数理解析研究所講究録1089 巻 1999 年 1-14

1

An Overview of T.he Studi.es on
Catalytic Stochastic Processes*)
KU (J‘g-エ
Isamu
Department of Mathematics, Saitama University
Urawa 338-8570 Japan$\mathrm{D}\hat{\mathrm{O}}$1$\ovalbox{\tt\small REJECT}$

Prologue

The aim of this expository article is to give an overview of comparatively recent resultson critical continuous super-Brownian motions $X$ in catalytic media, which have been
. Dawson and K. Fleischmann (cf. [DFG95]). Herestudied and developed by chiefly
catalytic media means that the branching rate for measure-valued processes is given by ag.e.neralized function. The most typical example is an extremely simplified, in which there. In this case, branching occurs only at position
case, namely,
nt catalyst with infinite rate; while outside only the heat flow isis the so-called single$\mathrm{D}.\mathrm{A}$$\rho$$\mathrm{o}\mathrm{n}\mathrm{e}- \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{S}\mathrm{i}_{0}\mathrm{n}\mathrm{a}1$$c\in \mathrm{R}$$\rho=\delta_{c}$$c$$\mathrm{p}\mathrm{o}.\mathrm{i}$predominant.
.
Generally, the superprocess is a measure-valued branching process, and it may be obtained from certain specific branching particle systems via high density limit procedure[D93]. It is known that many new distinct remarkable phenomena are observable for the
single point catalytic super-Brownian motion (SBM).
(a) Jointly continuous super-Brownian motion local times$y:=\{y_{t}(a);t>0, a\in \mathrm{R}\}$

exists, but $y(c):=\{y_{t}(c))t>0\}$ is only singularly continuous for the catalyst point .
(b) The super-Brownian local time $y(c)$ at can be alternatively constructed at the total.
stable motion on
occupation time measure of a one-sided(c) With an excursion type formula, we can define the mass density field
$c$

$c$

$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{e}\mathrm{r}-\frac{1}{2}$$\mathrm{R}_{+}$$x:=\{x_{t}(a);t>0, a \neq c\}$*Research supported in part by JMESC$\mathrm{C}\mathrm{R}(\mathrm{A})(1)$

09304022,$\mathrm{C}\mathrm{R}(\mathrm{A})(1)$

10304006.

$\mathrm{c}_{\mathrm{r}\mathrm{a}\mathrm{n}}\mathrm{t}-\mathrm{i}\mathrm{n}$-Aid SR(C) 07640280 and also by JMESC-Aid$\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{t}- \mathrm{i}\mathrm{n}$2of single point catalytic super-Brownian motion $X$ by employing $y(c)$ , with the result that
we can show that it solves the heat equation and also that it is.
So much for interesting properties that the single point catalytic super-Brownian motion $X$ possesses, another stimulating problem is the construction of higher dimensional
catalytic super-Brownian motions with absolutely continuous states, in contrast to the constant medium as . In so doing, the main analytical tool would be the theory of nonlinear
reaction diffusion equations in which -functions enter in various ways.
Remark. The intuitive interpretation behind the phenomenon described above in (a) is thatthe catalyst normally kills off theby the infinite branching rate, however, branchingdoes occur occasionally, at exceptional times of full dimension.
$C^{\infty}$$\mathrm{c}$$\mathrm{e}$$\delta$$\mathrm{m}\mathrm{a}s\mathrm{s}$2

Ekom Branching Diffusion to SBM

2.1 The Feller Branching Diffusion
A. The $BGW$ Process. First of all we consider the idealized simplest model where thespace consists of a single point, say, $E=\{x\}$ . This means that the positions of the particles
are not distinguishable and their motions are completely neglected. Let$z^{(\rho)}=\{z_{t}^{(\rho)};t\geq 0\}$be a continuous-time critical binary Bienayme’-Galton-Watson (BGW) process with branching rate $\rho>0$ . The model starts at time $t=0$ , and $z_{0}>0$ indicates the number of initial
particles of the model. Here particles evolve independently in accordance with the law thateach particle dies with rate and splits into two particles with rate ; i.e.,$\rho$$\rho$,

with rate2,

with rate$0$$1arrow\{$

Note that this$z^{(\rho)}$$\rho$$\rho$,(1).

is nothing but a birth and death process.

B. The Scded $BGW$ Process. Now let $N>>1(N\in \mathrm{N})$ , and let$z_{0}:=[N(_{0}]$$\zeta_{0}>0$

be fixed. Put.

This time, let the model start withinitial particles, and give each particle the small
mass $1/N$ (because $N$ is a large number). We consider the situation of speeding up theprocess by switching to the large branching rate
. Then we define the scaled BGWprocess by$z_{0}$$N\rho$$\rho$$\{(_{t\}_{t\geq}:=}^{(N)}0\{\frac{1}{N}z^{(N\rho)}t|_{Z}0=[N\zeta_{0}]\}_{t\geq 0}$

Then this$\zeta^{(N)}$determines a sequence of$\mathrm{R}_{+}$-valued continuous-time processes.

3

C. The Feller Branching $Diff_{LL}sion$. We consider the limit procedure of the above.
mentioned scaled model. It is well-known (cf. Feller (1951), [Fe51]) that by passage to$t\geq 0$ converges in distribution to thethe limit as $Narrow\infty$ , the scaled BGW model:
critical branching diffusion process$\cdot$.

$\{..\zeta_{t}^{(N)}\},$$\zeta=\{\zeta_{t}\}$$(Narrow\infty)$$\zeta^{(N)}\Rightarrow\zeta$.

The limiting process is called Feller’s branching diffusion. This diffusion process on
can also be obtained as a solution of the stochastic differential equation of the form$\zeta$$\mathrm{R}_{+}$$\zeta$$d\zeta_{t}=\sqrt{2\rho\zeta_{t}}dB_{t}$(2),$t>0$ ,,$\zeta_{0}\geq 0|$

where $B=\{B_{t}\}$ is a one-dimensional standard Brownian motion$(\mathrm{B}\mathrm{M})$.

2.2 Super-Brownian Motion
D. A Prototype of Superprocess. Since the superprocess is a certain measure-valued
process, adding a spatial concept we may introduce$X_{t}:=\zeta t\delta_{0}$,$t\geq 0$.

The location just corresponds to the single point of the space in question. In the above,
thought out is a new notion to combine the population mass together with the location.
Note that there is no notational distinction between the -measure and the Dirac -function
as a superprocess in zero dimensions.
as its generalized derivative. We can interpret thisSymbolically, we may have the following formal expression$0$$\zeta_{t}$$\delta$$\delta$$X_{t}$$dX_{t}=\sqrt{2\rho X_{t}}dB_{t}$,$t>0$ ,

$X_{0}\geq 0$(3).

be a critical
Approximation. Let
and
E. Branching $BM$ inof \S 2.1. Suppose now that the particles act according
binary BGW process given in. Here newly born particles start
to law of independent standard Brownian motions inat their parents’ position. Then we get the critical binary branching Brownian motion inwith branching rate$R^{d}$$\{z_{t}^{(\rho)}; t\geq 0\}$

$Diff_{LS}\iota ion$”$\mathrm{A}$”$\mathrm{R}^{d}$$\mathrm{R}^{d}$$\rho$$\Phi_{t}^{(\rho)}:=\sum_{i=1}^{z_{t}^{(\rho)}}\delta_{w_{t}^{i}}$,$t\geq 0$,

denotesis the number of particles at time , acting as the BGW process, andwhere
the position of the i-th particle at time , which arises from some Brownian path whereis described in terms of counting
these paths are not independent any more. The statemeasures at time . We consider the same type of diffusion approximation. Let $N(>>1)$
be the number of the initial particles situated at , and let the model start with $N$ particles.
$w_{t}^{i}$$t$$z_{t}^{(\rho)}$$t$$\Phi_{t}^{(\rho)}$

$t$

$x$

4

Let each particle branch with the large rate
to say, this leads to the scaled process

and possess the small mass $1/N$ . That is$N\rho$$X_{t}^{(N)}:=\{^{\frac{1}{N}\Phi_{t}^{(N\rho}})|\Phi_{0}=N\delta x\}$.

By passage to the limit $Narrow\infty$ , we obtain$X_{t}^{(N)}\Rightarrow X_{t}$,$(Narrow\infty)$(cf S. Watanabe (1968), [W68]), where the limiting process
is called the critical continuous super-Brownian motion with branching rate (cf. Dawson-Watanabe superprocess[Dy94] . Consequently, the measuredescribes the population at time , and can beinterpreted as a cloud of mass. Heuristically, $X=\{X_{t}\}$ can be regarded as a solution ofsymbolic stochastic equation$X_{t}$$\rho$$X_{t}$$)$$t$$dX_{t}= \frac{1}{2}\Delta X_{t}dt+\sqrt{2\rho X_{t}}dW_{t}$where$\Delta$

is the Laplacian and$\dot{W}$,$t>0$ ,

$X_{\mathrm{o}}=\delta_{x}$,$x\in \mathrm{R}^{d}$,(4)

is a spacetime white noise.

Remark. (i) The equation (4) consists of two components, in other words, it seems to be
a combined version of Eq. (2) and Eq. (3). This suggests that, as to the $(2\rho x_{t})1/2dW_{t}$ part,the population grows at each pointaccording to Feller’s branching diffusion, while,
as to the (1/2)
part, the population mass is smeared out by the heat fiow.
(ii) It is known that in the case $d=1,$ $X$ lives on the space of absolutely continuous
measures with probability one, i.e., there exists the density fieldsuch that$x\in \mathrm{R}^{d}$$\Delta X_{t}dt$$x_{t}(a)$$X_{t}(da)=x_{t}(a)da$Then$x_{t}(a)$,$t>0$ .

solves the stochastic partial differential equation (SPDE):$dx_{t}(a)= \frac{1}{2}\Delta x_{t}(\mathit{0})dt+\sqrt{2\rho x_{t}(a)}dW_{t}(a)$,$T>0$ ,

$a\in \mathrm{R}$.

(5)

Note thatacts on the space variable only. In fact,
has a rigorous meaning forthe one-dimensional case, which is due to e.g. Konno-Shiga(1988) [KS88].
$\Delta$3$a$$\mathrm{E}\mathrm{q}.(5)$Catalyst and Catalytic Media

3.1 Particular Situation in Irregular Media

Consider the SBM with branching rate
, which was introduced in theprevious section. If this parameter is not a constant any longer, and if it varies in space
and take different values respectively at each point, namely,$\rho(=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t})>0$$\rho$$\rho=\rho(a)$,$a\in \mathrm{R}^{d}$,5then the situation will be described by the so-called model in a varying medium. Then thesituation implies that there exists some place in the model in which the branching occurs
with a larger rate than at others. If a generalization of is taken into consideration, it ispossible to think of extensions of to wider classes of functions, in accordance with theinclusion of functional spaces, such as$\rho$$\rho$$C^{\infty}\subset C^{m}\subset \mathcal{B}$,denote the space of measurable functions. Furthermore, may possibly be awheregeneralized function. Such a model is called the model in an irregular medium. One of thetypical examples in the irregular medium model is the case$B$$\rho$(a Ditac$\rho=\delta_{c}$$\delta$-function).

That is to say, it exhibits the particular situation that branching occurs only at the single
point with an infinite rate [DFG95].
$c$Definition 1 (Single Point Catalyst) In this case, we say that a point catalyst is located at .
$c$

This single point catalyst controls the branching at , whereas only the heat fiow acts andis dominant outside .
$c$

$c$

3.2 Approximation and Local Timewith
Now let us consider some approxim‘ation of a Dirac -function by step function. As the particle level, this means that a particie may branchand height
width
only if it lies within the -vicinity of the point . Then the integral$\delta$$\varphi_{\epsilon}$$\epsilon>0$$1/\epsilon$$c$$\epsilon$$\int_{0}^{t}$I$\{|w_{s}-C|\leq\frac{\epsilon}{2}\}ds$

represents the occupation time by time . This gives us the interpretation that the particle(at )during that period. Then the Brownian local time
will branch with rate
arises naturally in the limiting procedure [IW81]:$t$$L^{\mathrm{c}}(t)$$1/\epsilon$$\frac{1}{\epsilon}\int_{0}^{t}$I$\{|w_{c}-c|\leq\frac{\epsilon}{2}\}dsarrow L^{c}(t)$,$(\epsilonarrow 0)$$c$.

On this account, it seems quite reasonable to think of point catalysts only in dimension
one, since Brownian local times at points make proper sense only in dimension one. (cf.
[DFG95]$)$

3.3 Studies on Catalysts

Some other related studies or fields on catalyst are the followings [DF97]:

6

1. Fractal catalysts from a physical point of view (Sapoval, 1991), where
the objects are higher dimensional catalytic media.
2. Catalytic reaction diffusion equations via PDE method (Bramson-Neuhauser,
1992), (Chan-Fung, 1992), where the nonlinear differential equation of the form$-\partial_{t}u=(1/2)\Delta u+\rho_{t}\cdot R(u)$ is considered.
3. Catalytic chemical systems (Chadam-Yin, 1994), where various kinds ofchemical reactions are considered.
4. Catalytic biological systems, where considered is the model of biochemical
reactions that glycosis enzymes are serving as catalysts on a filament network.

One of the main reasons to study branching models in varying media, in particular, in
irregular media is to search for new mathematical phenomena caused by the medium itself.
Superprocess in irregular media was introduced by,[DFR91]. The next theorem is a simple example of a new effect.
$\mathrm{D}\mathrm{a}\mathrm{w}\mathrm{s}\mathrm{o}\mathrm{n}- \mathrm{F}\mathrm{l}\mathrm{e}\mathrm{i}_{\mathrm{S}\mathrm{C}}\mathrm{h}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{n}-\mathrm{R}\mathrm{o}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{y}(1990)$

Theorem 1 Let $X=\{X_{t}\}$ be a super-Brownian motion. The total mass process {$t\geq 0\}$ is a Feller branching
diffusion if and only if the medium is constant, $i.e.,$$X_{t}(R^{d}),\cdot$$\rho=$

4

const.

Single Point Catalytic SBM

We can observe many interesting and stimulating phenomena in the mathematical modelof one-dimensional super-Brownian motion $X$ with a single point catalyst, whichis the extremely simplified model among catalytic superprocesses. First of all, we observe
some basic properties on the density field of singi..e point catalytic SBM.
Let
be the catalyst’s position and fixed once and for all. A single point catalyst
provides with the point-catalytic medium for superprocess. We denote by $M_{F}=$the set of finite measures on R. We consider below the one-dimensional SBM$\rho=\delta_{\mathrm{c}}$$c\in \mathrm{R}$$\rho=\delta_{c}$$M_{F}(\mathrm{R})$$X=\{x_{t}, \mathrm{p}_{\mu}, \mu\in M_{F}\}$

with a single point catalyst, where is the initial stateof $X$ , namely, $X_{0}=\mu$
holds
, and
is the law of $X$ . The existence of such a finite measure-valued Markov
process $X$ is intuitively
ed on the existence of Brownian local times in one dimension.
There are two distinct methods to show its existence. One is a strong construction of $X$
by regularization of , which is due to Dawson-Fleischmann (1991) [DF91]. The other isa quite different method to construct $X$ by making use of the Brownian local time at via
additive functional approach, which is due to Dynkin (1991) [Dy91].
Letdenote the space of weakly continuous finite measure-valued trajectories
satisfying $X_{t}(\{c\})=0$ for all $t>0$ .
is the space of all non-negative continuous functions
on having a Gaussian decay, i.e.,$\rho=\delta_{c}$$\mathrm{a}.\mathrm{s}.$$X_{0}$$\mu$$\mathrm{P}_{\mu}$$\mathrm{b}\mathrm{a}s$$\delta_{c}$$c$$C(\mathrm{R}_{+}, M_{F})$$\mathcal{G}_{+}$$\mathrm{R}$$\varphi$$|\varphi(z)|\exp\{C\varphi^{Z\}}2$,$z\in \mathrm{R}$,7. $\{S_{t};t\geq 0\}$ denotes the Brownian semigroup, i.e., itis bounded for some constantand transition density $p=p(t, z)$ . The
is the Markov semigroup with generator (1/2)
next theorem asserts the path continuity of catalytic SBM.
$c_{\varphi}>0$$\Delta$

Theorem 2 ([DF94], Theorem 1.2.1, p.6) The $time-homogeneo’\iota\iota S$ Markov process $X=$
determined by equation$\{X_{\ell}, \mathcal{F}_{t}, t\geq 0, P_{\mu}, \mu\in M_{F}\}$$\frac{\partial}{\partial t}u(t, z)=\frac{1}{2}\Delta u(t, z)-\delta_{\mathrm{C}}(_{Z})u^{2}(t, z)$,$t>0$ ,,

$z\in R$$z\in R$

,

(6)

$\{$

$u(0, z)=\varphi(Z)$,$\varphi\in \mathcal{G}+$,via the Laplace transition functional$E\{\exp\langle X_{t}, -\varphi\rangle|X_{s}=\mu\}=\exp\langle\mu, -u(t-s)\rangle$,$0\leq s\leq t,$$z\in \mathcal{G}_{+},$$\mu\in M_{F}$,(7)

can be constructed on $C(R_{+}, M_{F})$ .
Moreover, the following expectation and covariance formulae hold.

Proposition 1 For $0\leq s\leq t,$ $\mu\in M_{F},$$\varphi,$$\psi\in \mathcal{G}$

,
(8)

,

$E_{\mu}\langle X_{t}, \varphi\rangle$$=$$\langle\mu S_{t}, \varphi\rangle$$c_{ov_{\mu}}[\langle xs’\varphi\rangle, \langle x_{t}, \psi\rangle]$$=$$2 \int\mu(da)\int_{0}^{S}p(r, c-a)S_{S}-r\varphi(C)S_{t-}r\psi(_{C})dr$(9).

The next result shows that the catalytic SBM $\{X_{t};t>0\}$ lives on the space of absolutely
can be chosen properly to becontinuous measures, and also that the density field
jointly continuous on $\{t>0\}\mathrm{x}\{z\neq c\}$ .
$\{x_{t}\}$,Theorem 3 ([DF94], Theorem 1.2.2, p.7) $(a)$ There is a version of $X$ (with$c\in R)$ such that there ex\’ists a sample $joinu_{y}$ continuous random field $x=\{x_{t}(Z);t>0$ ,$z\neq c\}$ satisfying$\rho=\delta_{c}$$X_{t}(d_{Z})=x_{t}(z)dZ$$(b)$

The state$x_{t}$

,

for

all

$t>0$ ,

$P_{\mu}-a.s.$,$\mu\in M_{F}$at time $t>0$ of the time-homogeneovs Markov process$x$.

has the Laplace

functions,$E_{\mu} \exp\{-\sum_{i=1}^{k}xt(Zi)\theta_{i\}}=\exp\langle\mu, -u(t)\rangle$$t>0$ ,$\theta_{i}\geq 0$,$z_{i}\neq c$

where $u(\geq 0)$ solves the equation,$\frac{\partial}{\partial t}u=\frac{1}{2}\Delta u-\delta_{c}u^{2}$$u|_{t=0}+= \sum_{i=1}^{k}\theta_{i}\delta z\dot{.}$.

,$1\leq i\leq k$,

8Moreover, for fixed $t>0$ and$c\in \mathrm{R}$$a,$the following estimate is obtained:as$Var[x_{t}(Z)|X_{0}=\delta_{a}]\sim \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{S}\mathrm{t}.|\log|z-c||$$zarrow c$.

Thus immediately we get

Proposition 2 (Blow-Up of the Variance)$Var_{\mu}[_{X_{t}}(z)]arrow\infty$,$(zarrow c)$.

The variance of the continuous density
blows up as approaches the catalyst’s position. Roughly speaking, it suggests that the density ofis highly fluctuating in the vicinityof the catalyst. However, opposed to this, the following is true.
$x_{t}(z)$$z$$X_{t}$$c$

Theorem 4 (Vanishing Density at the Catalyst’s Position) For fixed $t>0$ , bypassage to the limit $zarrow c$ ,$x_{t}(z)arrow 0$

5

,

in

$P_{\mu}$-probability,$\mu\in M_{F}$.

Super-Brownian Local Time

By the sample path continuity of the $X$ -process stated in the previous section, we may
introduce the occupation time process $Y=\{Y_{t}; t\geq 0\}$ related to $X$ , that is,$\langle Y_{t}, \varphi\rangle:=\int_{0}^{t}\langle X_{s}, \varphi\rangle ds$

Since

$Y$

,

$\varphi\in \mathcal{G}_{+}$.

is smoother than $X$ by the integration,$y_{t}(z):= \int_{0}^{t}x_{t}(z)d_{S}$,$t\geq 0$,$z\neq c$,(10)$\mu\in M_{F}$ .
is jointly continuous on
Thisyields an occupation
density field of $Y$ , which is also called the super-Brownian local time (SBLT) related to
X. Our main concern is to study the behavior of SBLT when approaching the catalyst’s
position. The following theorem implies the non-degeneracy of SBLT at .
$\mathrm{R}_{+}\cross\{z\neq c\},$$\mathrm{P}_{\mu}- \mathrm{a}.\mathrm{s}.,$$y_{t}$$c$

Theorem 5 ([DF94], Theorem 1.2.4, p.8) $(a)$ There is a version of $X$ such that theoccupation density field of $Y$ defined by Eq. (10) extends continuously to all of $R_{+}\cross R$ .
$(b)$ Moreover, the following moment
formvlae hold for $0\leq s\leq t<s’\leq t’,$ $z’\in R,$
:
$y$

$z,$

$\mu\in$$M_{F}$$E_{\mu}[y_{t}(z)]$$=$$\int\mu(da)\int_{0}^{t}p(_{S}, Z-a)dS$

,

(11)

$Var_{\mu}[yt(Z)-ys(z)]$$=$2 $\int\mu(da)\int_{0}^{t}d\tau p(\tau, c-a)\{\int_{\tau\vee S}^{t}p(r-\mathcal{T}, z-C)dr\}$ .

(12)

The expectation formula implies that even at the catalyst’s position the occupation density
cannot be identically , which is in contrast to the . vanishing random density
at for fixed in the sense of Theorem 4. Note also that the variance of remains finite
even at the catalyst’s position, opposed to the blow-up effect described in Proposition 2.
$y_{t}(c)$

$c$

$0$

$t$

$x_{t}(c)$$\mathrm{a}.\mathrm{s}$

$y$

9

Singularity at The Catalyst

6

The occupation density field is monotone increasing in the time variable , and hence:on
it defines some locally finite continuous random measurefor each$t$$y$$\lambda^{z}$$z\in \mathrm{R}$$z\in R$$\lambda^{z}(dt):=dy_{t}(_{Z)},$.

We call it the occupation density measure at . By he definit.ion (10).’,are . absolutely continuous as long as $z\neq c$ .
$z$$\mathrm{R}_{+}$$.\mathrm{t}$t.h.eSe m.easures$\lambda^{z}$$\mathrm{a}.\mathrm{s}$. The occupaTheorem 6 ([DFLM95], Theorem 1.1.4, p.38) Assume thatat the catalyst’s position is with probability one a singular diffuse
tion density measure.
random measure on$X_{0}=\delta_{c}$$\lambda^{c}$$R_{+}$

The approach to the above theorem, adopted in [DFLM95], is very unique. They firstconsider an enriched version of $X$ , namely, the historical point catalytic super-Brownian
at time keeps track of the entire history of. Here the statemotionses alive at and their family relationships. In addition, it arises as thethe population
diffusion limit of the reduced branching tree structure associated with the approximating
son-Perkins (1991) [DP91]; and also see Dynkin (1991)
branching particle system (cf.
.
, whereis replaced
In this setting, the occupation density measure$\tilde{X}=\{\tilde{X}_{t};t\geq 0\}$$\tilde{X}_{t}$

$t$

$t$

$\mathrm{m}\mathrm{a}s$$\mathrm{D}\mathrm{a}\dot{\mathrm{w}}$$[\mathrm{D}\mathrm{y}9\mathrm{l}\mathrm{a}])$$\mathrm{b}^{f}\mathrm{y}\tilde{\lambda}^{c}(d[r, w])$$\lambda^{c}(dr)$$\tilde{\lambda}^{\mathrm{c}}([r_{1,2}r]\mathrm{x}B)$

due to paths in
exposes the contribution to the occupation density increment
on $[0, r]$ , which start at time at and also
the subset $B$ of -Brownian bridge paths.
end up in at time
-Brownian bridge paths on $[0, r]$ can be interpreted as
Remark. The
the trajectories of particles in question that contributed to the occupation density increment$\tilde{\lambda}^{c}([r_{1}, r_{2}])$

$r,$

$c$

$0$

$w$

$c$

$(r_{1}\leq r\leq r_{2})$$w$$\mathrm{a}\mathrm{b}\mathrm{o}\mathrm{v}\epsilon\succ \mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{d}C$$\lambda^{c}([r_{1}, r_{2}])$$c$.

We fix a closed finite time interval $I:=[0, T],$ $0<T<\infty$ . For a path $w\in C=C(I, \mathrm{R})$ ,we write for the set of all stopped paths . Given a path $w\in C$ , we interpret$C^{t}$$\tilde{w}_{t}$$\tilde{w}:=\{\tilde{w};t\in I\}$

as a path trajectory. In addition, we introduce the set$C^{t,z}:=\{w\in C^{t}, w_{t}=z\}$of continuous paths on stopped at timeat time is a unit measureofat the catalyst. Also, let$I$$\tilde{X}$$s$$c_{s,w}^{r,C}*:=$

$t$

,

is

,

$z\in R$,at . For $s\in I$ , the starting measure, a path stopped at timeat$\mu\in M_{F}^{s}$$z$$\delta_{w}*\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}${ $w\in C^{r,c};w$$t\in I$$w^{*}\in C^{s,c}$a continuous extension of$w^{*}$

}

$s$

10

for $0\leq s\leq t\leq T,$$w^{*}\in C^{s,c}$. We define a subset$A\cross B\wedge$

of $A\cross B$ by$A \cross\wedge B^{\cdot}:=\{[a, b];a\in A, b\in B^{a}\}=\bigcup_{a\in A}\{a\}\mathrm{X}B^{a}$.

Analogously to the standard theory (e.g. [DP91]), even in [DFLM95] the infinite divisalso allows us to use the framework of theibility of the law of the random measureso-called L\’evy-Khintchine representation. That is,$\tilde{\lambda}^{c}$Lemma 1 ([DFLM95], Lemma 3.3.1, p.47) For $s\in$ I and $w^{*}\in C^{s,c}$ , there is aunique -finite measuredefined on the set of all nonvanishing finite measuresonsuch that$Q_{s,w^{*}}$$\sigma$$\chi$$[s, T]\cross\wedge C_{s’,w^{\star}}^{c}$$\tilde{P}_{s,w}*\exp\langle\tilde{\lambda}_{s,\tau}^{\mathrm{C}}, -\psi\rangle=\exp\{-\int(1-\exp\langle x, -^{\psi}\rangle)Qs,w^{*}(d\chi)\}$for$\psi\in \mathcal{B}([_{S,T}]\hat{\cross}C_{s,w}^{\cdot}’ c*, R)$

,

(13)

.

Moreover, we may disintegrate the-Khintchine measure
relative to its intensity(cf. Lemma 3.3.6 in [DFLM95]) to obtain its Palm distribution
measure.
is the law of a canonical cluster . Then it is easy to show
Roughly speaking,that a Palm representation formula holds in terms of the Brownian local time measure$(w, dt)$ atof the given bridge path . That is to say,$Q_{s,w^{*}}$$\mathrm{L}\mathrm{e}\mathrm{v}\mathrm{y}\text{ノ}$$\overline{Q}_{s,w^{*}}$$Q_{S_{)}w}^{r,w}*(d\chi)$$Q_{s,w^{*}}^{r,w}(d\chi)$$\chi$$L^{c}$$w$$c$

Theorem 7 ([DFLM95], Theorem 3.3.9, p.48) Let $s\in I$ and $w\in Cs,$ . For
almost all, the Palm distribution $Q_{s}r,,w*whas$ Laplace functional$c$$\overline{Q}_{s,w}*-$$[r, w]\in[s, T]\cross\wedge C_{s,w^{*}}’ C$$\int\exp\langle\chi, -\psi\rangle Q_{s,w^{*}}r,w(d\chi)=\exp\{-2\int_{s}^{r}u_{\psi_{C}},(t, w.\wedge t, T)L^{c}(w, dt)\}$

,

(14)

, whereis the unique bounded non-negative solution
forof a historical version of the nonlinear singular equation$\psi\in \mathcal{B}_{+}([0, T]\hat{\cross}C^{\cdot},c, R)$$u_{\psi,c}(\cdot, \cdot,\tau)$$- \frac{\partial}{\partial r}u=\frac{1}{2}\Delta u+f\delta_{c}-\delta u^{2}c$’$r>0$ .

On this account, Theorem 6 can be readily obtained by employing the aforementioned
results. More precisely, it is interesting to note that the key step of the proof of Theorem
6 is to demonstrate that the random measure $\chi(d[r’, w’])$ distributed according tohas with probability 1 at the Palm point an infinite left upper density with respect to
the Lebesgue measure $dr’$ (cf. Theorem 4.2.2, p.52, [DFLM95]). The mathematical tools
exploited in [DFLM95] are of strong independent interest for the author in connection withhistorical stochastic calculus (e.g. [P95]).
So much for Theorem 6, we lastly introduce another different behavior of the superBrownian local time measure
at the catalyst . Recall the definition of the HausdorffBesicovitch dimension $d^{*}=\dim(A)\in[0,1]$ of a subsetof R. It is defined by therequirement that$Q_{s,w}^{r,w}*$$r$$\lambda^{a}$

$c$

$A$

$\lim_{\deltaarrow 0}\inf+\{\sum_{k}($diam$(B_{k}))^{\rho}; \bigcup_{k}B_{k}\supset A$, diam $(B_{k})<\delta\}$

11

is a countable
for $\rho\in(0, d^{*})$ wheras it vanishes for $\rho\in(d^{*}, 1]$ . Hereequalswith diameter smaller than . Then the following
covering of by closed intervals
result holds:$\{B_{k}\}$$+\infty$$A$$\delta$$B_{k}$. The occupation density
Theorem 8 ([DF94], Theorem 1.2.5, p.9) Assume
at the catalyst’s position has a. . carrying Hausdorff-Besicovitch dimension
measureone.
$X_{0}=\delta_{c}$$s$$\lambda^{c}$

Consequently, the super-Brownian local time is singular continuous at (cf. Theorem
in Section
6). Recall that this is in a sharp contrast to the constant medium case
2. But nevertheless production of population mass occurs on a time set of full dimension(cf. Theorem 8).
Remark. It is quite interesting to compare the result obtained in Theroem 8 with the usual
Brownian local time, which determines a singular random measure with carrying dimension
1/2. See, for instance, It\^o-McKean (1974) [IMc74; \S 2.5, pp.5054.].
$c$$\mathrm{E}\mathrm{q}.(5)$7

Total Mass Extinction

In [FLG95] Fleischman and Le Gall (1995) has proposed a new approach to SBM $X$ withas branching rate, and has proved that the occupation density
a single point catalyst
of $X$ at the. catalyst is distributed as the total occupation time measure ofmeasureby an explicit representation formula, whereis determined from, and also th. at
is a superprocess with constant branching rate and spatial motion by the 1/2-stableis provided
subordinator. Moreover, a new derivation of the singularity of the measure
in [FLG95] as well.
Recall that the stable subordinator with index 1/2 is the L\’evy process on the real line
whose transition probabilities are given by$\delta_{c}$$\lambda^{c}$

$c$

$U$

$\lambda^{c}$$X_{t}$$U$$\lambda^{c}$$s>0$ ,,$q(s, b):= \mathrm{I}_{\{0}b>\}\frac{s}{\sqrt{2\pi b^{3}}}\exp\{-\frac{s^{2}}{2b}\}$$b\in \mathrm{R}$.

can also be interpreted as the density function of the first hitting time
Notice thatof the point by a linear Brownian motion started at the origin. The next result is adensity field via the SBLT measure .
representation of the$q(s, \cdot)$$s$$\mathrm{m}\mathrm{a}s\mathrm{s}$$\lambda^{c}$$x$

Theorem 9 ([FLG95], Theorem 1 (b), p.67) Withsity field can be represented as$P_{\delta_{c}}$-probability one the mass den-$x$$x_{t}(a)= \int_{0}^{t}q(t-S, |a-C|)\lambda \mathrm{C}(ds)$,$t>0$ ,

$a\neq c$

.

(15)

12

Now assume for the moment that $X$ starts off at time with the Lebesgue measure, namely, $X_{0}(dZ)=m(dz)$ . Then we already know that
denoted bysuffers local
extinction. That is, as,$0$$\mathrm{d}\mathrm{m}$$X_{t}$$tarrow\infty$

stochastically,$\langle X_{t}, \varphi\ranglearrow 0$

for each$\varphi\in \mathcal{G}_{+}$.

In fact, according to [DF94], we have

Proposition 3 ([DF94], proposition 1.3.1, p.ll) For all, we have $\int u(t, z)dZ$
as $tarrow\infty$ , where $u(\geq 0)$ is the solution to the integral equation$\varphi\in \mathcal{G}_{+}$$arrow 0$,$u(t, z)=s_{t} \varphi(z)-\int_{0}^{t}p(t-r, c-z)u^{2}(r, c)dr$$t\geq 0$,$z\in R$.

Actually, the representation formulain the above theorem is a very powerful tooland has interesting applications. For example, the next result is a complement to theabove-mentioned local extinction proposition by a total extinction property, which is due
to [FLG95].
$\mathrm{E}\mathrm{q}.(15)$Proposition 4 (Total Mass Extinction)pressed by$(a)$

The total mass of $X$ at time

$X_{t}(R)= \int^{t}0\sqrt{\frac{2}{\pi(t-s)}}\lambda^{c}(ds)$$(b)$$P_{\delta_{c}}$This total mass is strictly-probability as $tarrow\infty$ .

$po\mathit{8}itive$

8$\mathrm{a}.\mathrm{s}.$is ex-.

for every $t\geq 0a.s$ . andThat is to say, although $X_{t}(\mathrm{R})>0$ holds for $t\geq 0,$
mass survives as $tarrow\infty$ becomes very small.

$t$$X_{t}(R)$converges to$\mathit{0}$in, the probability that some total

Support Property

We consider the closed supports of the states of super-Brownian motions $X$ in catalytic
media. It is known (e.g. [Is88]) that the support property is valid in the constant medium
case, i.e.,
Theorem 10 (Iscoe, 1988) Let $X$ be a super-Brownian motion without
in theconstant medium ($i.e.$ , the caseconst.). If the initial mesure $X_{0}=\in M_{F}$ has compact
support, then so too does $X_{t}(t>0)$ , whatever the dimension is.
$cataly\mathit{8}t$$\rho=$On the contrary, for catalytic SBM we have
Theorem 11 (Dawson-Mueller, 1993) Let $X$ be a single point catalytic $SBM$. If, then the support of$t>0$ is the whole space .

$0$$X_{t},$$X_{0}\neq$$R$

The above theorem indicates that the compact support property is obviously violated. The
question arises whether one can formulate criteria for the compact support property to holdfor super-Brownian motions in catalytic media.

13

9

Epilogue

Dawson and Fleischmann (1997) have recently studied a catalytic SBM in a superBrownian medium. As a matter of fact, in [DF97] a continuous super-Brownian motion
is constructed in which branching occurs only in the presence of catalysts which evolve
themselves as a continuous super-Brownian motion with constant branching rate. Moreprecisely, there the Brownian collision local time plays an important role, that is, the colof an underlying Brownian motion path $W$ with the catalytic mass
lision local timeprocess governs the branching of new system. Furthermore, in the one-dimensional case,new types of limit behaviors are discovered. In fact, almost sure convergence of the total
mass process is proved with preservation of the mean and also with a non-degenerate limit,
and for the catalytic SBM starting with a Lebesgue measure $m$ , stochastic convergence ofto $m$ is proved as well when time tends to infinity. For the details, see
which
is an expository article of [DF97].
$X^{\rho}$$p$$L_{[W,\rho]}$$\rho$$X^{\rho}$$t$$[\mathrm{D}\mathrm{o}\mathrm{K}\mathrm{j}99]$

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$6\lambda 82$