数理解析研究所講究録 1253 巻 2002 年 26-37

26Weighted Monotone Fock Space andABrownian Motion with
the Distribution of Bozejko-Leinert-Speicher
NAOFUMI MURAKI
Mathematics Laboratory, Iwate Prefectural University
Takizawa, Iwate 020-0193, Japan

Abstract. We construct on aweighted monotone Fock spacewith weight
sequence $w=$
an example of noncommutative Brownian motion
such that the distributionof an increment $Q_{t}-Q_{s}$ , $0<s<t$ , coincides
with the distribution of-Leinert-Speicher but that the processis
not isomorphic to the -free Brownian motion of-Leinert-Speicher.
$\Phi_{w}$$(1, c, c^{2}, c^{3}, \cdots)$$\{Q_{t}\}_{t\geq 0}$$\mu_{s,t}$$\{Q_{t}\}_{t\geq 0}$$\mathrm{B}\mathrm{o}\dot{\mathrm{z}}\mathrm{e}\mathrm{j}\mathrm{k}\mathrm{o}$$\{\overline{Q}_{t}\}_{t\geq 0}$$\mathrm{B}\mathrm{o}\dot{\mathrm{z}}\mathrm{e}\mathrm{j}\mathrm{k}\mathrm{o}$$\mathrm{c}$1. Weighted Monotone Fock Space

Aweighted monotone Fock spaceis adeformation of the monotone Fock space 4
through aweight sequence, $w_{n}>0$ . It is aspecial case of interacting
Fock spaces [, ALV]. The usual monotone Fock space corresponds to the caseof trivial weight sequence $w_{n}:=1$ , $n\geq 1$ [Lu, Mul, Mu2]. Let us give the precise
definitions.
Letbe the set of all strictly positive real numbers $s>0$ . Denote by
the set of all monotone sequences $\sigma=(s_{n}>s_{n-1}>\cdots>s_{1})$ of length from , which
are increasing to the left. For each $n\geq 1$ ,is the measure space equipped withthe (induced) Lebesgue measure c&.
is the singleton consisting of thenull sequence Awith the point mass (measure). Denote by
the complex-spacewith anew inner product$\Phi_{w}$$w=\{w_{n}\}_{n=0}^{\infty}$$\mathrm{A}\mathrm{c}\mathrm{B}$$T=\mathrm{R}_{+}^{*}$$\Sigma_{n}$

$n$

$T$

$\Sigma_{n}$$\Sigma_{0}=\{\Lambda\}$$=\mathrm{D}\mathrm{i}\mathrm{r}\mathrm{a}\mathrm{c}$$\mathrm{L}^{2}$$\mathcal{H}_{n}$$\mathrm{L}^{2}(\Sigma_{n})$$<u|v>$

This Hilbert space$=$$(u, v\in \mathcal{H}_{n})$$w_{n} \int_{\Sigma_{h}}d\sigma\overline{u(\sigma)}v(\sigma)$$?t_{n}:=(\mathrm{L}^{2}(\Sigma_{n}), w_{n})$$\Phi_{w}$

$:=$

.

is called the -particle space. Then we put$n$$\mathrm{C}\oplus \mathcal{H}_{1}\oplus \mathcal{H}_{2}\oplus\cdots\oplus \mathcal{H}_{n}\oplus\cdots$.

27
and call it aweighted monotone Fock space with aweight sequence.
Here we identified
withthrough the identification of the function 1:
with the unit 1of C. Also we assume that $w_{0}=1$ . We denote by $\Omega:=1$ the vacuum
vector.
For each one-particle vector, the creation operatoronis defined,as the left multiplication operator, by$w=\{w_{n}\}_{n=0}^{\infty}$$\Lambdaarrow 1$$\mathrm{C}$$\mathcal{H}_{0}$$(\in?t_{0})$$f\in \mathcal{H}_{1}$$(\delta_{f}^{+}u)(t>\sigma)$$f(t)u(\sigma)$$=$$(u=\mathrm{u}(\mathrm{a})\in \mathcal{H}_{n})$The annihilation operator
is defined as the adjoint of, we have. The concrete action of on$\delta_{f}^{-}$$\Omega$$\delta_{f}^{-}\Omega=0$$=$

So we put$r_{n}:= \frac{w_{n}}{w_{n-1}}$$\delta_{f}^{+}$$u\in \mathcal{H}_{n}$$\delta_{f}^{-}$$(\delta_{f}^{-}u)(\sigma)$.

. For the vacuum vector, $n\geq 1$ , is given by$\frac{w_{n+1}}{w_{n}}\int_{t>\sigma}dt\overline{f(t)}u(t>\sigma)$.

, n $=1,$ 2,3, \cdots , then aweight sequence w

sponds to asequence r$=\{r_{n}\}_{n=1}^{\infty}$$\Phi_{w}$$\delta_{f}^{+}$$=\{w_{n}\}_{n\ovalbox{\tt\small REJECT}}^{\infty}$

corre-

in the bijective way:

w $=(w_{0}, w_{1}, w_{2},$\cdots ) – r\cdots )$=(r_{1}, r_{2}, r_{3},$under the assumption $w_{0}=1$ .
Let
be the C’-algebra generated by the creation and annihilation operators, and letthe vacuum state over
. We
will working on this C’-probability space. We often use the short notation$<\cdot>:=\phi(\cdot)$ to mean the expectation w.r.t. agiven stateover aC’-algebra.
$A_{w}$$\phi_{w}(\cdot)=<\Omega|\cdot\Omega>\mathrm{b}\mathrm{e}$$\{\delta_{f}^{+}, \delta_{f}^{-}|f\in \mathcal{H}_{1}\}$$A_{w}$$(A_{w}, \phi_{w})$$\phi$

2. Brwonian motion
For each
specialization, the canonical operator
is defined bywith $t\geq 0$ , we obtain the creation process
annihilation process, and the pair of canonical processes$f\in \mathcal{H}_{1}$$Q_{f}$$Q_{f}=\delta_{f}^{+}+\delta_{f}^{-}$$f:=\chi_{(0,c\mathrm{l}}$. By the, the$D_{t}^{+}=\delta_{\chi_{(0.t\mathrm{l}}}^{+}$$D_{t}^{-}=\mathrm{C}5_{\chi_{(0,t\mathrm{l}}}^{-}$$Q_{t}=D_{t}^{+}+D_{\overline{t}}$and $P_{t}=\sqrt{-1}(D_{t}^{+}-D_{t}^{-})$ . Heredenotes the indicator function of an intervalon the real line. We are interested in the probability law of the canonical process.
At first let us consider the independence structure of the process. Let aC’-probability space $(A, \phi)$ and astochastic process
be given. Let 72
be the ring generated by all the semi-closed interval $(s, t]$ with $0<s<t$ . For each, letbe the C’-algebra generated by the increments $\{X_{t}-X_{s}|(s, t]\subset I\}$
supported in . Then aprocess
is said to be aprocess with independent
increments if, for each increasing finite sequence $I_{1}<I_{2}<\cdots<I_{n}$ of elements of, we have$I$$\chi_{I}$$\{Q_{t}\}_{t\geq 0}$$\{Q_{t}\}_{t\geq 0}$$\{X_{t}\}_{t\geq 0}\subset A$$I\in \mathcal{R}$$A_{I}$$I$$\{X_{t}\}_{t\geq 0}$$\mathcal{R}$$\phi(A_{1}A_{2}\cdots A_{n})=\phi(A_{1})\phi(A_{2})\cdots$ $\phi(A_{n})$28
for all
all t$A_{:}\in A_{I}.$$\in J$, i $=1,$ 2, \cdots , n. Of course I $<J$ means that s $<t$ for all s$\in I$

and.

Proposition 2.1.

$\{Q_{t}\}_{t\geq 0}$

is a process with independent increments.

This is acorollary of the following Propisition 2.2. For each
, letbe
the C’-algebra generated by. Then the following iseasily shown.
$A_{t}^{(w)}$$I\in \mathcal{R}$$\{\delta_{f}^{+},\delta_{f}^{-}|f\in \mathcal{H}_{1};\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(f)\subset I\}$

Proposition 2.2. Let
be the system of C’-subalgebras ofdefined above. Then for each increasing finite sequence $I_{1}<I_{2}<\cdots<I_{n}$ ofelements of $R$ , we have$\{A_{I}^{(w)}|I\in \mathcal{R}\}$$(A_{w}, \phi_{w})$$\phi_{w}(A_{1}A_{2}\cdots A_{n})=\phi_{w}(A_{1})\phi_{w}(A_{2})\cdots$

for all$A_{:}\in A_{I_{}}^{(w)}$, $i=1,2$ ,$\cdots$, .
$n$

Proposition 2.2 means that also the pair processincrements process.
Proposition 2.3.

$\{Q_{t}\}_{t\geq 0}$$\phi_{w}(A_{n})$$(Q_{t}, P_{t})_{t\geq 0}$

is an independentis a process with stationary increments, $i.e$ .

$\phi_{w}((Q_{t+\mathrm{u}}-Q_{s+u})^{p})=\phi_{w}((Q_{t}-Q_{s})^{p})$for all $0\leq s<t$ , all $u>0$ and all $p=1,2,3$ ,$\cdots$.

The proof of Proposition 2.3 will become ovbious in the later. Now we know thatis aprocess with independent and stationary increments. Besides
is shown to be ascale-invariant process in the following sense.
(resp.
) be astochastic process on aC’-probability
Let
’
’
$(A,
\phi)$$(\{X_{t}|t\in T\})$ andspace(resp. ( , )). Put$T\})$ where
denotes the C’-subalgebra generated by asubset $E$ . Then aprocess
is said to be isomorphic to aprocess
if there exists some$t\geq0$
isomorphic :such thatfor all
and that $\psi(\pi(X))=$$\phi(X)$ for all $X\in A_{0}$ . Under this definition, we have$\{Q_{t}\}_{t\geq 0}$$\{Q_{t}\}_{t\geq 0}$$\{X_{t}\}_{t\geq 0}\subset A$$\{\mathrm{Y}_{t}\}_{t\geq 0}\subset B$$B$$A_{0}:=\mathrm{C}$$\psi$$B_{0}:=\mathrm{C}$$(\{\mathrm{Y}_{t}|t\in$$\mathrm{C}’(E)$$\{X_{t}\}_{t\geq 0}$$C^{*}$$\{\mathrm{Y}_{t}\}_{t\geq 0}$$\pi$$A_{0}arrow B_{0}$Theorem 2.4. For eachto the original process$\pi(X_{t})=\mathrm{Y}_{t}$$\lambda>0$$\{Q_{t}\}_{t\geq 0}$, the scaled process$\lambda>0$’: u$\Phi_{w}arrow\Phi_{w}$$\in \mathcal{H}_{1}$$f( \cdot)\mapsto\prime f_{\lambda}(\cdot)=\frac{1}{\sqrt{\lambda}}f(\frac{1}{\lambda}\cdot)$

Also for each -particle vector$n$

is isomorphic.

, let us define amap
, we put
For each one-particle vector f

Proof. For each fixed$\{Q_{t\tau_{\lambda}^{1}}’:=Q_{\lambda t}\}_{t\geq 0}$$u\in \mathcal{H}_{n}$.

, we put$u( \cdot)\mapsto\prime u_{\lambda}(\cdot)=(\frac{1}{\lambda})^{\frac{n}{2}}u(\frac{1}{\lambda}\cdot)$.

$\vdash’arrow u’=u_{\lambda}$

as follows.

29
Besides for the vacuum vector, we put$\Omega\mapsto’\Omega$

Then this map$u\mapsto u’’=u_{\lambda}$$<u’|v’>$.

defines aunitary operator on$=$$\Phi_{w}$, because we have$\overline{u’}(\sigma)v’(\sigma)$$w_{n} \int_{\sigma}d\sigma$$=$$w_{n} \int_{\sigma}d\sigma(\frac{1}{\lambda})^{\frac{\overline n}{2}}u(\frac{1}{\lambda}\sigma)(\frac{1}{\lambda})^{\frac{n}{2}}v(\frac{1}{\lambda}\sigma)$$=$$w_{n} \int_{\sigma}\frac{1}{\lambda^{n}}d\sigma u(\frac{1}{\lambda}\sigma)v(\frac{1}{\lambda}\sigma)$

$=$

$=$

$w_{n} \int_{\tau}d\tau\overline{u(\tau)}v(\tau)$$<u|v>$ ,.
because of, and used
where in the 4th equality we put
This unitary operator $\Phi_{w}\ni u\vdasharrow u’\in\Phi_{w}$ naturally induces the transformation ofoperators $T-*T’$ as$d\sigma=\lambda^{n}d\tau$$\tau=\frac{1}{\lambda}\sigma$’’$\Phi_{w}\downarrow\tauarrow$$\Phi_{w}\downarrow\tau’$’$arrow$$\Phi_{w}$Then first we know$(\delta_{f}^{+})’=\delta_{fx}^{+}$$((\delta_{f}^{+})’u’)(t>\sigma)$

$=$

$=$

$((\delta_{f}^{-})’u’)(\sigma)$

$=$

$=$

$=$

$(\delta_{f}^{-})’=\delta_{f_{\lambda}}^{-}$v$u_{\mathrm{I}^{\tau’}}’$’$arrow$$v’$, because we have$=$Besides we know$\Phi_{w}$$uarrow\downarrow\tau$$(\delta_{f}^{+}u)’(t>\sigma)$$( \frac{1}{\lambda})^{\frac{n+1}{2}}(\delta_{f}^{+}u)(\frac{1}{\lambda}(t>\sigma))$$( \frac{1}{\lambda})^{\frac{1}{2}}f(\frac{1}{\lambda}t)\cdot(\frac{1}{\lambda})^{\frac{n}{2}}u(\frac{1}{\lambda}\sigma)$$=$$f_{\lambda}(t)\cdot u_{\lambda}(\sigma)$$=$$(\delta_{f_{\lambda}}^{+}u_{\lambda})(t>\sigma)$$=$$(\delta_{f_{\lambda}}^{+}u’)(t>\sigma)$.

, because we have$(\delta_{f}^{-}u)’(\sigma)$$( \frac{1}{\lambda})^{\frac{n-1}{2}}(\delta_{f}^{-}u)(\frac{1}{\lambda}\sigma)$$( \frac{1}{\lambda})^{\frac{n-1}{2}}\frac{w_{n}}{w_{n-1}}dt\overline{f}(t)u(t>t>\frac{\int_{1}}{\lambda}\sigma\frac{1}{\lambda}\sigma)$$\sigma=\lambda\tau$

30
$=$

$( \frac{1}{\lambda})^{-1}\frac{w_{n}}{w_{n-1}}t>\frac{\int_{1}}{\lambda}\sigma\frac{1}{\lambda}d(\lambda t)(\frac{1}{\lambda})^{\frac{1}{2}}\overline{f}(\frac{1}{\lambda}\cdot\lambda t)\cdot(\frac{1}{\lambda})^{\frac{n}{2}}u(\frac{1}{\lambda}(\lambda t>\sigma))$$=$$\frac{w_{n}}{w_{n-1}}\int_{s>\sigma}ds\overline{f_{\lambda}}(s)\cdot u_{\lambda}(s>\sigma)$$=$$(\delta_{f_{\lambda}}^{-}u_{\lambda})(\sigma)$$=$$(\delta_{f_{\lambda}}^{-}u’)(\sigma)$,

where in the 5th equality we put, we getization$s=\mathrm{X}\mathrm{t}$. So we have$(Qf)’=Q_{f_{\lambda}}$. By the special-$f:=\chi_{\iota 0.t)}$$( \chi_{\iota \mathrm{O},t)})_{\lambda}(s)=\frac{1}{\sqrt{\lambda}}\chi_{\iota 0.t)}(\frac{1}{\lambda}s)=\frac{1}{\sqrt{\lambda}}\chi_{\mathrm{l}0.\lambda t)}(s)$,and hence$Q_{(\chi_{\mathrm{l}0,\iota)})_{\lambda}}=Q_{T^{1}x^{\chi_{\mathrm{l}0.\lambda C)}}}= \frac{1}{\sqrt{\lambda}}Q_{\chi_{\iota 0.\lambda t)}}$.

So we obtain$(Q_{t})’$Besides it is easy to see that$\phi_{w}(T’)=\phi_{w}(T)$ .

$T\vdasharrow T’$$=$$\frac{1}{\sqrt{\lambda}}Q_{\lambda t}$.

is aC’-algebra automorphism of$A_{v}$

satisfying$\square$Corollary 2.5. For $eac/i$$t_{1}$, ,$t_{2}$$\cdots$, $t_{l}\in T$ , we have$<Q_{t_{1}}Q_{t_{2}}\cdots Q_{t_{l}}>=<Q_{t_{1}}’Q_{t_{2}}’\cdots Q_{t_{l}}’>$.

$Pro\mathrm{o}/$$T\Delta$$T’$

is aC’-algebra automorphism of$A_{w}$.

satisfying $\phi_{w}(T’)=\phi_{w}(T)$ .

$\square$Proposition 2.6.

for , $t\in T$ .

$<Q_{s}Q_{t}>= \min\{s, t\}$$s$

By Propositions 2.1, 2.3 and 2.4, it is natural to interpret the process
as anoncommutative analogue of Brownian motion.

$\{Q_{t}\}_{t\geq 0}$

3. Moments of Canonical Operators
Let aweighted monotone Fock spacewith the weight sequence
be given. In this section, we derive some recurrence relations concerning the
moments of the distributionon
under, of the canonical operator
the vacuum state. For simplicity we assume that. Put.
Let us treat the moments of :$\Phi_{w}$$w=\{\mathrm{w}\mathrm{n}\}(rightarrow \mathrm{r}=$$\{r_{n}\})$$Q_{f}$$\mu_{[,w}=\mu_{f\mathrm{r}}$$\phi_{w}$$||f||_{\mathrm{L}^{2}}=1$$\mu:=\mu_{f,\mathrm{r}}$$\mu$$m_{p}$$=$$\phi_{w}(Q_{f}^{p})$$= \int_{-\infty}^{+\infty}x^{p}d\mu$,$p=0,1,2,3$ ,$\Phi_{w}$$\cdots$

.

31
The

moment$p\mathrm{t}\mathrm{h}$$m_{p}$$m_{p}$

can be expanded as$=$$=\phi((D_{f}^{+}+D_{f}^{-})^{p})$$\phi(Q_{f}^{p})$$=$$\sum_{\epsilon_{1},\epsilon_{2},\cdots,\epsilon_{p}\in\{+,-\}}\phi(D_{f}^{\epsilon_{\mathrm{p}}}\cdots D_{f}^{\epsilon_{2}}D_{f^{1}}^{\epsilon})$$=$$\sum_{\epsilon_{1},\epsilon_{2},\cdots,\epsilon_{\mathrm{p}}\in\{+,-\}}<\Omega|D_{f}^{\epsilon_{\mathrm{p}}}\cdots D_{f}^{\epsilon_{2}}D_{f}^{\epsilon_{1}}\Omega>$.

Besides it is easy to see that the contributing terms in the last expression are given
satisfying the following two conditions
by the sequences of signatures$(\epsilon_{p}, \cdots, \epsilon_{2}, \epsilon_{1})$$\#\{i|1\leq i\leq l, \epsilon_{i}=+\}$$\geq$$\#\{i|1\leq i\leq l, \epsilon_{i}=-\}$$\#\{i|1\leq i\leq p, \epsilon_{i}=+\}$$=$$l=1$ , $\cdots,p$ ,

,

$\{$

$\#\{i|1\leq i\leq p, \epsilon_{i}=-\}$.

correspond to the noncrossing pair partitions
Such sequencesof the points set $\{p,p-1, \cdots, 2,1\}$ in the bijective way. Besides the noncrossing
pair partitions are identified with the noncrossing diagrams in the natural way. For
example we have$(=\mathrm{N}\mathrm{C}\mathrm{P}\mathrm{P})$$(\epsilon_{p}, \cdots, \epsilon_{2}, \epsilon_{1})$$p$(-, -, -,$+, +, +,$-, -,$+,$-,$+, +)$For anoncrossing diagram, we putfor the even moments$(\epsilon_{p}, \cdots, \epsilon_{2}, \epsilon_{1})$$g$–which is corresponding to asequence of signatures. Then we obtain aformula$Vr(g):=<\Omega|D_{f}^{\epsilon_{\mathrm{p}}}\cdots D_{f}^{\epsilon_{2}}D_{f^{1}}^{\epsilon}\Omega>$$m_{2k}$$m_{2k}$$\sum$$=$$g$$V_{\Gamma}(g)$:NCPP ofpoints set$2k$

Besides we can see that the following recurrence formula for \langle$g\rangle_{\mathrm{r}}:=V_{\mathrm{r}}(g)$hold.

Recurrence relations
(i)
(ii)

(iii)

$\langle\bigcap_{g_{1}}g\bigcap_{2}$...

$\bigcap_{g_{j}\rangle_{\mathrm{r}}}=$$\langle\cap g\rangle_{\mathrm{r}}=\frac{r_{1}}{|g|+1}$$\langle\bigcap_{g_{1}\rangle_{\mathrm{r}}\langle}g\bigcap_{2}\rangle_{\mathrm{r}}\cdots\langle\bigcap_{g_{j}\rangle_{\mathrm{r}}}$$\langle g\rangle_{\mathrm{r}’}$$\langle\cap\rangle_{\mathrm{r}}=r_{1}$Here denotes the number of lines in adiagram . denotes the sequence obtained. For example, the, that is,by the shift of.
following figure explains the rule for the calculation of$\mathrm{r}’$

$|g|$

$g$

$\mathrm{r}=(r_{1}, r_{2}, r_{3}, \cdots)$$\mathrm{r}’:=(r_{2}, r_{3}, r_{4}, \cdots)$$V_{\mathrm{r}}(g)$32

Let us write
to suggest explicitly the dependence on.
Using the recurrence relations for, themomentcan be rewrittenas$m_{p}=m_{p}(\mathrm{r})$$\mathrm{r}=(r_{1}, r_{2}, r_{3}, \cdots)$$V_{\mathrm{r}}(g)$$m_{2k}(\mathrm{r})$$=$$\sum$$2k\mathrm{t}\mathrm{h}$$m_{2k}(\mathrm{r})$$V_{\mathrm{r}}(g)$9: NCPP of
Points set$2k$$\sum k$$=$$V_{\mathrm{r}}( \bigcap_{\mathit{9}1}\bigcap_{\mathit{9}2}$$\sum$$j=1$...

$\bigcap_{g_{j}})$

9: NCPP with{connectedcomponents} $=j$$\sum k$$=$$\sum$$j=1$$\sum$$k_{1}+\cdots+k_{\mathrm{j}}=k$$k_{1}\geq 1$,$\cdots$,$=k_{\mathrm{j}}-1$$\sum$$\sum$$k_{1}+\cdots+k_{j}=k$$|g_{1}|=k_{1}-1$$k_{1}\geq 1$,$\cdots$,$V_{\mathrm{r}}( \bigcap_{\mathit{9}j})$$|g_{1}|=k_{1}-1$$\sum k$

$j=1$

...

$k_{\mathrm{j}}\geq 1$$|g_{\mathrm{j}}|$$=$$V_{\mathrm{r}}( \bigcap_{\mathit{9}1})$$\frac{r_{1}}{k_{1}}V_{\mathrm{r}’}(g_{1})$...

$\frac{r_{1}}{k_{j}}V_{\mathrm{r}’}(g_{j})$$k_{j}\geq 1$$|g_{\mathrm{j}}|=k_{\mathrm{j}}-1$$= \sum_{\mathrm{j}=1}^{k}k_{1}\geq 1,,\cdots k_{j}\geq 1\sum_{k_{1}+\cdots+k_{\mathrm{j}}=k}(\sum_{|g_{1}|=k_{1}-1}\frac{r_{1}}{k_{1}}V_{\mathrm{r}’}(g_{1}))$...

$( \sum_{|g_{\mathrm{j}}|=k_{\mathrm{j}}-1}\frac{r_{1}}{k_{j}}V,( g_{j}$$= \sum_{j=1}^{k}k_{1}\geq 1,\cdots k_{\mathrm{j}}\geq 1\sum_{k_{1}+\cdots+k_{\mathrm{j}}=k},\frac{r_{1}}{k_{1}}m_{2(k_{1}-1)}(\mathrm{r}’)\cdots\cdot\cdot\frac{r_{1}}{k_{j}}m_{2(k_{\mathrm{j}}-1)}(\mathrm{r}’)$)).

That is we get the recurrence formula$m_{2k}(\mathrm{r})$$= \sum_{j=1}^{k}k_{1}\geq 1,\cdots k_{\mathrm{j}}\geq 1\sum_{k_{1}+\cdots+k_{\mathrm{j}}=k},\frac{r_{1}}{k_{1}}m_{2(k_{1}-1)}(\mathrm{r}’)\cdots\cdot\cdot\frac{r_{1}}{k_{j}}m_{2(k_{\mathrm{j}}-1)}(\mathrm{r}’)$

.

(3.1)

Another form of recurrence formula is also useful:$m_{2k}( \mathrm{r})=\sum_{j=0}^{k-1}\frac{r_{1}}{j+1}m_{2j}(\mathrm{r}’)m_{2(k-1-j)}(\mathrm{r})$

.

(3.1)

33
Also we note here that,in variables , ,is ahomogeneous polynomial of degree
moment. So we have for each $c>0$$2k\mathrm{t}\mathrm{h}$$k$$r_{1}$$r_{2}$$\cdots$$r_{k}$$m_{2k}(\mathrm{r})$$m_{2k}(cr_{1}, cr_{2}, cr_{3}, \cdots, cr_{n}, \cdots)$$=$$c^{k}m_{2k}(r_{1}, r_{2}, r_{3}, \cdots, r_{n}, \cdots)$.

Let us derive afunctional equation satisfied by the generating function $f(s)=$:of the distribution
for the even moments$\{m_{2k}(\mathrm{r})\}$$f(s;\mathrm{r})$$\mu=\mu_{\mathrm{r}}=\mu f,\mathrm{r}$.

$f(s; \mathrm{r})=\sum_{k=0}^{\infty}m_{2k}(\mathrm{r})s^{k}$

Using the recurrence relations for the moments (3.1), the generating function $f(s;$ r)
can be rewritten as$f(s;\mathrm{r})$$=$$\sum_{k=0}^{\infty}m_{2k}(\mathrm{r})s^{k}$$=$$1+ \sum_{k=1}^{\infty}\sum_{j=1}^{k}k_{1}\geq 1,\cdots k_{j}\geq 1\sum_{k_{1}+\cdots+k_{\mathrm{j}}=k},\frac{r_{1}}{k_{1}}m_{2(k_{1}-1)}(\mathrm{r}’)s^{k_{1}}\cdots\cdot\cdot\frac{r_{1}}{k_{j}}m_{2(k_{\mathrm{j}}-1)}(\mathrm{r}’)s^{k_{\mathrm{j}}}$$=$Now we put$1+ \sum_{j=1}^{\infty}(\sum_{k=1}^{\infty}\frac{r_{1}}{k}m_{2(k-1)}(\mathrm{r}’)s^{k)^{j}}$$g(s; \mathrm{r}):=\sum_{k=1}^{\infty}\frac{r_{1}}{k}m_{2(k-1)}(\mathrm{r}’)s^{k}$, then this quantity satisfies$f(s; \mathrm{r})=\frac{1}{1-g(s,\mathrm{r})}.\cdot$

Also this quantity$g(s;\mathrm{r})$can be rewritten as$g(s;\mathrm{r})$$=$$\sum_{k=1}^{\infty}\frac{r_{1}}{k}m_{2(k-1)}(\mathrm{r}’)s^{k}$$=r_{1} \sum_{k=1}^{\infty}\int_{0}^{s}dsm_{2(k-1)}(\mathrm{r}’)s^{k-1}$$=r_{1} \int_{0}^{s}ds\sum_{l=0}^{\infty}m_{2l}(\mathrm{r}’)s^{l}$$=r_{1} \int_{0}^{s}ds\sum_{k=0}^{\infty}m_{2k}(\mathrm{r}’)s^{k}$$=$$r_{1} \int_{0}^{s}dsf(s;\mathrm{r}’)$.

So we get$g(s;$

r)$=r_{1} \int_{0}^{s}dsf(s;\mathrm{r}’)$

.

(3.3)

34
Therefore the moment generating function
equation:$f(s;\mathrm{r})$$f(s;\mathrm{r})$satisfies the following functional$=$$\frac{1}{1-r_{1}\int_{0}^{S}dsf(s,\mathrm{F})}.$

’

$\{$

$f(0)$

$1$

$=$

.

4. An example –the distribution of-Leinert-Speicher$\mathrm{B}\mathrm{o}\dot{\mathrm{z}}\mathrm{e}\mathrm{j}\mathrm{k}\mathrm{o}$In this section, we give an example of weighted monotone Fock space
the probability distributionof its associated Brownian motion
explicitly obtained. This example corresponds to the weight sequence$\mathrm{r}=(r_{1},r_{2},r_{3}, \cdots):=(1,c, c,c, \cdots)$$g(s;\mathrm{r})$such thatcan begiven by$\{Q_{t}\}_{t\geq 0}$$\mu_{t}$In this case, the quantity$\Phi_{w}$$w$.

is given by$g(s;1, c, c, c, \cdots)$

$=$

1

$=$

$\int_{0}^{s}ds\sum_{k=0}^{\infty}m_{2k}(c,c, c, \cdots)s^{k}$$\cdot\int_{0}^{s}dsf(s;c, c, c, \cdots)$from (3.3). By the way, since in general thehomogeneous polynomial of degree$k$$m_{2k}(c, c, c, \cdots)$

in

$k$

$=$

moment,variables , ,$m_{2k}(r_{1},r_{2},r_{3}, \cdots)$$2\mathrm{f}\mathrm{c}\mathrm{t}\mathrm{h}$$\mathrm{r}\mathrm{x}$$r_{2}$$\cdots$$c^{k}m_{2k}(1,1, 1, \cdots)$$r_{k}$

is a, we have.

Note that $a_{2k}:=m_{2k}$ (1, 1, 1, ) is just the
moment of the arcsine law withmean 0and variance 1because the weight sequence $(1, 1, 1, \cdots)$ corresponds to theusual monotone Fock space [Mul, Mu2]. Now
can be rewritten as$\cdots$$2\mathrm{f}\mathrm{c}\mathrm{t}\mathrm{h}$$g(s;\mathrm{r})$$g(s;1, c, c, c, \cdots)$$=$$\int_{0}^{s}ds\sum_{l=0}^{\infty}m_{2l}(1,1,1, \cdots)(cs)^{l}$$=$$\int_{0}^{s}dsf(cs;1,1,1, \cdots)$

where $f(s;1,1,1, \cdots)$ is just the generating function
moments of the arcsine law. Hence we have$g(s;1, c, c, c, \cdots)$$=$$\int_{0}^{s}dsa(cs)$$=$$[- \frac{1}{c}(1-2cs)^{\frac{1}{2}}]_{0}^{s}$,$a(s):=\ovalbox{\tt\small REJECT}^{1}1-s$

for the even$= \int_{0}^{s}ds\frac{1}{\sqrt{1-2cs}}$$=$$\frac{1}{c}-\frac{1}{c}(1-2cs)^{\frac{1}{2}}$.

35
Using the basic relation $f(s)= \frac{1}{1-g(s)}$ , we obtain the explicit form of the generating, associatedfunction $f(s)=f(s;$ r) for the even moments of the distribution jl
with the weight sequence r (1, c, c, c, \cdots ), as
to the Fock space$=\mu f^{\mathrm{r}}$$=$$\Phi_{w}$$f(s)$

Then the Cauchy transform$=$$\frac{(c-1)-\sqrt{1-2cs}}{(c-2)+2s}$.

of the measure$G_{\mu}(z)= \int_{-\infty}^{+\infty}\frac{1}{z-\xi}d\mu(\xi)$$G_{\mu}(z)= \frac{1}{z}f(\frac{1}{z^{2}})=\frac{(1-c)z+\sqrt{z^{2}-2c}}{(2-c)z^{2}-2}$$\mu$

is given by

.

(4.1)

Here we remark that the expression (4.1) is obtained as the specialization :=1&
-Leinertof the distribution ofof the Cauchy transform, which is defined as follows [BLS]:
Speicher$\alpha$$\mathrm{B}\mathrm{o}\dot{\mathrm{z}}\mathrm{e}\mathrm{j}\mathrm{k}\mathrm{o}$$G_{\nu_{\alpha,\beta}}(z)$$\beta:=\sqrt{\frac{c}{2}}$$\nu_{\alpha,\beta}$$\nu_{\alpha,\beta}=\tilde{\nu}_{\alpha,\beta}+a(\delta_{x_{1}}+\delta_{x_{2}})$,,($d \tilde{\nu}_{\alpha,\beta}(x)=\chi_{1-2\beta,2\beta \mathrm{l}}(x)\frac{1}{2\pi}\frac{\alpha^{2}\sqrt{4\beta^{2}-x^{2}}}{\alpha^{4}-(\alpha^{2}-\beta^{2})x^{2}}dx$$x_{1}=- \frac{\alpha^{2}}{\sqrt{\alpha^{2}-\beta^{2}}}$,$x_{2}= \frac{\alpha^{2}}{\sqrt{\alpha^{2}-\beta^{2}}}$. conti. part)(atomic part),$0 \leq\frac{\beta^{2}}{\alpha^{2}}\leq\frac{1}{2})$$a=\{$$\mathrm{a}\mathrm{b}\mathrm{s}$,$\frac{1}{02}\frac{\alpha^{2}-2\beta^{2}}{\alpha^{2}-\beta^{2}}$$\{$$\frac{1}{2}\leq\frac{\beta^{2}}{\alpha^{2}})$The Cauchy transform of$\nu_{\alpha,\beta}$$G(z)$.

is known to be$=$.

$\frac{z(\frac{1}{2}\alpha^{2}-\beta^{2})+\frac{1}{2}\alpha^{2}\sqrt{z^{2}-4\beta^{2}}}{z^{2}(\alpha^{2}-\beta^{2})-\alpha^{4}}$Hence we obtain the explicit form of the measure$\mu=\tilde{\mu}+b(\delta_{\xi_{1}}+\delta_{\xi_{2}})$$\mu$

as follows.

,,$d \tilde{\mu}(x)=\chi_{1-\sqrt{2\mathrm{c}},\Gamma 2\mathrm{c}\mathrm{l}}(x)\frac{1}{\pi}\frac{\sqrt{2c-x^{2}}}{2+(c-2)x^{2}}dx$$\xi_{1}=-\sqrt{\frac{2}{2-c}}$,$\xi_{2}=\sqrt{\frac{2}{2-c}}$,($\mathrm{a}\mathrm{b}\mathrm{s}$. conti. part)(atomic part)$b=\{\begin{array}{l}\frac{1-c}{2-c}(0\leq c\leq 1)0(1\leq c)\end{array}$, put $f=$
. For general
Now let us remove the assumption of, then we have $<Q_{f}^{p}>=(||f||_{\mathrm{L}^{2}})^{p}<Q_{u}^{p}>$ . Put
with.
equals to, and hence, then we see that$f\in \mathcal{H}_{1}$$||f||_{\mathrm{L}^{2}}=1$$||f||_{\mathrm{L}^{2}}.u$$||u||_{\mathrm{L}^{2}}=1$$\mu_{t}(dx)=\mu(\frac{dx}{\sqrt{t}})$$\mu_{t}:=\mu_{\chi_{(0,t\mathrm{l}},\mathrm{r}}$$\mu_{t}$$\nu_{\sqrt{t},\sqrt{\frac{\mathrm{c}t}{2}}}$

36
Since the distribution of
dependes only on ||f$Q_{t}-Q_{s}$incrementcoinsides with.
After all we have$Q_{f}$$||_{\mathrm{L}^{2}}$, the distribution$\mu_{s,t}$of an$\mu_{t-s}$

Proposition 4.1. Let
be the canonical process on a weighted monotone
Fock spacewith weight sequence $w=$ $(1, c,c^{2}, c^{3}, \cdots)$ . Then, under the vacuum
state , the probability distributionof an increment $Q_{t}-Q_{s}$ , $0<s<t$ , of theprocessis the distribution of $Bo\dot{z}ejko- Leine\hslash$ -Speicher
with parameter$\alpha=\sqrt{t-s}$ and.
$\{Q_{t}\}_{t\geq 0}$$\Phi_{w}$$\phi_{w}$$\mu_{s,t}$$\{Q_{t}\}_{t\geq 0}$$\nu_{\alpha,\beta}$$\beta=\sqrt{4^{t}\lrcorner-s2}$

Remark 4.2. Note that, by the specializations $c=1$ and $c=2$ for , we get thearcsine law and the Wigner semicircle law, respectively.
$\mu$(arcsine law)$c=1$$\Rightarrow p(x)=\frac{1}{\pi}\frac{1}{\sqrt{2-x^{2}}}$$c=2$$\Rightarrow p(x)=\frac{1}{\pi}\sqrt{1-(\frac{x}{2})^{2}}$$\{$(Wigner semi-circle law)

Remark 4.3. The distribution of-Leinert-Speicher,
was obtained in
BLS] as the central limit distribution in the -free central limit theorem. Also the
distribution of its associated Brownian motion
is given by the distributionof-Leinert-Speicher. We remark here that our Brownian motion
is not isomorphic to the -ffee Brownian motion of-Speicherin[BSp] although they have the same distribution, for each time $t\geq 0$ ,with aand. The reason is that the correlation function ofisdifferent from the correlation function of. For our Brownian motion,
the correaltion
not symmetric in two variables and . Indeed,
for $0<s<t$ , we have$\mathrm{B}\mathrm{o}\dot{\mathrm{z}}\mathrm{e}\mathrm{j}\mathrm{k}\mathrm{o}$$[\mathrm{B}\mathrm{S}\mathrm{p}$$\nu_{\alpha,\beta}$$c$$\{\tilde{Q}_{t}\}_{t\geq 0}$$\mathrm{B}\mathrm{o}\dot{\mathrm{z}}\mathrm{e}\mathrm{j}\mathrm{k}\mathrm{o}$$\{Q_{t}\}_{t\geq 0}$$\{\overline{Q}_{t}\}_{t\geq 0}$$\mathrm{B}\mathrm{o}\dot{\mathrm{z}}\mathrm{e}\mathrm{j}\mathrm{k}\mathrm{o}$$\mathrm{c}$$\mu_{t}=\nu_{\alpha,\beta}$$=\sqrt{t}$$\beta=\sqrt{\frac{d}{2}}$$\{Q_{t}\}_{t\geq 0}$$\{\tilde{Q}_{t}\}_{t\geq 0}$$\{Q_{t}\}_{t\geq 0}$$<Q_{s}Q_{t}Q_{t}Q_{s}>\mathrm{i}\mathrm{s}$

$t$

$s$

$<Q_{s}Q_{t}Q_{t}Q_{s}>=w_{2} \{\frac{1}{2}s^{2}+s(t-s)\}+w_{1}^{2}s^{2}$

,

(4.2)

whereas we have$<Q_{t}Q_{s}Q_{s}Q_{t}>=w_{2} \{\frac{1}{2}s^{2}\}+w_{1}^{2}s^{2}$

.

(4.3)

Hence we know $<Q_{s}Q_{t}Q_{t}Q_{s}$ $>\neq<Q_{t}Q_{s}Q_{s}Q_{t}$ , and recognize the non-symmetricin the roles played by the past and the furture $t(0\leq s<t)$ . On the other
hand, for the -free Brownian motion of-Speicherin [BSp], it canbe checked that
$>$

$s$

$\mathrm{c}$$\mathrm{B}\mathrm{o}\dot{\mathrm{z}}\mathrm{e}\mathrm{j}\mathrm{k}\mathrm{o}$$\{\tilde{Q}_{t}\}_{t\geq 0}$$<\tilde{Q}_{t}\tilde{Q}_{t}\tilde{Q}_{t}\tilde{Q}_{s}>=<\tilde{Q}_{t}\tilde{Q}_{s}\tilde{Q}_{s}\tilde{Q}_{t}>$for $0<s<t$ . This concludes thatis not isomorphic tothe expressions (4.2) and (4.3) hold for the Brownian motion$\{Q_{t}\}_{t\geq 0}$$\{\tilde{Q}_{t}\}_{t\geq 0}$. Note that$\{Q_{t}^{(w)}\}_{t\geq 0}$

of genera

37
be the interacting free Fock space over the oneweight sequence w. Now let( , A2, A3, \cdots ), such
, with the weight sequenceparticle space.
coincides withof its canonical operator
that the probability measure
Such asequence Aalways exists (see [AcB]). Then we can check that$\Phi^{(\lambda)}$$\lambda=$$\mathcal{H}_{1}:=\mathrm{L}^{2}(\mathrm{R}_{+})$$\overline{Q}_{1}$$\mu^{(\lambda)}$$<\overline{Q}_{s}\overline{Q}_{t}\overline{Q}_{t}\overline{Q}_{s}>$$\lambda_{1}$$\mu_{w}$$=$$<\overline{Q}_{t}\overline{Q}_{s}\overline{Q}_{s}\overline{Q}_{t}>$.

on
for $0<s<t$ . So we observe that also, for each , the Brownian motion
on the corresponding interactis not isomorphic to the Brownian motion
for each
ing free Fock space
although they have the same distribution
.
time
$w$

$\{Q_{t}^{(w)}\}$$\{\overline{Q}_{t}^{(\lambda)}\}$$\Phi_{w}$$\Phi^{(\lambda)}$$\nu_{\sqrt{t},\sqrt{\frac{\mathrm{c}t}{2}}}$$t$$\geq 0$

References[AcB] Accardi L., Bozejko, M.: Interacting Fock spaces and gaussianization of probability
measures. Infinite Dim. Anal. Quantum Prob. 1, 663-670 (1998)[ALV] Accardi, L., Lu, Y. G., Volovich, I.: Interacting Fock spaces and Hilbert module
extensions of the Heisenberg commutation relations. Publ. HAS, Kyoto: HAS,

1997

[BLS] Bozejko, M., Leinert, M., Speicher, R.: Convolution and limit theorems for conditionally free random variables. Pacific J. Math. 175, 357-388 (1996), M., Speicher, R.: -independent and symmetrized white noises. In: Ac[BSp]cardi, L. Quantum Probability and Related Topics VI. Singapore: World Scientific,
1991, pp.219-236.
[Lu] Lu, Y. G.: An interacting free Fock space and the arcsine law. Prob. Math. Statist.
17, 149-166 (1997)[Mul] Muraki, N.: Anew example of noncommutative “de Moivre-Laplace theorem.”
In: Watanabe, S., Fukushima, M., Prohorov, Yu. V., Shiryaev, A. N. (eds.) Pro bability Theory and Mathematical Statistics. Proceedings of Seventh Japan-Russia
Symposium, Tokyo 1995, Singapore: World Scientific, 1996, pp.353-362[Mu2] Muraki, N.: Noncommutative Brownian motion in monotone Fock space. Commun. Math. Phys. 183, 557-570 (1997)[Spe] Speicher, R.: Anew example of ‘independence’ and ‘white noise.’ Prob. Th. Rel.
Fields 84, 141-159 (1990)[VDN] Voiculescu, D. V., Dykema, K. J., Nica, A.: Free Random Variables. CRM MonOgraph Series, Providence, RI:Amer. Math. Soc, 1992
[Voi] Voiculescu, D. V.: Symmetries of some reduced free product C’-algebras. In: Araki,
H., Moore, C. C., Stratila, \S ., Voiculescu, D. V. (eds.) Operator Algebras and their
Connections with Topology and Ergodic Theory. Proceedings, Bugteni, Romania
1983, Lecture Notes in Math. 1132, Berlin, Heidelberg, New York: Springer, 1985,
pp.556-588$\mathrm{B}\mathrm{o}\dot{\mathrm{z}}\mathrm{e}\mathrm{j}\mathrm{k}\mathrm{o}$$\psi$