Matsumoto–Yor Process and Infinite Dimensional Hyperbolic Space

Abstract

The Matsumoto–Yor process is $\int _{0}^{t}\exp (2B_{s} - B_{t})\,ds,t \geq 0$, where (B _t ) is a Brownian motion. It is shown that it is the limit of the radial part of the Brownian motion at the bottom of the spectrum on the hyperbolic space of dimension q, when q tends to infinity. Analogous processes on infinite series of non compact symmetric spaces and on regular trees are described.

A la mémoire de Marc Yor, avec admiration

Appendix: Asymptotics of Spherical Functions in Rank One

We describe the needed asymptotic behaviour of spherical functions on SO(1, q), SU(1, q) and Sp(1, q) when $q \rightarrow +\infty$. We adapt Shimeno [38] to this setting (which only considers the real split case) and Oshima and Shimeno [32]. The advantage of this approach is that it is adapted to higher rank cases. The details of the computations are quite long but straightforward. Therefore we only indicate the main points of the proof where they differ from [38].

We adapt the notions of Sect. 5.1 to the rank one case. In this case there are at most two roots, α and 2α and we may suppose that α(r) = r when $r \in \mathfrak{a} = \mathbb{R}$. Their multiplicity are $(m_{\alpha },m_{2\alpha }) = (q - 1,0)$ for SO(1, q), (2(q − 1), 1) for SU(1, q) and (4(q − 1), 3) for Sp(1, q). Let

$$\displaystyle{\varrho _{q} = \frac{1} {2}(m_{\alpha } + 2m_{2\alpha }),}$$

and

$$\displaystyle{ \delta _{q}(r) = (e^{r} - e^{-r})^{m_{\alpha }}(e^{2r} - e^{-2r})^{m_{2\alpha } }. }$$

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Then (see (5.1))

$$\displaystyle\begin{array}{rcl} H_{CMS}& =& \delta _{q}^{1/2} \circ \{\varDelta _{ R} +\rho _{ q}^{2}\} \circ \delta _{ q}^{-1/2} {}\\ & =& \frac{d^{2}} {dr^{2}} -\frac{m_{\alpha }(m_{\alpha } + 2m_{2\alpha } - 2)} {\sinh ^{2}r} -\frac{m_{2\alpha }(m_{2\alpha } - 2)} {\sinh ^{2}2r}. {}\\ \end{array}$$

For $\lambda \in \mathbb{C}$, the spherical function $\varphi _{\lambda }$ satisfies

$$\displaystyle{\varDelta _{R}\tilde{\varphi }_{\lambda } = (\lambda ^{2} -\varrho _{ q}^{2})\tilde{\varphi }_{\lambda },}$$

where $\tilde{\varphi }_{\lambda }(r) =\varphi _{\lambda }(D_{1}(r))$, $r \in \mathbb{R}$, therefore,

$$\displaystyle{H_{CMS}(\delta _{q}^{1/2}\tilde{\varphi }_{ \lambda }) =\lambda ^{2}\delta _{ q}^{1/2}\tilde{\varphi }_{ \lambda }.}$$

There exists a unique function $\varPsi _{CMS}(\lambda,q,r),r \in \mathbb{R}$, of the form

$$\displaystyle{ \varPsi _{CMS}(\lambda,q,r) =\sum _{n\in \mathbb{N}}b_{n}(\lambda,q)e^{(\lambda -n)r},\quad b_{ 0}(\lambda,q) = 1, }$$

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such that

$$\displaystyle{ H_{CMS}\varPsi _{CMS} =\lambda ^{2}\,\varPsi _{ CMS}, }$$

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(see [38, (17)]). When $q \rightarrow +\infty$,

$$\displaystyle{\lim H_{CMS}(r +\log m_{\alpha }) = H_{T},}$$

where H _T is the Toda type Hamiltonian

$$\displaystyle{H_{T} = \frac{d^{2}} {dr^{2}} - e^{-2r}.}$$

There is also a unique function $\varPsi _{\text{T}}(\lambda,r),r \in \mathbb{R}$, of the form

$$\displaystyle{ \varPsi _{T}(\lambda,r) =\sum _{n\in \mathbb{N}}b_{n}(\lambda )e^{(\lambda -n)r},\quad b_{ 0}(\lambda ) = 1, }$$

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such that

$$\displaystyle{H_{T}\varPsi _{T} =\lambda ^{2}\varPsi _{ T},}$$

([38], notice that this is the function denoted $\varPsi _{T}(-\lambda,-r)$ in [38]).

Lemma 9

If $\lambda \in \mathfrak{a}_{\mathbb{C}}^{{\ast}}$ and $2\lambda \not\in \mathbb{Z}^{{\ast}}$ , then

$$\displaystyle{\lim _{q\rightarrow +\infty } \frac{1} {m_{\alpha }^{\lambda }}\varPsi _{CMS}(\lambda,q,r +\log m_{\alpha }) =\varPsi _{T}(\lambda,r)}$$

Proof

The proof is similar to the one of Proposition 1 in [38].

Let

$$\displaystyle{ a(q) = \frac{\varGamma (m_{\alpha }/2)^{2}} {\varGamma (m_{\alpha })2^{1+3m_{2\alpha }/2}}, }$$

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and

$$\displaystyle{ g_{q}(\lambda,r) = a(q)(\delta _{q}^{1/2}\tilde{\varphi }_{ \lambda })(\log m_{\alpha } + r) - K_{\lambda }(e^{-r}). }$$

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Proposition 11

Uniformly on $\lambda$ in a small neighborhood of 0 in $\mathbb{C}$ , $\lambda g_{q}(\lambda,r)$ and its derivatives in $\lambda$ converge to 0 as $q \rightarrow +\infty$ .

Proof

Using the Harish Chandra spherical function expansion of $\varphi _{\lambda }$ (see [18, Theorems IV.5.5 and IV.6.4] or [16]), when $\lambda$ is for instance in the ball $\{\lambda \in \mathbb{C};\vert \lambda \vert \leq 1/4\}$ and $\lambda \not =0$, one has

$$\displaystyle{\delta _{q}^{1/2}(r)\tilde{\varphi }_{\lambda }(r) = c(\lambda )\varPsi _{ CMS}(\lambda,q,r) + c(-\lambda )\varPsi _{CMS}(-\lambda,q,r),}$$

where

$$\displaystyle{c(\lambda ) = \frac{2^{\frac{1} {2} m_{\alpha }+m_{2\alpha }-\lambda }\varGamma (\frac{1}{2}(m_{\alpha } + m_{2\alpha } + 1))} {\varGamma (\frac{1} {2}(\frac{1} {2}m_{\alpha } + 1+\lambda ))\varGamma (\frac{1} {2}(\frac{1} {2}m_{\alpha } + m_{2\alpha }+\lambda ))}.}$$

It follows from Lemma 9 that

$$\displaystyle{\lim _{q\rightarrow +\infty }a(q)c(\lambda ) \frac{1} {m_{\alpha }^{\lambda }}\varPsi _{CMS}(\lambda,q,r +\log m_{\alpha }) =\varGamma (\lambda )2^{\lambda -1}\varPsi _{ T}(\lambda,r).}$$

Hence

$$\displaystyle{\lim _{q\rightarrow +\infty }a(q)(\delta _{q}^{1/2}\tilde{\varphi }_{ \lambda })(r +\log m_{\alpha }) =\varGamma (\lambda )2^{\lambda -1}\varPsi _{ T}(\lambda,r) +\varGamma (-\lambda )2^{-\lambda -1}\varPsi _{ T}(-\lambda,r).}$$

The limit can be expressed in terms of the Whittaker function for $Sl(2, \mathbb{R})$ as in [38, Theorem 3]. Due to the relation between Whittaker and Macdonald functions (see, e.g., Bump [6]), one finds that this limit is $K_{\lambda }(e^{-r})$, thus $g_{q}(\lambda,r)$ tends to 0 when $q \rightarrow +\infty$.

Now we remark that, for r fixed, the functions $\lambda g_{q}(\lambda,r)$ are analytic functions in $\lambda$ in the ball $\{\lambda \in \mathbb{C};\vert \lambda \vert \leq 1/4\}$ which are uniformly bounded in q by the computations of [38, Proposition 1]. Notice that we have to multiply by $\lambda$ to avoid the singularity at 0 of the Harish Chandra c function. The uniform convergence of $\lambda g_{q}(\lambda,r)$ and its derivatives in $\lambda$ thus follows from Montel’s theorem.

Corollary 4

As $q \rightarrow +\infty$ , $g_{q}(\lambda,r)$ and all its derivatives at $\lambda = 0$ converge to 0.

Proof

This follows from the proposition and the fact that

$$\displaystyle{\frac{d^{n}} {d\lambda ^{n}}g_{q}(\lambda,r)_{\{\lambda =0\}} = \frac{1} {n + 1} \frac{d^{n+1}} {d\lambda ^{n+1}}(\lambda g_{q}(\lambda,r))_{\{\lambda =0\}}.}$$

Remark 7

A small mistake in the constant in [38, Example 1] is corrected in [32].