s,x)-\phi_{j}(s,x))\bigr{)}^{2}}\Biggr{)}ds$$

$$\frac{1}{x}=\frac{1}{x_0}-\frac{x-x_0}{x_0^2}+\frac{(x-x_0{)}^2}{{\tilde{x}}^3}$$

$$x=X_{s,k}^i-X_{s,k}^j\text{and}{x}_0=\sqrt{k}({\varphi }_i(s,x)-{\varphi }_j(s,x)),$$

$$R_{t,k}^i$$

$$\displaystyle=-\int_{0}^{t}\Biggl{(}\sum_{j\neq i}\frac{R_{s,k}^{i}-R_{s,k}^{j}}{(\phi_{i}(s,x)-\phi_{j}(s,x))^{2}}+H_{s,k}^{i}\Biggr{)}ds$$

$$H_{s,k}^{i}=k\sum_{j\neq i}\frac{\Bigl{(}(X_{s,k}^{i}-\sqrt{k}\phi_{i}(s,x))-(X_{s,k}^{j}-\sqrt{k}\phi_{j}(s,x))\Bigr{)}^{2}}{\Bigl{(}\sqrt{k}(\phi_{i}(s,x)-\phi_{j}(s,x))+D_{i,j}(s)\Bigr{)}^{3}}$$

Full text

1. Introduction

Integrable interacting particle systems on $\mathbb{R}$ of Calogero-Moser-Sutherland type with $N$ particles can be described as multivariate Bessel processes on appropriate closed Weyl chambers in $\mathbb{R}^{N}$. These processes are time-homogeneous diffusions with well-known generators of the transition semigroups and transition probabilities, and they are solution of the associated stochastic differential equations (SDEs); see [CGY, GY, R1, R2, RV1, RV2, DV, A]. These multivariate Bessel processes $(X_{t,k})_{t\geq 0}$ are described via root systems, a possibly multidimensional multiplicity parameter $k$ and by their starting points $X_{0,k}:=x$. The multiplicies $k$ are coupling constants which describe the strength of interaction of the particles. We restrict our attention to the root systems $A_{N-1}$, $B_{N}$, and $D_{N}$ on $\mathbb{R}^{N}$ as the most relevant cases in view of interacting particle systems and random matrix theory.

We briefly recapitulate the most important cases $A_{N-1}$ and $B_{N}$. For $A_{N-1}$, we have a multiplicity $k\in]0,\infty[$, the processes live on the closed Weyl chamber

[TABLE]

and the generator of the transition semigroup is

[TABLE]

where we assume reflecting boundaries, i.e., the domain of $L$ is

[TABLE]

For $B_{N}$, we have the multiplicity $k=(k_{1},k_{2})\in]0,\infty[^{2}$, the processes live on

[TABLE]

and the generator is

[TABLE]

where we again assume reflecting boundaries.

By [R1, R2, RV1, RV2], the transition probabilities of $(X_{t,k})_{t\geq 0}$ have the form

[TABLE]

for $t>0$, $x\in C_{N}$, and $A\subset C_{N}$ a Borel set. For the root systems $A_{N-1}$ and $B_{N}$, we here have the weight functions $w_{k}$ of the form

[TABLE]

and $\gamma_{k}^{A}(k):=kN(N-1)/2$ and $\gamma_{k}^{B}(k_{1},k_{2}):=k_{2}N(N-1)+k_{1}N$ respectively. $w_{k}$ is homogeneous of degree $2\gamma_{k}$, and $c_{k}>0$ is a known normalization, and $J_{k}$ a multivariate Bessel function of type $A_{N-1}$ or $B_{N}$ with multiplicities $k$ or $(k_{1},k_{2})$ which is analytic on $\mathbb{C}^{N}\times\mathbb{C}^{N}$ with $J_{k}(x,y)>0$ for $x,y\in\mathbb{R}^{N}$. Moreover, $J_{k}(x,y)=J_{k}(y,x)$ and $J_{k}(0,y)=1$ for $x,y\in\mathbb{C}^{N}$. For this and further informations we refer e.g. to [R1, R2]. In particular, if $X_{0,k}=0$, then $X_{t,k}$ has the Lebesgue density

[TABLE]

on $C_{N}$ for $t>0$. Hence, for the root systems $A_{N-1}$, $B_{N}$, these processes $(X_{t,k})_{t\geq 0}$ (with proper parameters) are related to random matrix theory, as the distributions in (1.5) are those of the ordered eigenvalues of $\beta$-Hermite and $\beta$-Laguerre ensembles; see e.g. the tridiagonal random matrix models of Dumitriu and Edelman [DE1, DE2]. For other starting points, there exist further tridiagonal random matrix models for the distributions of $X_{t,k}$; see [AG, HP].

We here study limit theorems for $(X_{t,k})_{t\geq 0}$ when one or several components of $k$ tend to $\infty$ in a coupled way (we briefly write $k\to\infty$ by misuse of notation). In physics, this means freezing, and in random matrix theory, $k\to\infty$ roughly means $\beta\to\infty$ in [DE2]. Several limit theorems for $k\to\infty$ are known for fixed starting points $x\in\mathbb{R}^{N}$; see [AKM1, AKM2, AM, V] where for $x=0$ the results fit to [DE2]. In the present paper we regard the $(X_{t,k})_{t\geq 0}$ as solutions of the SDEs

[TABLE]

with starting points $X_{0,k}=\sqrt{k}\cdot x$ with $x$ in the interior of the Weyl chambers and an $N$-dimensional Brownian motion $(B_{t})_{t\geq 0}$. Note that by [CGY], see also [Sch] and [GM], under the condition E\Bigl{(}\int_{0}^{t}\nabla(\ln w_{k})(X_{s,k})\>ds\Bigr{)}<\infty, (1.6) has a unique (strong) solution $(X_{t,k})_{t\geq 0}$, the corresponding Bessel process. Moreover, if all components of $k$ are at least $1/2$, and $x$ is in the interior of $C_{N}$, then $(X_{t,k})_{t\geq 0}$ does not hit the boundary a.s..

The associated renormalized processes $(X_{t,k}/\sqrt{k})_{t\geq 0}$ then start in $x$ and satisfy SDEs with drifts independent of $k$ and vanishing diffusion part as $k\to\infty$. In [AV1], several strong, locally uniform limit theorems, of law of large number type, were derived. Moreover, only in a simple special case, a corresponding central limit theorem was derived in [AV1]. In the present paper we derive functional central limit theorems in general as follows: The renormalized processes $(X_{t,k}/\sqrt{k})_{t\geq 0}$ satisfy SDEs with vanishing Brownian parts for $k\to\infty$. Let $\phi:=(\phi(t,x))_{t\geq 0}$ be the solution of the associated deterministic limit differential equation with start in $x$. We shall present Gaussian diffusions $W:=(W_{t})_{t\geq 0}$ such that

[TABLE]

locally uniformly in $t$ with rate $O(1/\sqrt{k})$ a.s.. This $\phi$ plays an essential role in these limit theorems, especially in the covariance matrix of $W$. Unfortunately, up to particular examples, $\phi$ cannot be written down explicitly. However, $\phi$ can be described explicitly in some new coordinates using elementary symmetric polynomials; we shall present these results on $\phi$ in [VW]. We mention that for particular starting points $x\in C_{N}$, the functions $\phi$ are provided explicitly via the zeros of the Hermite or Laguerre polynomial. This connection with zeros of classical orthogonal polynomials already appeared in [AKM1, AKM2, AM, AV1].

This paper is organized as follows. In the next two sections we study Bessel processes of type $A_{N-1}$ for $k\to\infty$ and, in the $B_{N}$-case, for multiplicities $k=(k_{1},k_{2})=(\nu\cdot\beta,\beta)$ with $\nu>0$ and $\beta\to\infty$. Depending on $\nu$, here the zeros of the Laguerre polynomial $L_{N}^{\nu-1}$ play a prominent role. Section 4 is devoted to the root systems $D_{N}$. These results are then used in Section 5 to settle also the limits $k=(k_{1},k_{2})$ for $k_{1}\geq 0$ fixed and $k_{2}\to\infty$ in the $B_{N}$-case. As in the $B_{N}$-case the case $k=(k_{1},k_{2})$ for $k_{2}\geq 0$ fixed and $k_{1}\to\infty$ was already treated in [AV1], we skip this case here. In Section 6 we consider an extension of our results by adding an additional drift of the form $-\lambda X_{t,k}$, $\lambda\in\mathbb{R}$ to our SDEs. For $\lambda>0$ the resulting process is ergodic and mean reverting. For $N=1$ with $\lambda>0$, it is related to the Cox-Ingersoll-Ross process in finance.

2. The root system $A_{N-1}$

The SDE (1.6) for Bessel processes $(X_{t,k})_{t\geq 0}$ of type $A_{N-1}$ reads as

[TABLE]

with an $N$-dimensional Brownian motion $(B_{t}^{1},\ldots,B_{t}^{N})_{t\geq 0}$. Hence, the renormalized processes $(\tilde{X}_{t,k}:=X_{t,k}/\sqrt{k})_{t\geq 0}$ satisfy

[TABLE]

The solutions of (2.2) are closely related to limit $k=\infty$ in Lemma 2.1 of [AV1]. By [AV1] we know that for each starting point $x_{0}$ in the interior of the Weyl chamber $C_{N}^{A}$, the dynamical system

[TABLE]

with

[TABLE]

has a unique solution $\phi(t,x_{0})$ for $t\geq 0$. It admits an explicit solution

[TABLE]

for special starting points of the form $cz$, with $c>0$ and

[TABLE]

consisting of the ordered zeroes $z_{i}$ of Hermite polynomial $H_{N}$. We assume that the $(H_{N})_{N\geq 0}$ are orthogonal w.r.t. the density $e^{-x^{2}}$ as in [S].

It can be shown that the solutions $\phi$ in (2.4) are attracting in some way, and all solutions of the ODEs (2.3) can be determined explicitly after some transformation of coordinates; we shall discuss this in [VW].

We here only discuss the growth behavior of $\phi$. We observe that

[TABLE]

As $\|\phi(0,x)\|_{2}^{2}=\|x\|^{2}$, we see that for all $t\geq 0$ and all $x$,

[TABLE]

We now use $\phi$ to derive limit theorems for Bessel processes of type $A_{N-1}$ for $k\to\infty$. We have the following strong limit law from Theorem 2.4 of [AV1].

Theorem 2.1**.**

Let $x$ be a point in the interior of $C_{N}^{A}$ and $y\in\mathbb{R}^{N}$. Let $k_{0}\geq 1/2$ such that $\sqrt{k}\cdot x+y$ is in the interior of $C_{N}^{A}$ for $k\geq k_{0}$.

For $k\geq k_{0}$ consider the Bessel processes $(X_{t,k})_{t\geq 0}$ of type $A_{N-1}$ starting at $\sqrt{k}\cdot x+y$. Then, for all $t>0$,

[TABLE]

In particular, locally uniformly in $t$ a.s., $X_{t,k}/\sqrt{k}\to\phi(t,x)$ for $k\to\infty.$

We now turn to an associated functional central limit theorem. For this we again fix $x$ in the interior of $C_{N}^{A}$ and consider $t\mapsto\phi(t,x)$ ($t\geq 0$) as solution of (2.3). We also introduce the $N$-dimensional process $(W_{t})_{t\geq 0}$ as unique solution of the inhomogeneous linear SDE

[TABLE]

with initial condition $W_{0}=0$; notice that the denominator is $\neq 0$ for $t>0$. In matrix notation, (2.9) means that

[TABLE]

with the matrices $A(t,x)\in\mathbb{R}^{N\times N}$ with

[TABLE]

for $i,j=1,\ldots,N$, $i\neq j$. The process $(W_{t})_{t\geq 0}$ is obviously Gaussian and admits the explicit representation in terms of matrix-valued exponentials

[TABLE]

$(W_{t})_{t\geq 0}$ is related to the Bessel processes $(X_{t,k})_{t\geq 0}$ by the following functional CLT:

Theorem 2.2**.**

Let $x$ be a point in the interior of $C_{N}^{A}$ and $y\in\mathbb{R}^{N}$. Let $k_{0}\geq 1/2$ such that $\sqrt{k}\cdot x+y$ is in the interior of $C_{N}^{A}$ for $k\geq k_{0}$. For $k\geq k_{0}$ consider the Bessel processes $(X_{t,k})_{t\geq 0}$ of type $A_{N-1}$ starting at $\sqrt{k}\cdot x+y$. Then, for $t>0$,

[TABLE]

i.e., $\lim_{k\to\infty}\sqrt{k}(\frac{X_{t,k}}{\sqrt{k}}-\phi(t,x))=W_{t}$ locally uniformly in $t$ a.s. with rate $O(1/\sqrt{k})$.

Proof.

For $k\geq k_{0}$ consider the processes $(R_{t,k}:=X_{t,k}-\sqrt{k}\phi(t,x)-W_{t})_{t\geq 0}$ on $\mathbb{R}^{N}$. Then $R_{0,k}=0$, and, by the SDEs (2.9) and (2.1) and the ODE for $\phi$ in (2.3),

[TABLE]

for $i=1,\ldots,N$. We now use Taylor expansion for the function $1/x$ with Lagrange remainder around some point $x_{0}\neq 0$, i.e.,

[TABLE]

with some $\tilde{x}$ between $x\neq 0$ and $x_{0}\neq 0$ where $x,x_{0}$ have the same sign. Taking

[TABLE]

we arrive at

[TABLE]

with the error terms

[TABLE]

where, by the Lagrange remainder,

[TABLE]

By Theorem 2.1, this can be bounded by some a.s. finite random variable $D$ independent of $i,j$, $s\in[0,t]$, and $k\geq k_{0}$ where $D$ depends on $x,y,t$. Therefore,

[TABLE]

with some a.s. finite random variable $H$. In summary,

[TABLE]

and thus, for suitable norms and all $u\in[0,t]$,

[TABLE]

with $A:=\sup_{s\in[0,t]}\|A(s,x)\|<\infty$. Hence, by the classical lemma of Gronwall,

[TABLE]

for all $u\in[0,t]$. This yields the claim. ∎

Remark 2.3**.**

The processes $(X_{t,k})_{t\geq 0}$ of type A admit some algebraic properties which are related with corresponding properties of $\phi$ and $A(t,x)$:

1. (1)

$(X_{t,k})_{t\geq 0}$ has the same scaling as Brownian motions, i.e., for $r>0$, the process $(\frac{1}{r}X_{r^{2}t,k})_{t\geq 0}$ is also a Bessel process of type A with the same $k$. The corresponding relations for $\phi$ and $A$ are

[TABLE]

Moreover, for $(W_{t})_{t\geq 0}$ from (2.11), $(\frac{1}{r}W_{r^{2}t})_{t\geq 0}$ is also a process of this type where $x$ is replaced by $rx$ in Eqs. (2.9)–(2.11). 2. (2)

By (2.1), the center of gravity

[TABLE]

is a Brownian motion up to scaling. For $\phi$ and $(W_{t})_{t\geq 0}$ this means that

[TABLE]

for $t\geq 0$. This yields that the sums over all rows and columns of $A(t,x)$ are equal to 0, i.e., $A(t,x)$ is singular. 3. (3)

Let $(\widehat{X_{t,k}})_{t\geq 0}$ be the orthogonal projection of the Bessel process $(X_{t,k})_{t\geq 0}$ to the orthogonal complement $(1,\ldots,1)^{\bot}\subset\mathbb{R}^{N}$ of $(1,\ldots,1)$. Then $(\widehat{X_{t,k}})_{t\geq 0}$ is again a diffusion living on this $(N-1)$-dimensional subspace which is stochastically independent of the center-of-gravity-process $(\overline{X_{t,k}})_{t\geq 0}$ on $\mathbb{R}\cdot(1,\ldots,1)$. On the level of $\phi$ and $A$, we have the relations

[TABLE]

and $A(t,x+r\cdot(1,\ldots,1))=A(t,x)$ for $r\in\mathbb{R}$.

Remark 2.4**.**

Similarly as in Theorem 2.2 we may deduce functional central limit theorems for the powers $X_{t,k}^{p}$ ($p\in\mathbb{N}$) where these powers are taken in all coordinates. The most prominent examples appear for $p=2$. We have

[TABLE]

a.s. locally uniformly in $t$ with limit processes which solve

[TABLE]

with the matrices $A_{p}(t,x)\in\mathbb{R}^{N\times N}$ with

[TABLE]

[TABLE]

Remark 2.5**.**

Assume now that $N$ is odd, and that $(X_{t,k})_{t\geq 0}$ starts at $\sqrt{k}cz$ in the interior of $C_{N}^{A}$. In order to study the $\frac{N-1}{2}+1$-th component, we notice that $z_{\frac{N-1}{2}+1}=0$ and that hence for $p\geq 2$ we have $W_{t}^{\frac{N-1}{2}+1}=0$ which implies a degenerate normal limiting distribution. This suggests that for the $\frac{N-1}{2}+1$-th component we have a faster rate of convergence than $\sqrt{k}$. Indeed from Theorem 2.2 we see that as $k\to\infty$

[TABLE]

for some normal random variable $\mathcal{N}$ and hence $(X_{t,k}^{\frac{N-1}{2}+1})^{p}\stackrel{{\scriptstyle d}}{{\to}}{\mathcal{N}}^{p}$.

We finally calculate the covariance matrix of $W_{t}$ for the special solution $\phi$ given by (2.4). For this we introduce the matrix $A\in\mathbb{R}^{N\times N}$ with

[TABLE]

with $z$ as in (2.5). Moreover, let $E$ be the N-dimensional unit matrix. It is shown in [AV2] that the matrix $E-A$ has the eigenvalues $1,2,\ldots,N$.

Lemma 2.6**.**

Assume that $(X_{t,k})_{t\geq 0}$ starts in the interior of $C_{N}^{A}$ in $\sqrt{k}\cdot cz+y$ with $y\in\mathbb{R}$, $z$ as in (2.5) and $c>0$. Then the covariance matrices $\Sigma_{t}\in\mathbb{R}^{N\times N}$ for $t>0$ of the limit process $(W_{t})_{t\geq 0}$ are given by

[TABLE]

with eigenvalues $\lambda_{k}^{A}(t,c)=\frac{1}{2k}\frac{(2t+c^{2})^{k}-c^{2k}}{(2t+c^{2})^{k-1}}$ ($k=1,\ldots,N$), where $\lambda_{1}^{A}(t,c)=t$.

Proof.

For the special case $\phi(s,cz)=\sqrt{2s+c^{2}}z$, the matrix function $A(s,cz)$ satisfies $A(s,cz)=\frac{1}{2s+c^{2}}A$. Hence,

[TABLE]

Since $A$ is real and symmetric with eigenvalues $1,\ldots,n$, we have $A=UDU^{t}$ with an orthogonal matrix $U$ and the diagonal matrix

[TABLE]

Hence,

[TABLE]

with the rotated Brownian motion $(\tilde{B}_{t}:=U^{t}B_{t}U)_{t\geq 0}$. This, the Itô-isometry, and $d_{i}/2\neq 1$ for all $i$ yield

[TABLE]

As

[TABLE]

and

[TABLE]

we obtain by functional calculus that

[TABLE]

which yields the desired form of the covariance matrix. ∎

Remark 2.7**.**

1. (1)

The covariance matrix $(E-A)^{-1}$ above already appeared in [V] for the case $X_{0,k}=0$ and in [DE2] in the context of asymptotics for eigenvalues of Hermite ensembles. Note that though we assume $c>0$ we may formally set $c=0$ and obtain $\Sigma_{t}=t(E-A)^{-1}$ as in [V]. Since $\lim_{t\to\infty}\Sigma_{t}/t=(E-A)^{-1}$, we obtain asymptotically in $t$ the same result as starting in zero independent of the actual starting point $cz$. 2. (2)

As

[TABLE]

the diagonal elements have a behavior in $t$ which is different from the remaining entries of $\Sigma_{t}$. 3. (3)

Lemma 2.6 holds also for $t=0$ with $\Sigma_{0}=0$.

3. The root system $B_{N}$

We now turn to Bessel processes for the root systems $B_{N}$ with multiplicities $k=(k_{1},k_{2})=(\nu\cdot\beta,\beta)$ with $\nu>0$ fixed and $\beta\to\infty$. We first recapitulate some facts from [AKM2] and [AV1]. The SDE (1.6) of type $B_{N}$ here reads as

[TABLE]

for $i=1,\ldots,N$ with an $N$-dimensional Brownian motion $(B_{t}^{1},\ldots,B_{t}^{N})_{t\geq 0}$. The renormalized processes $(\tilde{X}_{t,k}:=X_{t,k}/\sqrt{\beta})_{t\geq 0}$ satisfy

[TABLE]

for $i=1,\ldots,N$. These processes are related with the deterministic limit case $\beta=\infty$; see Lemma 3.1 of [AV1]. By [AV1] we know that for $\nu>0$, for each starting point $x_{0}$ in the interior of $C_{N}^{B}$, the ODE

[TABLE]

with

[TABLE]

has a unique solution $\phi(t,x_{0})$ in the interior of $C_{N}^{B}$ for $t>0$. Furthermore, we obtain the special solution

[TABLE]

for the special starting points $cy$, with $c>0$ and $y\in C_{N}^{B}$ given by the vector with

[TABLE]

Here $z_{1}^{(\nu-1)},\ldots,z_{N}^{(\nu-1)}$ denote the ordered zeros of the Laguerre polynomials $L_{N}^{(\nu-1)}$.

For this we recapitulate that for $\alpha>0$ the Laguerre polynomials $(L_{n}^{(\alpha)})_{n\geq 0}$ are orthogonal w.r.t. the density $e^{-x}\cdot x^{\alpha}$.

As in Section 2, the solutions $\phi$ in (3.4) are attracting in some way, and all solutions of the ODEs (3.3) can be determined explicitly after some transformation of coordinates; see [VW]. Moreover, similar to the preceding section,

[TABLE]

As $\|\phi(0,x)\|^{2}=\|x\|^{2}$, we obtain for $t\geq 0$ that

[TABLE]

The solutions $\phi$ appear in the following strong limit law from [AV1].

Theorem 3.1**.**

Let $\nu>0$. Let $x$ be a point in the interior of $C_{N}^{B}$, and $y\in\mathbb{R}^{N}$. Let $\beta_{0}\geq 1/2$ with $\sqrt{\beta}\cdot x+y$ in the interior of $C_{N}^{B}$ for $\beta\geq\beta_{0}$. For $\beta\geq\beta_{0}$, consider the Bessel processes $(X_{t,k})_{t\geq 0}$ of type B with $k=(k_{1},k_{2})=(\beta\cdot\nu,\beta)$, which start in $\sqrt{\beta}\cdot x+y$. Then, for all $t>0$,

[TABLE]

In particular, $\lim_{\beta\to\infty}X_{t,(\nu\cdot\beta,\beta)}/\sqrt{\beta}=\phi(t,x)$ locally uniformly in $t$ a.s..

We now turn to an associated functional central limit theorem. We fix $x$ in the interior of $C_{N}^{B}$ and consider the associated $\phi(t,x)$. We also introduce the process $(W_{t})_{t\geq 0}$ on $\mathbb{R}^{N}$ as unique solution of the inhomogeneous linear SDE

[TABLE]

for $i=1,\ldots,N$ with initial condition $W_{0}=0$. Notice that all denominators are $\neq 0$ for $t>0$. (3) may be written in matrix notation as

[TABLE]

with the matrices $A_{\nu}(t,x)\in\mathbb{R}^{N\times N}$ with

[TABLE]

for $i,j=1,\ldots,N$. The process $(W_{t})_{t\geq 0}$ is given by

[TABLE]

This process is obviously Gaussian; we describe it more closely below.

Theorem 3.2**.**

Let $\nu>0$. Let $x$ be a point in the interior of $C_{N}^{B}$ and let $y\in\mathbb{R}^{N}$. Let $\beta_{0}\geq 1/2$ such that $\sqrt{\beta}\cdot x+y$ is in the interior of $C_{N}^{A}$ for $\beta\geq\beta_{0}$.

For $\beta\geq\beta_{0}$ consider the Bessel processes $(X_{t,k})_{t\geq 0}$ of type $B_{N}$ with $k=(\nu\beta,\beta)$ starting at $\sqrt{\beta}\cdot x+y$. Then, for all $t>0$,

[TABLE]

i.e., $\lim_{\beta\to\infty}X_{s,k}-\sqrt{\beta}\phi(s,x)=W_{t}$ locally uniformly in $t$ a.s. with rate $O(1/\sqrt{\beta})$.

Proof.

For $\beta\geq\beta_{0}$ consider the processes

[TABLE]

on $\mathbb{R}^{N}$ with $R_{0,\beta}=0$. Then by (3), (3.1) and the ODE (3.3), $R_{t,\beta}^{i}$ is eqaul to

[TABLE]

The Taylor expansion for $1/x$ with Lagrange remainder at $x_{0}\neq 0$ yields

[TABLE]

with some $\tilde{x}$ between $x\neq 0$ and $x_{0}\neq 0$ which have the same signs. Taking

[TABLE]

we get

[TABLE]

with the error terms

[TABLE]

where, by the Lagrange remainders in the 3 Taylor expansions above,

[TABLE]

and

[TABLE]

By Theorem 3.1, the terms $|D_{i,j}^{\pm}|,|D_{i}|$ can be bounded by some a.s. finite random variable $D$ independent of $i,j$, the sign, $s\in[0,t]$, and $\beta\geq\beta_{0}$ where $D$ depends on $x,y,t$. Therefore, for all $i=1,\ldots,N$,

[TABLE]

with some a.s. finite random variable $H$. In summary,

[TABLE]

and thus, for suitable norms and all $u\in[0,t]$,

[TABLE]

with $A:=\sup_{s\in[0,t]}\|A_{\nu}(s,x)\|<\infty$. Hence, by the classical lemma of Gronwall,

[TABLE]

for all $u\in[0,t]$. This yields the claim. ∎

We next calculate the covariance matrix of $W_{t}$ for the special solution $\phi$ of (3.4). For this we introduce the matrices $A_{\nu}=(A_{\nu,i,j})_{i,j}\in\mathbb{R}^{N\times N}$ with

[TABLE]

for $i,j=1,\ldots,N$, $i\neq j$ and $y$ as in (3.5). By [AV2], $E-2A_{\nu}$ has the eigenvalues $2,4,\ldots,2N$ independent of $\nu$. With these notations we have:

Lemma 3.3**.**

Let $\nu>0$. Assume that the Bessel process $(X_{t,k})_{t\geq 0}$ of type B with $k=(\nu\beta,\beta)$ starts in the point $\sqrt{\beta}\cdot cy+w$ in the interior of $C_{N}^{B}$ with $w\in\mathbb{R}$, $y$ as in (3.5), and $c>0$. Then, the covariance matrices $\Sigma_{\nu,t}\in\mathbb{R}^{N\times N}$ for $t>0$ of the limit Gaussian process $(W_{t})_{t\geq 0}$ are given by

[TABLE]

with eigenvalues $\lambda_{k}^{B}(t,c)=\frac{1}{2k}\frac{(t+c^{2})^{2k}-c^{4k}}{(t+c^{2})^{2k-1}}$ $(k=1,\ldots,N$).

Proof.

For the special case $\phi(s,cy)=\sqrt{s+c^{2}}y$ with $c>0$ and the vector $y$ in (3.5), the matrix function $A_{\nu}(s,cz)$ has the form $A_{\nu}(s,cy)=\frac{1}{s+c^{2}}A_{\nu}$. Hence,

[TABLE]

Since $A_{\nu}$ is real and symmetric with eigenvalues $2,4,\ldots,2N$, we may write $A_{\nu}$ as $A_{\nu}=UDU^{t}$ with an orthogonal matrix $U$ and the diagonal matrix

[TABLE]

Hence,

[TABLE]

with the rotated Brownian motion $(\tilde{B}_{t}:=U^{t}B_{t}U)_{t\geq 0}$. Now similar arguments as in the proof of Lemma 2.6 yield the desired form of the covariance matrix. ∎

Remark 3.4**.**

The eigenvalues of $\Sigma_{t}$ in the cases $A_{2N-1}$ and $B_{N}$ are related by

[TABLE]

independent of $\nu$. This seems to be connected in some way with the formula $H_{2N}(x)=const.(N)\cdot L_{N}^{(-1/2)}(x^{2})$ for $\nu=1/2$.

All preceding results hold for $\nu>0$. We show in Section 4 that most results are also valid for $\nu=0$ with some modifications.

4. The root system $D_{N}$

We now briefly study limit theorems for Bessel processes of type $D_{N}$. We recapitulate that the associated closed Weyl chamber is

[TABLE]

i.e., $C_{N}^{D}$ is a doubling of $C_{N}^{B}$ w.r.t. the last coordinate. We have a one-dimensional multiplicity $k\geq 0$, and the SDE (1.6) for the processes $(X_{t,k})_{t\geq 0}$ of type $D$ is

[TABLE]

for $i=1,\ldots,N$ with an $N$-dimensional Brownian motion $(B_{t}^{1},\ldots,B_{t}^{N})_{t\geq 0}$. The renormalized processes $(\tilde{X}_{t,k}:=X_{t,k}/\sqrt{k})_{t\geq 0}$ satisfy

[TABLE]

For $k=\infty$ we have from Lemma 4.1 of [AV1] that for each starting point $x_{0}$ in the interior of $C_{N}^{D}$, the ODE

[TABLE]

with

[TABLE]

has a unique solution $\phi(t,x_{0})$ in the interior of $C_{N}^{D}$ for $t\geq 0$. Again, as in Section 4 of [AV1], a special solution associated to the zeros of Laguerre polynomials may be derived. Using the representation

[TABLE]

of the Laguerre polynomials according to (5.1.6) of Szegö [S], we form the polynomial $L_{N}^{(-1)}$ of order $N\geq 1$ where, by (5.2.1) of [S],

[TABLE]

We now denote the $N$ zeros of $L_{N}^{(-1)}$ by $z_{1}>\ldots>z_{N}=0$ and define

[TABLE]

Then we obtain for the starting points $cr$ with $c>0$, the particular solutions

[TABLE]

of the ODE (4.3); cf. [AV1].

Again, the solutions (4.6) are attracting in some way. Moreover, as in the preceding sections, we have

[TABLE]

Beside the particular solutions given by (4.6) we have the following observation which fits with Eq. (4.4).

Lemma 4.1**.**

Let $x$ be a point in the interior of $C_{N}^{D}$ with $x_{N}=0$. Then the associated solution of the ODE (4.3) satisfies $\phi(t,x)_{N}=0$ for all $t$, and the first $N-1$ components $(\phi(t,x)_{1},\ldots,\phi(t,x)_{N-1})$ solve the ODE of the B-case in (3.3) with dimension $N-1$ and $\nu=2$.

Proof.

If $x_{N}=0$, then by the ODE in (4.3), $\frac{d}{dt}\phi(t,x)_{N}=0$. ∎

We next recapitulate the following strong limit law; see Theorem 5.5 of [AV1]:

Theorem 4.2**.**

Let $x$ be a point in the interior of $C_{N}^{D}$, and $y\in\mathbb{R}^{N}$. Let $k\geq 1/2$ with $\sqrt{k}\cdot x+y$ in the interior of $C_{N}^{B}$ for $k\geq k_{0}$. For $k\geq k_{0}$, consider the Bessel processes $(X_{t,k})_{t\geq 0}$ of type $D_{N}$ starting in $\sqrt{k}\cdot x+y$. Then, for all $t>0$,

[TABLE]

In particular, $X_{t,k}/\sqrt{k}\to\phi(t,x)$ for $k\to\infty$ locally uniformly in $t$ a.s..

We now turn to an associated functional CLT. We fix some $x$ in the interior of $C_{N}^{D}$ and consider the associated solution $\phi(t,x)$. We also introduce an $N$-dimensional process $(W_{t})_{t\geq 0}$ as the unique solution of the inhomogeneous linear SDE

[TABLE]

for $i=1,\ldots,N$ with initial condition $W_{0}=0$. (4.8) may be written as

[TABLE]

with the matrix $A(t,x)\in\mathbb{R}^{N\times N}$ with

[TABLE]

for $i,j=1,\ldots,N$. The process $(W_{t})_{t\geq 0}$ is Gaussian and given by

[TABLE]

It is related to the Bessel processes $(X_{t,k})_{t\geq 0}$ of type $D$ by the following result. As the proof is completely analogous to that of Theorem 3.2, we omit the proof.

Theorem 4.3**.**

Let $x$ be a point in the interior of $C_{N}^{D}$ and $y\in\mathbb{R}^{N}$. Let $k_{0}\geq 1/2$ such that $\sqrt{k}\cdot x+y$ is in the interior of $C_{N}^{D}$ for $k\geq k_{0}$. For $k\geq k_{0}$ consider the Bessel processes $(X_{t,k})_{t\geq 0}$ starting at $\sqrt{k}\cdot x+y$. Then, for all $t>0$,

[TABLE]

i.e., $X_{s,k}-\sqrt{k}\phi(s,x)\to W_{s}$ locally uniformly in $s$ a.s. with rate $O(1/\sqrt{k})$.

Remark 4.4**.**

Consider the Bessel processes $(X_{t,k})_{t\geq 0}$ of Theorem 4.3 which start in $\sqrt{k}\cdot x$ for $x$ in the interior of $C_{N}^{D}$ with $x_{N}=0$. Then, by Lemma 4.1, $\phi(t,x)_{N}=0$ for all $t\geq 0$, and the matrix function $A$ from (4) satisfies $A(t,x)_{N,N}=0$.

We next calculate the covariance matrix of $W_{t}$ for the special solution $\phi$ in (4.6). For this we introduce the matrix $A\in\mathbb{R}^{N\times N}$ with

[TABLE]

for $i,j=1,\ldots,N$, $i\neq j$ and the vector $r$ as in (4.5). By [AV2], $E-2A$ has the eigenvalues $2,4,\ldots,2N$. We obtain the following result. As its proof is again analog to that of Lemma 3.3, we skip the proof.

Lemma 4.5**.**

Assume that the Bessel processes $(X_{t,k})_{t\geq 0}$ of type $D_{N}$ start in the points $\sqrt{k}\cdot cr+w$ in the interior of $C_{N}^{D}$ with $w\in\mathbb{R}^{N}$, $r$ given in (4.5), and $c>0$. Then, the covariance matrices $\Sigma_{t}\in\mathbb{R}^{N\times N}$ for $t>0$ of the limit Gaussian process $(W_{t})_{t\geq 0}$ are

[TABLE]

Remark 4.6**.**

Let $x$ be in the interior of $C_{N}^{D}$ with $x_{N}=0$. Then, by Lemma 4.1, $A(t,x)_{N,j}=A(t,x)_{j,N}=0$ for $j\neq N$ and $t>0$ in (4). Hence, by Eq. (4.11), that the $N$-th component $(W_{t}^{(N)})_{t\geq 0}$ of $(W_{t})_{t\geq 0}$ is independent from $(W_{t}^{(1)},\ldots,W_{t}^{(N-1)})_{t\geq 0}$. This appears in particular in the setting of Lemma 4.5.

5. Further limit theorems for the case B

Section 4 is closely related to limit results for Bessel processes of type B for $(k_{1},k_{2})=(0,k)$ for $k\to\infty$ which was excluded in Section 3. To explain this, we recapitulate some facts from [AV1], [V]. Let $(X_{t,k}^{D})_{t\geq 0}$ be a Bessel process of type D with multiplicity $k\geq 0$ on $C_{N}^{D}$ with starting point $x$ in the interior of $C_{N}^{D}$. It follows from the generator of the associated semigroup that then the process $(X_{t,k}^{B})_{t\geq 0}$ with

[TABLE]

is a Bessel process of type B with the multiplicity $(k_{1},k_{2}):=(0,k)$ and starting point $(x_{1},\ldots,x_{N-1},|x_{N}|)\in C_{N}^{B}$ with $x_{1}>\ldots>x_{N-1}>|x_{N}|\geq 0$. Notice that $(X_{t,k}^{B})_{t\geq 0}$ is a diffusion with reflecting boundary where the boundary parts with the $N$-th coordinate equal to zero are attained.

We now translate the results of Section 4. For this we consider the solutions $\phi(t,x)$ of the ODE (4.3) in the following two particular cases for $t\geq 0$:

1. (1)

If $x$ is in the interior of $C_{N}^{B}$, then $\phi(t,x)$ is also in the interior of $C_{N}^{B}$. 2. (2)

If $x_{1}>\ldots>x_{N-1}>x_{N}=0$, then $\phi(t,x)_{1}>\ldots>\phi(t,x)_{N-1}>\phi(t,x)_{N}=0$.

Case (2) appears in particular for $\phi(t,x)=\sqrt{t+c^{2}}\cdot r$ for $c>0$ and the vector $r$ as in (4.6) with $r_{N}=0$.

Theorem 4.2 now reads as follows for the B-case with $(k_{1},k_{2})=(0,k)$ for $k\to\infty$:

Theorem 5.1**.**

Let $x$ be as described above in (1) or (2). For $k\geq 1/2$, consider the Bessel processes $(X_{t,(0,k)})_{t\geq 0}$ of type $B_{N}$ starting in $\sqrt{k}\cdot x$. Then, for $t>0$,

[TABLE]

We next consider the Gaussian processes $(W_{t})_{t\geq 0}$ of Eq. (4.11). Theorem 4.3 now leads to functional CLTs where the cases (1) and (2) have to be treated separately for geometric reasons. For the case (1) we have the following result:

Theorem 5.2**.**

Let $x$ be a point in the interior of $C_{N}^{B}$. For $k\geq 1/2$ consider Bessel processes $(X_{t,(0,k)})_{t\geq 0}$ of type B starting at $\sqrt{k}\cdot x$. Then, for all $t>0$,

[TABLE]

Proof.

As $x$ is in the interior of $C_{N}^{B}$, we obtain that for each $t>0$ and almost all $\omega\in\Omega$, the path $(\sqrt{k}\phi(s,x)-W_{s}(\omega))_{s\in[0,t]}$ is arbitrarily far away from the boundary of $C_{N}^{B}$ whenever $k$ is sufficiently large. This, the connection between the D- and B-case, and Theorem 4.3 thus lead to the theorem. ∎

Theorem 5.2 corresponds to Theorem 3.2 for $\nu=0$. We next turn to case (2):

Theorem 5.3**.**

Let $x\in C_{N}^{B}$ with $x_{1}>\ldots>x_{N-1}>x_{N}=0$. For $k\geq 1/2$ consider Bessel processes $(X_{t,(0,k)})_{t\geq 0}$ of type B starting at $\sqrt{k}\cdot x$. Then, for the process $(\tilde{W}_{t}:=(W_{t}^{(1)},\ldots,W_{t}^{(N-1)},|W_{t}^{(N)}|))_{t\geq 0}$, and all $t>0$,

[TABLE]

Proof.

This follows immediately from Theorem 4.3, the connection between the D- and B-case, and from $||a|-|b||\leq|a-b|$ for $a,b\in\mathbb{R}$. ∎

For the process $(\tilde{W}_{t})_{t\geq 0}$, the first $N-1$ components form a Gaussian process which is independent from $(|W_{t}^{(N)}|)_{t\geq 0}$ by Remark 4.6. The variables $|W_{t}^{(N)}|$ are one-sided normal distributed. Therefore, Theorems 5.2 and 5.3 lead to a discontinuity (or phase transition) in the limit depending on the starting points here.

6. Extensions to multi-dimensional Bessel processes with an additional Ornstein-Uhlenbeck component

In this section we consider an extension of our previous models by adding an additional drift coefficient of the form $-\lambda x$, $\lambda\in\mathbb{R}$, i.e., a component as in a classical Ornstein-Uhlenbeck setting

[TABLE]

If $\lambda>0$, we obtain a mean reverting ergodic process with speed of mean-reversion $\lambda$. For $\lambda\leq 0$ the process is non-ergodic. For $N=1$ and $\lambda>0$ the squared process is the well-known Cox-Ingersoll-Ross process from mathematical finance.

We derive the results for the root system $A_{N-1}$ only as the same technique also holds for the other root systems. We consider processes $(Y_{t,k})_{t\geq 0}$ of type $A_{N-1}$ as solutions of

[TABLE]

with an $N$-dimensional Brownian motion $(B_{t}^{1},\ldots,B_{t}^{N})_{t\geq 0}$. Itô’s formula and a time-change argument show that $Y$ is a space-time transformation of the original $X$ (with $\lambda=0$), namely

[TABLE]

For a proof based on the generators cf. [RV1]. A similar relation holds for the associated ODEs.

Lemma 6.1**.**

Let $\phi(t,x)$ be a solution of the dynamical system $\frac{dx}{dt}(t)=H(x(t))$ with starting point $x$ in the interior of $C_{N}^{A}$ as in Section 2. Then

[TABLE]

solves the ODE $\frac{dx}{dt}(t)=H(x(t))-\lambda x(t)$ with starting point $x$.

Proof.

This follows from the space-time homogeneity in Remark 2.3. ∎

With the techniques in Theorem 2.1 and Theorem 2.2 we obtain a functional CLT for $(Y_{t,k})_{t\geq 0}$:

[TABLE]

for $k\to\infty$ locally uniformly in $t$ a.s. with rate $O(1/\sqrt{k})$, where

[TABLE]

with the matrices $A^{\lambda}(t,x)\in\mathbb{R}^{N\times N}$ with

[TABLE]

[TABLE]

for $i,j=1,\ldots,N$, $i\neq j$ and with initial condition $W_{0}=0$. The process $(W_{t})_{t\geq 0}$ admits the explicit representation

[TABLE]

Note that due to the constant term in the diagonal of $A^{\lambda}(t,x)$ for $\lambda\neq 0$, we obtain a linear time-dependence in the exponential of the matrix exponential which dominates the long-term behaviour of the covariance matrix. In particular:

Lemma 6.2**.**

Assume that $(Y_{t,k})_{t\geq 0}$ starts in the interior of $C_{N}^{A}$ in $\sqrt{k}\cdot cz+y$ with $y\in\mathbb{R}$, $z$ as in 2.5 and $c>0$. Then the covariance matrices $\Sigma^{\lambda}_{t}\in\mathbb{R}^{N\times N}$ for $t>0$ of the limit process $(W_{t})_{t\geq 0}$ are given by

[TABLE]

where $A$ is defined by (2.15).

Proof.

For the special starting points $cz$ we obtain the special solution

[TABLE]

Hence the matrix function $A^{\lambda}(s,cz)$ has the simple form with the same time-dependence for each entry

[TABLE]

where $A$ is given by (2.15). This yields the process

[TABLE]

Now we may proceed as in Lemma 2.6 to calculate the covariance matrix. ∎

Remark 6.3**.**

The long-term behaviour of the covariance matrix is inherited by the long-term behaviour of $Y$. In the ergodic case for $Y$, i.e. $\lambda>0$, we obtain $\lim_{t\to\infty}\Sigma^{\lambda}_{t}=\frac{1}{2\lambda}(E-A)^{-1}$. For $\lambda<0$ we need an exponential scaling

[TABLE]