$\mathfrak{B}\uparrow Ow\eta_{1}.c\uparrow|\eta$ $s_{\}\eta \mathrm{c}f}\mathrm{k}\mathrm{e}$
$[_{\backslash }^{-}.\mathrm{X}3$
ミ則度値 \epsilon
枝拡
k
過
\subset 0
瑣
A
法
$>_{\mathrm{X}}j\forall_{\backslash }^{-}$
$\Phi$ $\tilde{\prime}P\mathrm{x}\vee\backslash T\mathrm{L}^{7}J^{\frac{\backslash }{\overline{\overline{\mathrm{o}}}}}---(3\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{t}0\sim \mathrm{W}_{kT\mathrm{n}\mathrm{n}}a\mathrm{b}6)$
1
Intrc
$\mathrm{A}\dot{\iota}\iota \mathrm{c}\{\mathfrak{l}\mathrm{O}(\cap$ $\grave{\backslash }\backslash _{-T’\overline{\mathrm{n}}\mathbb{R}}\backslash \cdot\neq_{\mathrm{c}^{\iota}}‘ 3\backslash \sim\succeq \mathrm{t}\mathrm{f}\text{ノ}$J.
$(\vee aYr\mathrm{e}\gamma 1(.[\mathrm{v}_{\mathrm{A}\gamma}\mathrm{J})$丙
{
$\sim \mathrm{I}\not\in$ が $\dot{\Delta}\overline{\S}\eta.f^{\mathrm{i}}4^{\mathrm{g}_{-}}\tau\succ<>’\tau_{-}\iota,\backslash 5$.
$\ovalbox{\tt\small REJECT}^{\text{ノ}}|\text{は}$) $6\geq 0$ し $\mathrm{t}_{-}\tau 2$ $\text{ノ_{}1}^{u_{i}}\text{ノ_{}\pi}\lambda^{\wedge}X4‘ \mathbb{E}$
田
$\text{ノ}x*_{\backslash }$ $\mathrm{L}_{6}=$ $\mathrm{X}\frac{\mathrm{d}^{\mathit{1}}}{d_{\lambda}}2$. $- \Theta \mathrm{x}\frac{\mathrm{d}}{c\lambda_{\mathrm{L}}}$
(i)
が‘ $\text{ノ}|\mathrm{a}\phi 3$ $[^{arrow}0,$ $\infty)\succeq\emptyset\pi^{\backslash }\#\mathrm{L}_{L^{\iota}}\mathrm{e}_{\ovalbox{\tt\small REJECT}},\mathrm{a}(\dot{L}_{b}-\cdot r_{\wedge}^{\backslash \not\subset}\mathrm{R}^{\angle}\mathrm{C}’ \mathrm{L}^{\nu}\mathrm{E}*_{\mathrm{f}}^{\mathrm{o}}\text{と呼床_{}\searrow}.$ $\vee\backslash$ 小k
|客 $S^{\cdot}$
DE
$(|5)$ $\mathrm{q}-\grave{\mathrm{R}}A\mathrm{R}^{-\mathrm{g}}’ {}^{\mathrm{t}\urcorner}k\cdot 4\backslash 3r_{J_{\mathrm{A}}\mu^{\mathrm{i}}}\mathcal{K}\llcorner\# 7/\backslash \not\simeq \mathrm{H}\mathrm{f}_{\pm}$と云 $y$ て $k\mathrm{f}\mathrm{t}\eta)\dot{E}\yen$
又るヒ
$\nearrow$$\mathit{0}7\backslash$
ほ発書
3
$\llcorner_{\theta}$-#
散最薩
$\nearrow 4^{\theta\ell}\star$) $z\backslash \backslash \tau_{\mathit{3}}$
$\mathrm{c}_{t}Z^{\iota}$
$\backslash \iota_{7}*_{\backslash }^{\backslash }\mu\tau\pi,3$ $\mathrm{L}_{\mathit{0}}-\dot{\grave{\%}}\#\prime {}_{\overline{\mathrm{A}}}\mathrm{H}\mathrm{L}$
.
$\text{ノ}\mathrm{a}.\mu^{o}L+$) $\mathrm{f}j\nearrow X3_{\mathcal{F}^{\underline{\mathrm{B}}}\triangleleft \mathrm{t}}^{\mathit{2}}.\#>\backslash \backslash \sim \mathrm{i}^{\text{ノ_{}\mathrm{T}}}\triangleleft\backslash |\mathrm{i}\{_{\backslash }^{9}- \mathrm{c}$
$\tilde{\mathrm{b}}\mathrm{j}_{\nwarrow\sim}’\S$ $\backslash \vee \mathrm{b}$
も示し於
$\mathrm{L}_{\theta^{-}}$
律六散遷咋
[}
ま
$\supset$
]
々叉
$\circ$)連
j
屹、杖
$\mathrm{r}_{\wedge}’k\xi_{\vee}$分才
i
払 k
辷置殺
(C\S
$\cdot\cdot$-$d^{\sim}$
.
高 $\mathfrak{U}>\mathrm{i}\mathit{0}\eta$)
と呼ば捷し
,
$\mathrm{G}_{4}$}
切
$\psi 1-\mathrm{W}\text{厳}\backslash c\gamma_{1}$分蕨遮殺外極阪拡
$\ovalbox{\tt\small REJECT} \mathbb{R}_{\nu}’7:\mathrm{a}E$a\ddagger
$\dot{\mathrm{t}}$偽として $\dot{\mathrm{L}}$
A}
吋
\sim
七
\iota
$|^{\backslash }=\mathrm{f}2$ て $-\tau \mathrm{g}_{\lambda_{-}}/arrow*rr-\overline{\backslash }(_{\mathrm{C}}+$,
[L],
$[\mathrm{K}w]i$[S}V]
$)$、牛
\tau
$|_{\backslash }^{-}$ $\llcorner$ 。-庄
‘^
翫岨嘱
,
ば
$;_{\mathrm{e}p\ }^{-}-,\mathrm{c}^{1-\Sigma}\supset\sim\tau$ $\ovalbox{\tt\small REJECT}$ 入\approx 起几 $\theta$)で゛
,
$\mathrm{F}$$\mu^{\mathrm{t}^{0})}(\mathrm{t})$
if
$y$$\overline{/_{kA\mathrm{J}^{\backslash }\overline{\text{フ}}}}\dagger^{\mathcal{Z}}\approx^{\mp}\backslash \grave{\nabla}\grave{-}\mathrm{R}\ovalbox{\tt\small REJECT}\urcorner$
a
$F_{\circ\urcorner}i\iota^{\frac{\text{ノ}{1}}\beta\yen}\Phi\epsilon \mathrm{J}\mathrm{f}\neg\dot{\mathrm{t}}\mathrm{t}.f_{\backslash }^{-}i_{\Gamma^{\sim}\mathrm{R}}^{\psi\sim\emptyset}\tilde{\mathrm{O}}$$\mathrm{R}^{\zeta\lambda}\iota^{\sim}\mathrm{K}^{\gamma_{1}}@$
ht
a
$\ddagger \mathrm{a}$a
$\mathrm{P}_{\backslash \Gamma^{\overline{j}}}^{-}\grave{arrow}\succeq \mathfrak{l}x.\neq<*_{\mathfrak{l}\mathrm{l}}\grave{b}\nu\dot{\mathrm{h}}<\iota\gamma$\S
;
Warre
$\omega$ $|\S)\gamma \mathrm{t}^{(6)}(\star)(6>0)\mathrm{t}\delta 3T^{\iota}.\mathrm{h}E\mathrm{f}\mathrm{f}_{7}\mathcal{T}_{\backslash }^{-}\check{\text{ノ}\mathrm{A}}g_{\backslash }\mathrm{t}_{x\overline{7}^{\mathrm{t}}\grave{\text{フ}}\mathrm{g}_{\Re^{-}}^{3}}fi\overline{\neg}\neq \text{フ}..’\backslash$$|-\backslash \mathrm{f}\supset\tau \mathrm{R}a_{3^{-\mathrm{K}^{\gamma}}}|\mathrm{i}3^{\mathrm{h}}.T\mathrm{a}^{\mathrm{f}\mathrm{a}\xi \mathrm{f}\not\equiv}6,$
$k^{\llcorner}T\grave{.}\circ)20\emptyset \mathrm{R}^{c1}d- \mathrm{K}n\grave{\wp}\dot{\mathrm{h}}\mathrm{c}$
$\ovalbox{\tt\small REJECT}\iota \mathrm{R}^{\eta}\mathrm{t}|\mathrm{g}]|_{\overline{\backslash }}\mathrm{I}\mathrm{J},$
$\star 33\lambda p_{\backslash },\uparrow \mathrm{A}\mathrm{A}\neq 7^{\mathrm{t}\mathrm{I}}\overline{-7}\ddot{\eta}_{\vee^{\backslash }}.\mathrm{g}_{\overline{\mathrm{B}^{\backslash }}}^{\eta}f\mathrm{l}\backslash (x_{\ }^{*}\S|\mathrm{C}\epsilon_{J^{-}\backslash }^{arrow}.|\prime \mathrm{u}\mathrm{p}^{\mathrm{c}}$
$4^{7\text{フ}}E4\doteqdot\sim_{2}*_{\mathrm{f}\acute{\mathrm{x}}}\text{ノ}|\mathrm{f}-\grave{\ovalbox{\tt\small REJECT}}x\ovalbox{\tt\small REJECT}_{\backslash }\neq\cdot 7\mathrm{T}7^{\iota \mathrm{t}}\overline{.7}^{1\grave{7}}.\backslash ’\ovalbox{\tt\small REJECT}^{>_{p}}\overline{\mathrm{g}})\text{ノ}t^{\iota}\backslash ‘ \mathfrak{l}\wedge|\grave{r}_{\mathcal{I}}- LT\dot{k}^{\sigma})$
$\}$
\S
$4_{\text{ノ}l\backslash }^{Z\backslash \not\supset}’\backslash \cdot$
.
$5\grave{\mathrm{z}}\grave{>}$
A
$3\backslash .\succeq \mathfrak{F}_{J}\overline{\tau\backslash }\mathrm{b}7’-\backslash _{J}\backslash h\backslash \mathrm{I}E_{\text{ノ}}*\mathrm{f}4\mathrm{f}\text{ノ}\grave{\epsilon}^{*}\mathrm{g}\tau 7’\overline{7}^{\mathrm{I}}\grave{7}_{\vee^{\backslash }L}\not\in_{-}r$
$\frac \mathfrak{Y}^{d)}H\theta\not\in \mathrm{L}|\sim\backslash$
Na
$\mathrm{V}317‘ n^{\theta_{\backslash }\mathrm{p}_{\text{ノ}}}\ovalbox{\tt\small REJECT},’ \mathrm{x}\wedge \mathit{5}\prime \mathrm{g}/\neq x\mathrm{a}\mathrm{Q}\succeq\iota\tau-\sigma$) $\emptyset aX/\backslash \mathrm{t}\hslash 3$,
$-\mathfrak{F},$ $\mathrm{L}_{\theta^{-}}?^{f\mathrm{B}^{l_{\llcorner}}}\mathrm{A}\overline{\mathrm{A}}’\#\Phi^{\mathrm{I}}\mathrm{f}\backslash ’-\mathrm{A}|i)-\overline{\#}l\mathrm{g}_{\eta}\wedge\kappa\cdot i\mathrm{Z}^{\mapsto}\Phi- \mathrm{f}\mathrm{i}*_{\mathrm{A}}\nearrow \mathrm{x}\llcorner 6\wedge\supset(_{2}\mathrm{t}\overline{\overline{\overline{0}}}\overline{\mathrm{B}}\mathrm{S}\mathrm{t}\{\mathrm{p}w-\backslash \mathrm{w}$
,
$4_{\overline{\mathfrak{l}}^{\backslash }}\dagger \mathrm{f}\iota\lambda;io1’\mathrm{t}(_{\overline{\mathrm{C}}}+.[\mathrm{D}_{e\}’][_{\mathrm{D}_{\nearrow}}j])_{\mathit{0})}\mathrm{g}\#^{i}|\tau s^{\mu\Lambda}4\text{ノ}\iota \mathrm{D}\tau\backslash \mathrm{X}^{\ovalbox{\tt\small REJECT}}J_{\supset}\mathrm{s}($
$|\gtrless^{\urcorner}\mathrm{I}^{y}s\prime \mathrm{x}\mathrm{g}i^{\iota}\kappa$
$|X\star\hat{\ltimes}<\cdot 4|\urcorner \mathrm{t}_{\mathrm{I}\grave{\tau}_{)}}\Lambda\backslash \cdot$$3\backslash$ D
E
$1\backslash \backslash \mathrm{r}\backslash \backslash \mathrm{e}\mathrm{r}\eta 7\nearrow\acute{\mathrm{x}}3\#_{4}\triangle_{\tau}\grave{\iota}\mathfrak{F}s\overline{\mathrm{D}}’\mathrm{o}(.\backslash _{i}\mathrm{A}\mathrm{b}_{\sim_{j}}\#\sim\backslash _{\eta \mathrm{L}^{\circ_{)}}\exists}\mathrm{F}\ovalbox{\tt\small REJECT}\backslash$
$|\mathrm{H}^{\mathfrak{l}}\backslash |\grave{F}_{\lambda_{arrow}}\ell\wedge$
ノ
f
$\mathrm{i}*|\mathrm{x}[\mathit{0}_{i}\infty$)
$\mathfrak{x}_{-[\neg}\mathrm{E}-\overline{/\mathrm{T}}\mathrm{a}\prec-\dot{u}\iota 3$)
.
X
$\vee\backslash -\zeta\backslash ,,$$\backslash \mathrm{t}\mathit{1}aY\mathrm{v}\mathrm{c}\mathrm{e}\nu\uparrow‘ \text{ノ}\gamma_{\text{ノ}}\#\not\in\ovalbox{\tt\small REJECT}\hat{\leq}$
$’\iota_{\sim}^{-}\eta-\mathfrak{g}\hat{\mathrm{t}}\mathrm{r}\perp_{\eta_{\overline{\mathrm{o}}},)}\mathrm{H}^{j\backslash }\backslash .\backslash -\vee\zeta.|\mathrm{J}\,$ $i^{\zeta 1}\mathrm{t}\approx\sim\dot{\iota}\mathrm{J}^{\cdot}’ d$
j
ト
fZ‘\varpi l
$\dot{\mathrm{f}}1^{\tau i}\Re 3\text{ノ}\alpha\lambda l\tau\nearrow 1J\wedge l\mathcal{T}f\neq 7$ $\not\equiv_{J}\mathrm{b}-i^{(}4\mathrm{r}^{\mathrm{I}}\underline{\mathrm{t}}\iota \mathrm{k}3..\dot{\mathrm{F}}\grave{\tau}\mathrm{A}\mathrm{f}\mathit{4}\prime \mathrm{c}c\mathrm{f}\mathrm{f}\mathrm{i}\varphi \mathrm{A}(\llcorner- f\mathrm{A}\#\overline{\mathrm{A}}\underline{\ovalbox{\tt\small REJECT}}\mathrm{f}\{)\mathfrak{s}\Xi^{(}.+)\mathrm{p}_{\mathrm{x}}y$;
$\mathrm{S}$ }
$\mathrm{t}\acute{1}3$
cin
$d_{e\mathrm{v}}|_{\gamma\dot{\downarrow}}\rho 1t\zeta \mathrm{i}\{^{)}ro(-\epsilon \mathrm{S}S\mathrm{L}\iota^{-}\mathrm{C}\mathrm{f}^{\mathrm{g}}5$ノ
$<_{\mathrm{A}}‘ \mathrm{b}^{\gamma_{\mathcal{L}}\lambda \mathrm{n}}.c\mathrm{k}|\mathrm{n}\mathit{3}\dot{|}\eta ec\mathrm{A}\zeta t\uparrow \mathrm{t}1’\eta z\backslash \mathrm{x}1lq\mathit{2}$
14)
$=-z^{\mathrm{A}}\backslash$(2)
V,
$l^{\mathrm{x},\mathrm{Z}}$)
$—-\mathrm{z}^{2}-\theta’(_{X,)}.\mathrm{z}$ $(\ni)$$-\zeta^{\backslash }\Sigma\nwarrow\grave{<}\grave{\rangle}\ltimes 3$
$\sup^{\epsilon \mathrm{Y}^{--}C}\mathfrak{u}\mathrm{f}\mathrm{f}\iota 0\mathrm{s}\mathfrak{l}\mathit{0}\eta\parallel\ell+),$ $, \mathfrak{u}’(/+)\sigma)\nearrow_{\text{ノ}}\overline{\triangleright}4_{\eta}\bigwedge_{l\backslash }- \mathrm{w}_{\mathit{6}^{\gamma}’ r}\prime \mathrm{e}^{\eta}$
a
$\text{ノ}\#\yen$.
$\not\in\sigma$
) $arrow\phi J2\{_{\mathrm{L}}\mathrm{t}\backslash \sim \mathfrak{k}^{\Delta}-\sim 3\mathrm{b}^{L}\vdash 3$
.
$\cross_{\{\eta}|_{\text{ノ}\mathrm{T}}^{3\mathfrak{B}_{\backslash 4}}$Skix
ノ$\backslash \mathrm{v}c\backslash \mathrm{v}\mathrm{Y}^{\cdot}\mathrm{e}|\tau\emptyset\neq k\bigwedge_{\mathrm{D}}|\backslash -\overline{/\mathrm{x}}A^{\prime \mathrm{r}}\backslash$
$\grave{\mathcal{Z}}\grave{\xi}_{7^{1\mathrm{t}}\overline{7}\grave{7}\prime}\mathfrak{l}\backslash \mathrm{E}>\grave{\ovalbox{\tt\small REJECT}}|\mathrm{J}\mathrm{g}\eta \mathit{0}^{\text{ノ}/)}\rangle t^{q}\mathrm{E}\sim\not\in\ovalbox{\tt\small REJECT}\epsilon$
a
$\mathrm{g}\mathrm{t}\tau y\mathrm{L}_{\mathrm{e}C_{1^{c}\mathrm{c}}(}^{\cdot}|_{l^{-\mathrm{r}\lambda}}\sim.\epsilon_{\mathrm{T}}\mathrm{E}\wedge \mathit{2}-\subset\sim$$\Lambda_{\mathrm{L}}1’-\backslash \_{\backslash }rc\vee \mathrm{n}16\backslash ’\iota 1s$
ndte
$\mathrm{q}\ovalbox{\tt\small REJECT}\grave{\Re}_{-}^{\wedge}K^{\iota}(<^{-\not\in}-, \zeta_{\llcorner}\mathrm{J}\mathrm{J})\sim \mathrm{g}_{\mathfrak{l}}\not\in \mathrm{t}\iota 13^{\mathrm{c}}.\mathrm{b}\tau..\mathfrak{F}\{3$
2 Reflecting Brownian motions and $\mathrm{C}\mathrm{B}$-diffusions (Warren’s results)
We introduce the following three different reflecting Brownian motions $R,$$S$,A on
the positive halfline $[0, \infty)$:
(i) The standard refiecting Brownian motion $R=(R_{t})$ starting at $0$: it is well-known that $R$ is given, froma standard Brownian motion $B=(B_{t})$ on $\mathrm{R}$with
$B_{0}=0$, by
$R_{t}=B- \inf_{t{}^{t}0\leq s\leq}B_{s}$. (4)
(ii) For $\theta\geq 0$, let $S=(S_{t})$ be the reflecting Brownian motion starting at $0$ with aconstant drift $\theta/2$ towards the origin: $S$ is given, from a standard Brownian
motion $B=(B_{t})$ on $\mathrm{R}$ with $B_{0}=0$, by
$S_{t}=[B_{t}- \frac{\theta}{2}t]-\inf_{0\leq S\leq t}[B_{S^{-\frac{\theta}{2}S}}]$ . (5)
(iii) For $\theta\geq 0$, let $\Lambda=(\Lambda_{t})$ be the refiecting Brownian motion starting at $0$ with a
stickyboundary at $0$with the rate $2/\theta$: it is characterized by the followingSDE
for a nonnegative and continuous process A $=(\Lambda_{t})$ defined on a probability space with a filtration $\mathrm{F};\Lambda$ is $\mathrm{F}$-adapted, $B=(B_{t})$ is a standardF-Brownian
motion with $B_{0}=0$ and they satisfy
$\Lambda_{t}=\int_{0}^{t}1_{\{\Lambda_{S}>0}\}dBs+\frac{\theta}{2}\int_{0}^{t}1_{\{}\Lambda_{s^{=0\}}}ds$. (6)
It is well-known that asolution A exists and is unique in the law
sense.
Actually it is even known that the joint process $(\Lambda, B)$ is unique in the lawsense
for anysolution of (6), although the solution A can
never
be a strong solution. According to [War], this fact was first remarked by R. J. Chitashvili.When we would emphasize the parameter $\theta$, we write $S^{(\theta)}$ and $\Lambda^{(\theta)}$ for $S$ and $\Lambda$,
respectively. When $\theta=0$, then $S$ coincides with $R$ and A is trivial, i.e. $\Lambda_{t}\equiv 0$.
We give several apparently different but essentially equivalent ways of defining a joint law of $(R, S, \Lambda)$ for given constant $\theta\geq 0$
.
They are given by the following three theorems:Theorem 2.1. Let $S$ and A $be$ as above and assume that they are mutually
inde-$p$endent. Define $A=(A_{t})$ by
$A_{t}= \int_{0}^{t}1_{\{\}}\Lambda_{s}=0ds$, $t\geq 0$ (7)
and $R’=(R_{t}’)$ by
$R’=\Lambda_{t}+ttS_{A}$. (8)
Then, $R’=Rd$.
If we set $R:=R’$ in Theorem 2.1, then this determines uniquely a joint law of
Theorem 2.2. Let A satisfy the $SDE(\mathit{6})$ and, $\mathrm{u}$sing the same Brownian motion
$B=(B_{t})$ in (6), define $R=(R_{t})$ by (4). Then $the.j_{oi}\mathrm{n}t$ law of$(R,.\Lambda)$ is uniquely
determined. Define$A$ by (7) and se$t$
$S_{t}’=(R-\Lambda)A^{-1}t=R_{A_{t}}-1$, (9)
where $A_{t}^{-1}= \inf\{u|A_{u}>t\}$, so that $t[]arrow A_{t}^{-1}$ is the right continuous inverse of $t\mapsto A_{t}$
.
Then $S’=dS$ and, $S’$ and $\Lambda$ are mutually independent.Ifwe set $S:=S’$in Theorem 2.2, then the joint law $(R, S, \Lambda)$ is uniquelydetermined
and it coincides with the joint law of Theorem 2.1.
Theorem 2.3. Let $R=(R_{t})$ be given as above. Let $\kappa=(\kappa_{t})$ be a measurable
$\{0,1\}- val\mathrm{u}ed$process with the following $co\mathrm{n}d\mathrm{i}$tion$\mathrm{a}l$ law given $R$ so that the law of
the joint process $(R, \kappa)$ is uniq
uely.
determined: $\kappa_{0}=1,$ $a.s$. an$d$, for $0<t_{1}<t_{2}<$ $...<t_{n-1}<t_{n}$,$P(\kappa_{t_{1}}’=1, \kappa_{t_{2}t}=1, \ldots, \kappa n-1=1, \kappa_{t_{n}}=1|R)=e^{-M[}e-M[t_{1},t_{2}]\ldots e-M[tn-1t_{n}]0,t1],$,
(10) where
$M[s, t]= \theta(R_{tu}-\min_{us\leq\leq t}R)$, $0\leq s\leq t$. (11)
Set
$A_{t}’= \int_{0}^{t}\kappa_{s}ds$, $(A’)^{-}t1= \inf\{u|A_{u}>t\}$ and define $S’=(S_{t}’)$ and $\Lambda’=(\Lambda_{t}’)$ by
$S_{t}’=R_{(A’)_{t}^{-}}1$, $\Lambda’=R_{t}\iota-s’Att$.
Then $S’=dS$ and $\Lambda’=d\Lambda$, and $S’$ an$\mathrm{d}\Lambda’$ are independen$t$. Furthermore,
$\kappa_{t}=1_{\{\Lambda_{t}^{;}}=0\}$ foralmost all $t,$$a.s.$. (12)
Ifweset $S:=S’$ and$\Lambda$ $:–\Lambda’$ inTheorem 2.3, then the joint law $(R, S, \Lambda)$ is uniquely
determined and it coincides with the joint law of Theorem 2.1. Thus we have seen that there are three $\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{l}\mathrm{y}_{l}$different ways of determining the samejoint law of
$(R, S, \Lambda)$.
The joint process $(\Lambda, R)$ can be given explicitly as follows. Let $\Sigma^{2}=\{(\lambda, x)\in \mathrm{R}^{2}|x\geq 0,0\leq\lambda\leq x\}$ .
Theorem 2.4. $(\Lambda, R)$ is a time $ho\mathrm{m}$ogeneous diffusion on $\Sigma^{2}$ with
$\Lambda_{0}=R_{0}=0$
having the transition probability given by
where
$q_{a,b}^{(\lambda,)}x(d\lambda/dx’)$ (13)
$=$ 1$\{x-\lambda<a\}$ . $\delta x^{l}-x+\lambda(d\lambda/)\cdot\delta_{b}(dx^{J})$
$+$ $1_{\{\geq a}x-\lambda\}$
.
$1\{0<\lambda’<x-’\}a$ . $\theta\cdot e^{-\theta(a)}\cdot dx-’\lambda’-\lambda^{;}\cdot\delta b(dx’)$ $+$ $1_{\{\}}\cdot e-\theta(x’-a)$$\delta x-\lambda\geq a\mathrm{o}(d\lambda/)\cdot\delta_{b}(dX’)$.
and $_{t}^{x}(da, db)=P_{x}( \min_{0\leq}s\leq tR(S)\in da, R(t)\in db),$ $P_{x}$ being the pro\’oability law
governing the standard reflecting Brownian motion $R=(R(t))$ with $R(\mathrm{O})=x$: It is
given explicitly by .
$\Theta_{t}^{\dot{x}}(da,\cdot db)$
$=$ $\frac{2(_{X+b}-2a)}{\sqrt{2\pi t^{3}}}e^{-\frac{(x+b-2a)^{2}}{2t}}1_{\mathrm{t}^{0}<<bx\}}\wedge dadab$ (14)
$+$ $\sqrt{\frac{2}{\pi t}}e^{-\frac{(x+b\rangle^{2}}{2t}}1_{\{\}0}0<b\delta(da)db$.
For $\theta\geq 0$ and $\gamma\in[0, \infty)$, let $\mu^{(\theta)}=(\mu^{(\theta)}(t), P)\gamma$ be the $\mathrm{C}\mathrm{B}$-diffusion on $[0, \infty)$
with $\mu^{(\theta)}(0)=\gamma$ which is generated by the differential operator $L_{\theta}$ given by (1).
The origin $0$ is necessarily a trap. Equivalently, $\mu^{(\theta)}$ is
g.iven
by the unique strongsolution to the following SDE:
$d\mu(t)=\sqrt{2\mu(t)\vee 0}\cdot dB(t)-\theta\mu(t)dt$, $\mu(0)=\gamma$ (15)
where $B=(B(t))$ is the Brownian motion with $B(\mathrm{O})=0$. The connection of these
$\mathrm{C}\mathrm{B}$-diffusions with reflecting Brownian motions $R,$$S$ and A
can
be stated in thefollowing theorem ofthe Ray-Knight type.
Theorem 2.5. Let $R=(R_{t}),$ $S=(S_{t})$ and A $=(\Lambda_{t})$ be reflecting Brownian
motions as above and determine the joint law of$(R, S, \Lambda)$ by Theorem 2.1, or
equiv-alently, by Theorem 2.2 or Theorem 2.3. Let $l(t, a)$ and $l’(t, a)$ \’oe the local time
or
sojourn time density of$R$ and $S,$ respectively.
$l(t, a)= \lim_{\in\downarrow 0}\frac{1}{2\in}\int_{0}^{t}1[a,a+\epsilon](Rs)d_{S}$, $l’(t, a)= \lim_{\epsilon\downarrow 0}\frac{1}{2\in}\int_{0}^{t}1_{[\in}](a,a+s_{s})d_{S}$. (16)
Define, for $\gamma\in[0, \infty)$, continuous and nonnegative processes $\mu=(\mu(t))$ and $\mu’=$
$(\mu’(t))$ by ..-.
$\mu(t)=l(l^{-1}(\gamma, 0),$$t)$ and $\mu’(t)=l’(l^{\prime-1}(\gamma, \mathrm{o}),$ $t)$
. (17)
where $l^{-1}( \gamma, 0)=\inf\{t|l(t, 0)>\gamma\}$ and $l^{\prime-1}( \gamma, 0)=\inf\{t|l’(t, 0)>\gamma\}$. Then, $we$
have the following facts:
(i) $\mu=d$ the $CB$-diffusion $\mu^{(0)}$ starting at $\gamma$.
(ii) $\mu’=d$ the $CB$-diffusion $\mu^{(\theta)}st$artingat
(iii) It holds that $\mu’(t)=\int_{0}^{l^{-1}}(\gamma,0))\kappa(S\cdot l(dS, t)$ (18) where $\kappa(s)=1\{\Lambda_{s}=0\}$
.
(19) Since $\mu(t)=\int_{0}^{l^{-1}}(\gamma,0)l(dS, t)$,we may say that $\mu’$ is obtained from
$\mu$ by a killing determined by a $\{0,1\}$-valued
process $\kappa(s)$. This is what we called a”killing operation” in Introduction.
The results of this section are essentially due to Warren ([War]). We amplified them a little bit by adding Theorems
2.3
and 2.4 which are not stated explicitly in [War]. These results will be extended to the case of general super-diffusions in Section 4. In the next section, we recall the notion of Brownian snakes which will play a fundamental role in this extension.3 Brownian snakes
Throughout this paper, let $\xi=\{\xi(t), P_{x}\}$ be a nice diffusion process on a nice
manifold $M$ generated by a diffusion operator $L$ with $L1=0$. We call $\xi$ the
L-diffusion. This $L$-diffusion has been consideredas the underlyingdiffusion of
super-diffusions in Introduction.
In this section, we recall the notion of Brownian $\xi$-snake due to Le Gall $[\mathrm{L}1]$. It
is defined as a diffusion process with values in the space
of
stopped paths in $M$ sothat we introduce, first ofall, the following notations for several spaces of continuous paths in $M$ and continuous stopped paths in $M$:
(i) for $x\in M,$ $W_{x}(M)=\{w\in C([0, \infty)arrow M)|w(0)=x\}$,
(ii) $W(M)= \bigcup_{x\in}MWx(M)$,
(iii) for $x\in M$ and $t\geq 0$,
$\mathrm{w}_{x}^{(t)}(M)=$
{
$\mathrm{w}=(w,$$t)|w\in W_{x}(M)$ such that $w(s)\equiv w(s$ A $t)$},
(iv) $\mathrm{W}_{x}^{sto}p(M)=\bigcup_{t}\geq 0\mathrm{w}(x(t)M)$,
(v) $\mathrm{W}^{st_{\mathit{0}}p}(M)=\cup x\in M\mathrm{w}st_{\mathit{0}}p(xM)$.
For $\mathrm{w}=(w, t)\in \mathrm{w}^{sto\mathrm{p}}(M)$, we set $|\mathrm{w}|=t$ and call it the
lifetime
of$\mathrm{w}$. Thus wemay think of $\mathrm{w}\in \mathrm{W}^{stop}(M)$ a continuous path on $M$ stopped at
it..s
own lifetime$|\mathrm{w}|$. We endow a metric on $\mathrm{W}^{st_{\mathit{0}}p}(M)$ by
where $\rho$ is a suitable metric on $M$. Then, $\mathrm{W}^{stop}(M)$ is a Polish space and so is also
$\mathrm{W}_{x}^{st_{\mathit{0}}p}(M)$ as its closed subspace.
3.1 Snakes with $\mathrm{d}\mathrm{e}.\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}_{\mathrm{S}\mathrm{t}\mathrm{i}}\mathrm{C}$ lifetimes
Let $x$ be given and fixed. For each $0\leq a\leq b$ and $\mathrm{w}=(w, |\mathrm{w}|)\in \mathrm{W}_{x}^{st\varphi}(M)$ such that $a\leq|\mathrm{w}|$, define a Borel probability $Q_{a,b}^{\mathrm{w}}(d\mathrm{w}’)$
on
$\mathrm{w}_{x}^{sw_{p}}(M..)$.
by the followingproperty:
(i) $|\mathrm{w}’|=b\mathrm{f}\mathrm{o}\mathrm{r}Q^{\mathrm{W}}a,b-\mathrm{a}.\mathrm{a}\sim$. $\mathrm{w}’$,
(ii) $w’(s)=w(s),$$s\in[0, a]$, for $Q_{a,b}^{\mathrm{w}}-\mathrm{a}.\mathrm{a}$. $\mathrm{w}’$,
(iii) under$Q_{a,b}^{\mathrm{w}}$, the shiftedpath $\{(w’)_{a}+(s)=w’(a+s), s\geq 0\}$ is equally distributed
as the stopped path
{
$\xi$($s$ A $(b-a)$),$s\underline{>}\mathrm{o}$}
under $P_{w(a)}$.Let $\zeta(t)$ be a nonnegative continuous function of $t\in[0, \infty)$ such that $\zeta(0)=0$.
Define, for each $0\leq t<t’$ and $\mathrm{w}\in \mathrm{w}_{x}^{stop}(M)$, a Borel probability $P(t, \mathrm{w};t’, d\mathrm{W}’)$
on
$\mathrm{w}_{x}^{stop}(M)$ by$P(t, \mathrm{w};t^{;}, d_{\mathrm{W}}/)=Q^{\mathrm{W}}m\zeta[t,t’],\zeta(t’)(d\mathrm{W}’)$ (20)
$\mathrm{w}.$
he.r
$\mathrm{e}$
$m^{\zeta}[t, t]/= \min_{t\leq u\leq t},$$\zeta(u)$.
It is easy to seethat the family $\{P(t, \mathrm{W};t’, d\mathrm{W}’)\}$ satisfiesthe Chapman-Kolmogorov
equation
so
that itdefines afamily oftransition probabilitieson$\mathrm{W}_{x}^{stop}(M)$. Then,by the Kolmogorov extension theorem, we canconstruct atimeinhomogeneousMarkov process X $=\{\mathrm{X}^{t}=(X^{t}(\cdot), \zeta(t))\}$ on $\mathrm{W}_{x}^{st\circ p}(M)$ such that $\mathrm{X}^{0}=\mathrm{x}$ where $\mathrm{x}$ is theconstant path at $x:\mathrm{x}=(\{x(\cdot)\equiv x\}, 0)$. Note that $|\mathrm{X}^{t}|\equiv\zeta(t)$. If $\zeta(t)$ is
H\"older-continuous, then it
can
be shown that a continuousmodificationin$t$ of$\mathrm{X}^{t}$ exists (cf.Le Gall $[\mathrm{L}1])$. In the following, we always
assume
that $\zeta(t)$ is H\"older-continuous sothat $\mathrm{X}^{t}$ is continuous in $\mathrm{t},$ $\mathrm{a}.\mathrm{s}.$.
Definition 3.1. The $\mathrm{W}_{x}^{stop}(M)$-valued continuous process $\mathrm{X}=(\mathrm{X}^{t})$ is called the
$\xi$-snake starting at $x\in M$ with
th.
$\mathrm{e}$lif.etime
$\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}:.\mathrm{i}\mathrm{o}\mathrm{n}\zeta(t)$.
Its lawon.
$\cdot$$C([0, \infty)arrow$
$\mathrm{W}_{x}^{sto}p(M))$ is denoted by $\mathrm{P}_{x}^{\zeta}$.
We
can
easilyseethat the following three properties characterize the$\xi$-snake startingat $x\in M$ with the life time function $\zeta(t)$: (i) $|\mathrm{X}^{t}|\equiv\zeta(t)$ and, for each $t\in[0, \infty)$,
$X^{t}$ : $s\in[0, \infty)\mapsto X^{t}(s)\in M$
is an $L$-diffusion such that $X^{t}(0)=x$ and stopped at time $\zeta(t)$, (ii) for each $0\leq t<t’$,
(iii) for each $0\leq t<t’,$ $\{X^{t’}(s);s\geq m^{\zeta}[t, t’]\}$ and $\{X^{u}(\cdot);u\leq t\}$
are
independentgiven $X^{t’}(m^{\zeta}[t, t’])$.
3.2 Brownian snakes
In the following, we denote by RBM $([\mathrm{o}, \infty))$ a reflecting
Brownian
motion $R=$$(R(t))$ on $[0, \infty)$ with $R(\mathrm{O})=x$.
Definition 3.2. TheBrownian$\xi$-snake$\mathrm{X}=(\mathrm{X}^{t})$ starting at $x\in M$ isa$\mathrm{W}_{x}^{St\varphi}(M)-$
valued contin
uous
process with the law on $C([0, \infty)arrow \mathrm{w}_{x}^{stop}(M))$ given by$\mathrm{P}_{x}(\cdot)=\int_{C([0,\infty})arrow[0,\infty))\mathrm{P}_{x}\zeta(\cdot)PR(d\zeta)$ (21) where $P^{R}$ is the law on
$C([0, \infty)arrow[0, \infty))$ of$RBM^{0}([\mathrm{o}, \infty))$.
It is obvious that $\mathrm{X}^{0}=\mathrm{x},$ $\mathrm{a}.\mathrm{s}.$
.
Proposition 3.1. (Le Gall $([L\mathit{1}])$) $\mathrm{X}=(\mathrm{X}^{t})$ is a time homogeneous diffusion on
$\mathrm{W}_{x}^{stop}(M)$ with the transition probability
$P(t, \mathrm{w}, d_{\mathrm{W}}’)=\int\int_{0\leq a\leq b}<\infty a_{t}|\mathrm{w}|(d, db)Q_{a,b}^{\mathrm{w}}(d_{\mathrm{W}’})$ (22)
where$\Theta_{t}^{|\mathrm{w}|}(da, db)$is the joint law of
$( \min_{0\leq S}\leq tR(s), R(t)),$$R(t)$ being$RBM^{11}\mathrm{w}([0, \infty))$;
explicitly,
$_{t}^{|\mathrm{w}|}(da, db)$ $=$ $\frac{2(|\mathrm{w}|+b-2a)}{\sqrt{2\pi t^{3}}}e^{-\frac{(|\mathrm{w}1+b-2a)^{2}}{2t}}1_{\{0<a<}b\wedge|\mathrm{w}|\}$
(23)
$+$ $\sqrt{\frac{2}{\pi t}}e^{-\frac{(|\mathrm{w}|+b)^{2}}{2t}}1_{\{0}<b\}\delta 0(da)db$.
Thelifetimeprocess$\zeta(t):=|\mathrm{X}^{t}|$ isa$RBM^{0}([\mathrm{o}, \infty))$ and, conditioned on the process $\zeta=(\zeta(t))$, it is the$\xi$-snake with the deterministic lifetime fuction
$\zeta(t)$. 3.3 The snake description of$\mathrm{s}\mathrm{u}\mathrm{P}\mathrm{e}\mathrm{r}-\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{u}\mathrm{S}\mathrm{i}\mathrm{o}\mathrm{n}\{\mu(t), \mathrm{p}_{\mu}\}$
.
Let $x\in M$ and $\mathrm{X}=(\mathrm{X}^{t})$ be the Brownian $\xi$-snake starting at
$x$. Then $|\mathrm{X}^{t}|$ is a
$RBM^{0}([0, \infty))$. Let
$l(t, a)=1 \frac{\mathrm{i}}{\mathrm{c}}\downarrow \mathrm{m}_{0}\frac{1}{2\epsilon}\int_{0}^{t}1_{[}(|\mathrm{x}^{s}|)a,a+\epsilon)d_{S}$
(24) be its local time at $a\in[0, \infty)$.
Let $\mathcal{M}_{F}(M)$ be the space of all
finite
Borelmeasures
on $M$ with the topology of weak convergence and $C_{b}(M)$ be the space of all bounded continuousfunctions
on $M$. Introduce the usual notation$\langle\mu, f\rangle=\int_{M}f(X)\mu(dX)$, $\mu\in \mathcal{M}_{F}(M),$ $f\in C_{b}(M)$.
Let $(\mu(t), P)\mu$ bethe super-diffusion introduced inIntroduction. Recall
that.
this is givenas
follows:(i) the underlying process $\{\xi(t), P_{x}\}$ being given by the L-diffusion,
(ii) the branching mechanism given by
$\psi(x, z)=-c(X)z2$,
where $c(x)$ is a bounded and positive function on $M$,
$\mathrm{s}\mathrm{o}_{\vee}\mathrm{t}.\mathrm{h}\mathrm{a}\mathrm{t}\backslash$ the
$\log- \mathrm{L}\mathrm{a}\sim$place functional
$u(t, x)=-\log \mathrm{E}_{\delta_{x}}[\exp(-\langle\mu(t), f\rangle)]$
is the solution to the initial value problem
$\frac{\partial u}{\partial t}=Lu+\psi(\cdot, u)$, $u(0+, \cdot)=f$.
We
assume
that $c(x)\equiv 1$; a modification necessary to treat the general case ofpositive functions $c(x)$ has been studied in [Wat} (cf. [DS]).
Then, for $\gamma>0$ and $x\in M$, the process $\mu(t)$ under $P_{\gamma\cdot\delta_{x}}$ can be constructed
from the Brownian $\xi$-snake X $=(\mathrm{X}^{t}.)$ starting at $x$ in the following way: Define
$\mu(t)\in \mathcal{M}_{F}(M),$$t\geq 0$, by
$\langle\mu(t), f\rangle=\int_{0}^{l^{-1}}(\gamma,0)\mathrm{X}^{S}f(\langle\rangle)l(ds, t)$,
$f\in C_{b}.(M)$, (25)
where $\langle \mathrm{X}^{t}\rangle=X^{t}(|\mathrm{X}t|)\in M$: the position of$\mathrm{X}^{t}$ stopped at its lifetime $|\mathrm{X}^{t}|$ and
$l^{-1}( \gamma, \mathrm{O})=\inf\{u|l(u, 0)>\gamma\}$.
Theorem 3.1. (Le Gall$[L\mathit{1}]$) $\{\mu(t)\}$ defined by (25) is exactly the super-diffusion $\{\mu(t)\}$ under $P_{\gamma\cdot\delta_{x}}$.
Let $\mu’(t)$ be another super-process with the same underlying process as $\mu(t)$ but
with the branching mechanism
now
replaced by$\psi(_{X,Z})=-z^{2}-\theta(_{X)_{Z}}$.
It is intuitively obvious that the super diffusion$\mu’(t)$ is obtained from the
super-diffusion $\mu(t)$ by eliminating or killing some of its ”particles”; however, there is no
picture of particles in the usual formulation of super processes as measure-valued processes. We can, however, apply the snake description (25) to realize a killing operation; in the nextsection, we discuss how we can modify the expression (25) for
4 A killing operation
on
super-diffusions and subsnakesLet $\{\mu(t), \mathrm{p}_{\mu}\}$ and $\{\mu’(t), \mathrm{p}/\}\mu$ be super-diffusions as above. Then, for $x\in M$ and
$\gamma>0$, tfle measure-valued process $\mu(t)$ under $\mathrm{P}_{\gamma\cdot\delta_{x}}$ has the snake description given
by (25). We would obtain the process $\mu’(t)$ under $\mathrm{P}_{\gamma\cdot\delta_{x}}’$ in the form:
$\langle\mu’(t), f\rangle=\int_{0}^{l^{-1}}(\gamma,0))f(\langle \mathrm{x}^{S}\rangle)\kappa(sl(ds, t)$, $f\in C_{b}(M)$, (26)
where $\kappa(t)$ isacertain process taking values$0$ or 1. Soourproblem is concerned with the definition and the characterization of this process $\kappa(t)$ in terms ofthe Brownian
$\xi$-snake X and the function $\theta(x)$. Actually, we would associate to the snake X a
certain nonnegative and continuous process $\lambda(t)$ with $\lambda(0)=0$ so that the desired
process $\kappa(t)$ is given by
$\kappa(t)=1_{\{(}\lambda t)=0\}$, $t\geq 0$. (27)
We enlarge the stopped path space $\mathrm{W}_{x}^{st_{\mathit{0}}p}(M)$ to a larger space $[\mathrm{w}_{x}^{stp}o(M)]$ de-fined by
$[\mathrm{W}_{x}^{stop}(M)]=\{(\alpha, \mathrm{W})\in[0, \infty)\cross \mathrm{W}_{x}^{st_{\mathit{0}}p}(M)|0\leq\alpha\leq|\mathrm{w}|\}$ . (28)
We endow it with the topology induced from the product topology of $[0, \infty)\cross$
$\mathrm{W}_{x}^{stop}(M)$. Given abounded, nonnegative and continuousfunction$\theta=(\theta(x))$ on$M$, wedefine, for$t>0$and $(\alpha, \mathrm{w})\in[\mathrm{w}_{x}^{st_{\mathit{0}}p}(M)]$, aBorelprobability$\hat{P}(t, (\alpha, \mathrm{w}), d\alpha’d\mathrm{w}’)$ on $[\mathrm{W}_{x}^{sto}p(M)]$ by
$\hat{P}(t, (\alpha, \mathrm{w}), d\alpha’d_{\mathrm{W}}/)=\int\int_{0\leq a\leq<}b\infty)\theta|_{\mathrm{W}}t(|)da,$$dbqa,b(\alpha,\mathrm{w})(d\alpha d/\mathrm{W}’$ (29)
where $q_{a,b}^{()}(\alpha,\mathrm{W}d\alpha d/\mathrm{w}’),$ $0\leq a\leq b<\infty$, is defined by
$q_{a},((\alpha_{b},\mathrm{w})/d_{\mathrm{W})}/d\alpha$ $=$ $1_{\{\alpha<a\}}\delta\alpha(d\alpha/)Q^{\mathrm{w}}a,b(d\mathrm{W}’)$ (30)
$+1_{\{\alpha\geq a\}}1 \{a<\alpha’<|\mathrm{W}|’\}\theta(w’(\alpha’))\exp[-\int_{a}^{\alpha’}\theta(w(u)/)du]d\alpha/Q_{a}^{\mathrm{W}},b(d_{\mathrm{W}’})$
$+1_{\{\alpha\geq a\}} \exp[-\int_{a}^{|_{\mathrm{W}’}|}\theta(w/(u))du]\delta_{|\mathrm{w}}’|(d\alpha/)Q_{a,b}^{\mathrm{W}}(d\mathrm{w}’)$.
Theorem 4.1. The family $\{\hat{P}(t, (\alpha, \mathrm{w}), d\alpha’d_{\mathrm{W}}/)\}$ defines a system of $tr\mathrm{a}$nsition
probabilities on $[\mathrm{w}_{x}^{Sto}p(M)]$ and it determines a unique $tim\mathrm{e}ho\mathrm{m}$ogeneous diffu-sion $\overline{\mathrm{x}}=(\alpha^{t}, \mathrm{x}^{t})$ on $[\mathrm{W}_{x}^{st\sigma}p(M)]$.
In the following, we
assume
$\alpha^{0}=0$ and $\mathrm{X}^{0}=\mathrm{x}$:Definition 4.1. The diffusion$\overline{\mathrm{X}}=(\alpha^{t}, \mathrm{x}^{t})$on $[\mathrm{w}_{x}^{st_{\mathit{0}}p}(M)]$ with$\alpha^{0}=0$ and$\mathrm{X}^{0}=\mathrm{x}$
is called the $\theta$-subsnake of the Brownian
Obviously, the process $\mathrm{X}--(\mathrm{X}^{t})$ defined by the second component of
$\overline{\mathrm{X}}$
is
a
Brow-nian $\xi$-snake starting at $x$. ..
Define .
. $\lambda(t)=|\mathrm{X}^{t}|-\alpha^{t}$, $t\geq 0$. (31)
Since $|\mathrm{X}^{0}|=0$ and $\alpha^{0}=0$, we also have $\lambda(0)=0$.
Definition 4.2. The $di$ffusion $\overline{\mathrm{x}}=(\lambda(t), \mathrm{x}t)$ on [$\mathrm{w}_{x}^{sto}p(M)1$ is called the second
$\theta$-subsnake ofthe Brownian $\xi$-snake X starting at $x$.
Theorem 4.2. Let$\overline{\mathrm{X}}=(\lambda(t), \mathrm{X}^{t})$ be thesecond $\theta$-subsnake and define the process
$\kappa(t)$ by (27). Then the equation (26) determin
es
the superdiffusion$\mu’(t)$ under$\mathrm{P}_{\gamma\cdot\delta_{x}}’$.Thus, the killing operation (26) to obtain $\mu’(t)$ from $\mu(\underline{t)}$ through their snake
descriptions can be determined by the second $\theta$-subsnake X,
or
equivalently, bythe $\theta- \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{S}\mathrm{n}\mathrm{a}\mathrm{k}\mathrm{e}\overline{\mathrm{X}}$
. So we would like to characterize these snakes in terms of the Brownian $\xi$-snake and the function $\theta$ in a much simpler way.
Theorem 4.3. Let $\overline{\mathrm{X}}=(\lambda(t), \mathrm{X}^{t})$ be the second $\theta$-subsn$\mathrm{a}ke$ of the Brownian $\xi-$
sn$\mathrm{a}ke$ starting at $x\in M.$ Then, for
$0<s_{1}<s_{2}.<\ldots<s_{m-1}<s_{m}$ and $0<\mathrm{t}_{1}<$
.
..
$<t_{n}$, we have.
$P(\lambda(s_{1})=0,$ $\lambda(_{S_{2}})=0,$
$\ldots,$$\lambda.(_{S)=0,\lambda}m-1(s_{m})=0, \mathrm{x}t1\in d\mathrm{w}_{1}, \ldots, \mathrm{x}^{t_{n}}\in d\mathrm{w}_{n})$
$=$ $E$
(
$e^{-M[0,][}s_{1}e^{-}Ms1,s2]\ldots-M[s_{m}-1,s_{m}e;]\mathrm{X}^{t_{1}}\in d\mathrm{w}_{1},$.
. .,$\mathrm{X}^{t_{n}}\in d\mathrm{w}_{n}$)
(32)where
$M[s, t]= \int_{\min_{S\leq}|}^{|\mathrm{X}^{t}}|(\theta(xt)u)duu\leq t\mathrm{X}u|$
’ $0\leq s<t$. (33)
In other words, conditioned on the Brownian $\xi$-snake X, the joint law of the
$\{0,1\}-\mathrm{t}^{\mathit{7}}\mathrm{a}lued$process $\kappa(t)=1_{\{\lambda(t)=0}\}$ is given by
$P(\kappa(s_{1})=1, \kappa(s_{2})=1,$
$\ldots,$$\kappa(S_{m-1})=1,$$\kappa(s_{m}-)=1/\mathrm{X})$
$=$ $e^{-M[]-M}0,s1ee[s_{1},s_{2}]\ldots-M[sm-1,s_{m}]$. $\cdot$. (34)
Theorem 4.4. Let $\overline{\mathrm{X}}=(\lambda(t), \mathrm{x}t)$ be the second $\theta$-subsn$\mathrm{a}k\mathrm{e}$ ofthe Brownian $\xi-$
snake starting at$x\in M$. Define
$A(t)= \int_{0}^{t}1_{\mathrm{t}\lambda}(s)=0\}dS(=\int_{0}^{t}\kappa(s)d_{S})$ (35)
and let $A^{-1}(t)$ be the right-continuous inverse of$t\mapsto A(t)$. Define further
$S(t)=|\mathrm{X}^{A^{-1}()}t|$, $t\geq 0$
.
(36)Then $S(t)$ is a continuous process and the following identi$\mathrm{t}y$holds:
.$\cdot$..
Theorem 4.4 asserts that the process $(A(t), \mathrm{X}t)$ determines the $\theta$-subsnake $\overline{\mathrm{X}}$
or $\overline{\mathrm{X}}$
so that we only need to obtain the process $(A(t), \mathrm{X}t)$ in order to obtain the
$\theta- \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{S}\mathrm{n}\mathrm{a}\mathrm{k}\mathrm{e}\overline{\mathrm{x}}_{\mathrm{O}}\mathrm{r}\overline{\mathrm{X}}$. By
Theorem 4.3, we can obtain the process $(\kappa(t), \mathrm{x}t)$ uniquely in the lawsense and hence its measurableversion, so that the process $(A(t), \mathrm{X}t)$ can
be obtained uniquely in the law sense.
Another characterization of the second $\theta_{-\mathrm{S}}\mathrm{u}\mathrm{b}\mathrm{s}\mathrm{n}\mathrm{a}\mathrm{k}\mathrm{e}\overline{\mathrm{x}}=(\lambda(t), \mathrm{X}^{t})$ can be given
bymeans of a stochastic differentialequation (SDE)whichis
a
natural generalization ofSDE (6). First, we formulate a SDE.On asuitable probability space equipped with a filtration $\mathrm{F}=\{\mathcal{F}_{t}\}$, weconsider a continuous process $(\lambda(t), \mathrm{x}t)$ on $[0, \infty)\cross \mathrm{W}_{x}^{stop}$ with $\lambda(0)=0$ and $\mathrm{X}^{0}=\mathrm{x}$, which satisfies
,
$\mathrm{t}\mathrm{h}\mathrm{e}$ following conditions:
(i) the process $(\lambda(t), \mathrm{X}^{t})$ is $\mathrm{F}$-adaped and
$\mathrm{X}=\{\mathrm{X}^{t}\}$ is a Brownian$\xi$-snake
start-ing at $x$,
(ii) the Brownian motion $\{B(t)\}$ defined by
$B(t)=|\mathrm{X}^{t}|-l(t, \mathrm{o})$, (38)
is an $\mathrm{F}$-Brownian motion in the
sense
that $B(t)-B(s)$ is independent of$\mathcal{F}_{s}$
for every $0\leq s<t$,
(iii) $\{\lambda(t)\}$ satisfies the following stochastic differential equation:
$d \lambda(t)=1_{\{\lambda(t)0\}}>dB(t)+\frac{1}{2}1_{\{\lambda(t)\}}=0^{\cdot}\theta(\langle \mathrm{X}^{t}\rangle)dt$. (39)
Such a process $(\lambda(t), \mathrm{X}^{t})$ is called a solution ofSDE (39), or an $\mathrm{F}$-solution of SDE
(39) whenwewould refer to thefiltration$\mathrm{F}$, with initial values
$\lambda(0)=0$ and$\mathrm{X}^{0}=\mathrm{x}$.
Theorem 4.5. Let $\overline{\mathrm{X}}=(\lambda(t), \mathrm{X}^{t})$ be the secon$\mathrm{d}\theta$-subsnake of the Brownian
$\xi-$
snake starting at $x\in M.$ Then it is a solution to $\mathrm{S}DE\underline{(}\mathit{3}\mathit{9}$). Furthermore, the
uniq$\mathrm{u}$eness in law of solutions to $SDE(\mathit{3}\mathit{9})$ holds so that $\mathrm{X}=(\lambda(t), \mathrm{X}^{t})$ is
charac-terized completely by$\mathrm{S}DE(\mathit{3}\mathit{9})$.
In the definition of SDE (39), we assumed that the second component $\mathrm{X}^{t}$ of a
solution is a Brownian $\xi$-snake. If we rewrite a martingale problem for Brownian
$\xi$-snake studied by Dhersin and Serlet $([\mathrm{D}\mathrm{S}])$, we can also formulate a SDE for the
joint process $(\lambda(t), \mathrm{x}t)$ and characterize the second $\theta$-subsnake by its solution.
A proofof Theorem4.5 canbe given by showing the following: Set, fora bounded and continuous function $F(\alpha, \mathrm{w})$
on
$[\mathrm{w}_{x}^{stop}(M)]$ and $t\geq 0$,$H(t, ( \lambda, \mathrm{w}))=\int_{[\mathrm{W}_{x}^{sto}}p(M)]’-F(\alpha^{J/}\mathrm{w})\hat{P}(t, (|\mathrm{W}|\lambda, \mathrm{W}), d\alpha^{\prime/}d\mathrm{W})$
where $\hat{P}(t, (\alpha, \mathrm{w}), d\alpha d_{\mathrm{W}^{J}}’)$ is defined by (29). Then for any $\mathrm{F}$-solution $(\lambda(t), \mathrm{x}t)$ of SDE (39) and a fixed $T>0,$ $tarrow H(T-t, (\lambda(t), \mathrm{x}^{t}))$ is an F-martingale.
. , : ,.
$\cdot$
$*:-$ $\vee^{\backslash ^{\backslash }\cdot:}$ ’ $\mathrm{R}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{e}_{-}\mathrm{n}$
ces
$\wedge$ .$\cdot$.
..
$\backslash \backslash$ $.\mathrm{p}’$. ..
!
$.\theta$ :$\sim i$
.
$\cdot\vee$$-$
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