数理解析研究所講究録
1193 巻 2001 年 1-19

1

TWO TOPICS ON FLEMING-VIOT PROCESSES
東京理科大

平場

諭示 (SEIJI IHRABA)SCIENCE UNIVERSITY OF TOKYO

1. INTRODUCTION
For a Fleming-Viot process(which is a probability measure-valued process) on acompact metric space , it is well-known that if its mutation operator is bounded, thenis pure atomic for every $t>0$ (Ethier and Kurtz in [2], [4]). We shall extend this result
to some jump-type measure-valued processes which are called “jump-type Fleming-Viotprocesses” introduced by the author in [6].
It is also well-known that the normalized binary branching process is a time inhomogeneous Fleming-Viot process. We shall introduce another new class of probability
measure-valued diffusion, which are called “spaoe-time inhomogeneous Fleming-Viot processes” and show that the normalized space inhomogeneous binary branching process isa space-time inhomogeneous Fleming-Viot process.
$\mathrm{Y}_{t}$

$S$

$A$

$\mathrm{Y}_{t}$Let be a compact metric space, fix $r\geq 0$ and set $D_{r}=\mathrm{D}([r, \infty)arrow S)$ be a pathspace of right continuous functions with left-hand limit. Let $(w(t), P_{x})_{t}\geq r,x\in S$ be a Svalued Markov process starting from at $t=r$ with sample paths in. We denote the$D(A)$
transition semi-group by
and the generator by $(A, D(A))$ , whereis a domainof . We suppose thatis a Feller semi-group on $(C(S), ||\cdot||)$ , where $C(S)$ is a familyof continuous functions on anddenotes the supremum norm.
Letbe a family of finite Radon measures on with the weak topology,
that is,in.
for every $f\in C(S)$ , whereThen,
is a Polish space, i.e., complete separable metrizable space. The family of
probability measures on $S,$
, is a compact metric space (cf. Chap. 3
of [3] . For.
, we always denote the normalized measure as$S$$D_{r}$$x$$(P_{t})$$A$$(P_{t})$$S$$||\cdot||=||\cdot||_{\infty}$$S$$\mathcal{M}_{F}=\mathcal{M}_{F}(S)$$\langle\mu, f\rangle=\int fd\mu$$\mathcal{M}_{F}\Leftrightarrow\langle\mu_{n}, f\ranglearrow\langle\mu, f\rangle$$\mu_{n}arrow\mu$$\mathcal{M}_{F}$$\mathcal{M}_{1}=\mathcal{M}_{1}(S)\subset \mathcal{M}_{F}$$\mu\in \mathcal{M}_{F}\backslash \{0\}$

$)$

Let

$\overline{\mu}=\mu/\langle\mu, 1\rangle$

be a Fleming-Viot process on , with a mutation operator
is an
and
-valued process on such that$S$$(\mathrm{Y}_{t}, \mathrm{P}_{\mu}^{FV})$$(\mathrm{Y}_{t}, \mathrm{p}_{\mu}FV)$$S$$\mathcal{M}_{1}$$A$, that is,$\mathrm{P}_{\mu}^{FV}(\mathrm{Y}_{0}=\mu)=1$$\langle \mathrm{Y}_{t}, f\rangle=\langle \mathrm{Y}_{0}, f\rangle+\int_{0}^{t}\langle \mathrm{Y}S’ Af\rangle+Mt(f)$,where $\{M_{t}(f)\}$ is a continuous martingale with quadratic variation$\langle\langle M(f)\rangle\rangle t=C\int_{0}^{t}(\langle \mathrm{Y}s’ f2\rangle-\langle \mathrm{Y}_{s}, f\rangle 2)dS$$(C>0)$ ,is a generator (with a domain $D(A)\subset(C(S),$) of a conservative Feller process$(w(t), P_{x})_{t}\geq 0,x\in s$ ; a $S$-valued Markov process starting from
with$S)$ and the transition semi-group.
$f\in D(A)$ ,The generator of this process is given as, for$A$$||\cdot||)$$x$$w(\cdot)\in \mathrm{D}=\mathrm{D}([0, \infty)arrow$$(P_{t})$$\mathcal{L}$$\mathcal{L}e^{-\langle\cdot,f\rangle}(\eta)=-\langle\eta,$$\eta\in \mathcal{M}_{1},$A$f\rangle$$e^{-(J\rangle}+ \eta,\frac{c}{2}[\langle\eta, f^{2}\rangle-\langle\eta, f\rangle 2]e^{-\langle\eta,f})$.

2
is pure atomic for
It is-known that if its mutation operator is bounded, thenevery $t>0$ (Ethier and Kurtz in [2], [4], see also Th. 8.2.1 in [1]). In particular, if $A=0$
and if we denotecan be expressed as, for a function’ then the generator
of $\mathrm{m}=(m_{1}, m_{2}, \ldots)$ ,$A$$\mathrm{w}\mathrm{e}\mathrm{U}$$\mathrm{Y}_{t}$$\mathcal{L}$$\eta=\sum_{i}m_{i}\delta_{x}:$$\phi(\mathrm{m})$$G \phi(\mathrm{m})=\frac{c}{2}\sum m_{i}(\delta_{ij}-m_{\mathrm{j}})\partial i,\mathrm{j}ij2\phi(\mathrm{m})$.

The corresponding weight process $\{m_{i}(t)\}$ is given as$(i\in S)$$dm_{i}(t)= \sum_{j}(\delta_{i}\mathrm{j}-m_{i}(t))\sqrt{cm_{j}(t)}dB_{\mathrm{j}}(t)$,where $\{B_{j}(t)\}$ is a family of independent one-dimensional Brownian motions.
There is another well-known measure-valued process which is a branching process, that is,-valued process such that $\mathrm{P}_{\mu}(Z_{0}=\mu)=1$ and$(Z_{t}, \mathrm{P}_{\mu})$$\mathcal{M}_{F}$$\mathrm{P}_{\mu}[e^{-\langle z_{l},f)}]=e^{-\{Vf\rangle}\mu,\mathrm{c}$where$V_{t}f$,

is a unique solution to the following equation$V_{t}f(x)=P_{t}f(x)- \int_{0}^{t}dsP_{S}\Psi(Vt-Sf)(x)$,or$\partial_{t}V_{t}f(X)=AV_{t}f(x)-\Psi(Vtf)(x)$,$V_{0}f(X)=f(X)$with branching mechanism $\Psi(v)(x)$ ;,$\Psi(v)(x)=\frac{1}{2}c(X)v^{2}+\int_{0}^{\infty}[e^{-vu}-1+vu]\nu(x, du)(\geq 0)$

where $c(x)\geq 0$ is a bounded function and $\nu(x, du)$ is a kernel onthat( A ) $\nu(x, du)<\infty$ .
$\sup_{x\in S}\int_{0}^{\infty}$$u$$S\cross(\mathrm{O}, \infty)\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{S}\mathrm{p}_{\mathrm{i}}\mathrm{n}\mathrm{g}$$u^{2}$

In particular, if $\Psi(v)(x)=cv^{2}/2(c>0)$ , thenis called a binary branchingprocess or a binary branching measure-valued process, or a binary branching superprocess,or a Dawson-Watanabe process, etc.
Let $\tau_{0}=\inf\{t>0;\in Z_{t}.’ 1=0\}$ . For $t<\tau_{0}$ an.d $f\in D(A),$has the following
semi-martingale representation:$(Z_{t}, \mathrm{P}_{\mu})$$\langle Z_{t}, f\rangle$$\langle Z_{t}, f\rangle=\langle Z_{0}, f\rangle+\int_{0}^{t}\langle Z_{s}, Af\rangle ds+M_{t}^{c}(f)+M_{t}^{d}(f)$

where
that

$\{M_{t}^{c}(f)\}$is a continuous$L^{2}$-martingale with quadratic variation$\langle\langle M^{\mathrm{c}}(f)\rangle\rangle_{t}=\int_{0}^{t}\langle Zs’ cf2\rangle ds$,$\mathrm{a}\mathrm{n}\mathrm{d}_{\mathrm{t}}$$M_{t}^{d}(f)= \int_{0}^{t}\int_{\mathcal{M}_{F}}\langle\eta, f\rangle\overline{N}(ds, d\eta)$

where$\overline{N}(dS, d\eta)$,,

is a martingale measure with compensator$\overline{N}(dS, d\eta)=ds\int_{S}Z_{S}(dx)\int_{0}^{\infty}\nu(du)\delta_{u\delta_{x}}(d\eta)$.

$\langle\langle M^{c}(f)\rangle\rangle t$

such

3
The generator of this process$\mathcal{L}^{Z}$

is given as.

$\mathcal{L}^{z_{e^{-\langle\cdot,f}}}\rangle(\eta)=[-\langle\eta, Af\rangle+\langle\eta, \Psi(f)\rangle]e-\langle\eta,f\rangle$

It is also well-known that the normalized binary branching process is a time inhom
geneous Fleming-Viot process. More exactly, in 1991, Perkins [8] established that theconditional law of the binary branching process given the total mass process is a time
inhomogeneous Fleming-Viot process.
Fix $r\geq 0$ and let$(\succ$$C_{r,+}$$\equiv$$\{g:[r, \infty)arrow[0, \infty);g$

there is$\tau_{g}\in$

is continuous, andsuch that $g>0$ on$(r, \infty]$$[r, \tau_{g}),$$g=0$ on$[\tau_{g}, \infty)\}$.

$g\in C_{r,+}$ and $c>0$ .
Let
The time inhomogeneous Fleming-Viot processsatisfies the following:(i),(ii) For $f\in D(A),$has the following semi-martingale representation:$\mu\in \mathcal{M}_{1},$$(\mathrm{Y}_{t}, \mathrm{P}_{\mu}^{FV})$$\mathrm{Y}_{r}=\mu,$$\mathrm{Y}_{t}=\mathrm{Y}_{\tau_{g}-}(t\geq\tau_{g}),$$\mathrm{P}^{F}V\mathrm{a}r,\mu-.\mathrm{S}.$$\langle \mathrm{Y}_{t}, f\rangle$$\langle \mathrm{Y}_{t}, f\rangle=\langle \mathrm{Y}_{r}, f\rangle+\int_{r}^{t}\langle \mathrm{Y}_{S}, Af\rangle ds+M_{r,t}(f)$such that $\{M_{r},t(f)\}$ is a continuous$L^{2}$-martingale with quadratic variation$\langle\langle M(f)\rangle\rangle_{r},t=c\int_{r}^{t}g(_{S})-1[\langle \mathrm{Y}s’ f2\rangle-\langle \mathrm{Y}_{s}, f\rangle^{2}]I(s<\tau g)ds$

We denote normalized measure$\overline{\mu}=\mu/\langle\mu, 1\rangle$

Theorem 1 (Perkins ’91). Let
branching process, set$\mathrm{P}_{\mu}$

wherestarting from($( \mathrm{Y}_{t}, \mathrm{P}_{0}^{F},\frac{V}{\mu})$$\overline{Z.}\in B|$$\langle$Z.,$1\rangle$$\mu\in \mathcal{M}_{F}\backslash \{0\}$$\mu\in \mathcal{M}_{F}\backslash \{0\}$$x_{t}=\langle Z_{t}, 1\rangle$$(Z_{t}, \mathrm{P}_{\mu})$

for

and

and set,.

$y=\langle\mu, 1\rangle$$Q_{y}=\mathrm{p}_{\mu^{\mathrm{O}X}}.-1$$=g(\cdot))=\mathrm{p}_{0^{\frac{V}{\mu}}}^{F},(\mathrm{Y}\in B)$.

. For a binary. Then$Q_{y^{-}}a.a$. $g\in C_{0,+}$ ,is a time inhomogeneous Fleming-Viot process associated $wdh(A,g, c)$.

$\mathrm{Y}_{0}=\overline{\mu}$

We would like to extend these results to some wide class which include Fleming-Viotprocesses. The first one to “jump-type Fleming-Viot processes” introduced by the authorin [6], and second one to “space-dependent Fleming-Viot processes” intorduced by thispaper.
2. PURE ATOMIC JUMP-TYPE$\mathrm{F}\mathrm{L}\mathrm{E}\mathrm{M}\mathrm{I}\mathrm{N}\mathrm{G}-\mathrm{V}\mathrm{I}\mathrm{o}\mathrm{T}$PROCESSES

According to [6], we give characterizations of jump-type Fleming-Viot processes.
from the space of Dirac measures
For each $x\in S$ , we define an operator
toby,$T_{x}$$S$$\mathcal{M}_{1}$$\langle T_{x}\delta, f\rangle=\langle\delta_{x},Tf\rangle=Tf(x)=\int_{S}f(y)\tau(X, dy)$

where $T(x, dy)$ is a non-negative kernel on $S$ such that $T(x, s)=1$ .
We fix $\gamma>0$ and let $\nu(dv)$ be a measure on $(0, \infty)$ such that$\int_{0}^{\infty}$

( A
$v$

$v^{2}$) $\nu(dv)<\infty$ .

$\delta_{x}(dy)$on4be a jump-type Fleming-Viot process associated withLet
-valued process such thatfrom
. That is,is an
has the following semi-martingale representation:$(\mathrm{Y}_{t}, \mathrm{P}_{\mu})$$\mu\in \mathcal{M}_{1}$$(\mathrm{Y}_{l}, \mathrm{P}_{\mu})$$\{M_{t}^{c}(f)\}$starting$\langle \mathrm{Y}_{t}, f\rangle(f\in D(A))$$\mathcal{M}_{1}$$\langle \mathrm{Y}_{t}, f\rangle=\langle \mathrm{Y}_{0}, f\rangle+\int_{0}^{t}\langle \mathrm{Y}_{s},Af\rangle+M_{t}^{c}(f)+M_{t}^{d}(f)$where$(A,\gamma, \nu, \tau_{x})$,is a continuous martingale with quadratic variation$\langle\langle M^{C}(f)\rangle\rangle t=\gamma\int_{0}t(\langle \mathrm{Y}S’ f2\rangle-\langle \mathrm{Y}_{s}, f\rangle^{2})ds$$(\gamma>0)$,

and $\{M_{t}d(f)\}$ is a pure discontinuous martingale such that$M_{t}^{d}(f)= \int^{t}0\int_{\mathcal{M}p}\frac{\langle\eta,1\rangle}{1+\langle\eta,1\rangle}\langle\overline{\eta}-\mathrm{Y}S-, f\rangle\overline{N}(d_{S}, d\eta)$

where$\overline{N}(dS, d\eta)$,

is the martingale measure with compensator$\overline{N}(dS,d\eta)=ds\int_{S}\mathrm{Y}s(d_{X})\int_{0}^{\infty}\nu(dv)\delta v\tau_{x}\delta(d\eta)$

Remark 1. In [6] the process is defined only the case of $T_{x}=I$ , i.e.,
the extension is possible and easy.
The generator
given as$\mathcal{L}e^{-(\cdot,f\rangle}.(\mu)$$\mathcal{L}$$=$of this process for Laplace functionals$T_{x}\delta=\delta_{x}$. However$e^{-\langle\mu,f\rangle}(\mu\in \mathcal{M}_{1}, f\in D(A))$$- \langle\mu, Af\rangle e^{-}+\frac{\gamma}{2}\langle\mu,f\rangle[\langle\mu, f^{2}\rangle-\langle\mu, f\rangle^{2}]e-\langle\mu,f\rangle$$+ \int\mu(d_{X})\int_{0}^{\infty}\nu(dv)$$\{\exp[-\frac{v}{1+v}\langle\tau_{x}\delta-\mu, f\rangle]-1+\frac{v}{1+v}\langle\tau_{x}\delta-\mu, f\rangle\}e-(\mu,f)$

For a functional $F(\mu)$ , a derivative at $x\in S$ is defined by$\frac{\delta F(\mu)}{\delta\mu(x)}=\lim_{\epsilonarrow 0}\frac{1}{\epsilon}[F(\mu+\epsilon\delta_{x})-F(\mu)]$and higher order derivatives
Note that the generator can be expressed as$\delta^{2}F(\mu)/(\delta\mu(x)\delta\mu(y)),$(2.1)$\mathcal{L}F(\mu)$

$=$

.

$.$

,

(if exists),

are defined similarly.

$\langle\mu, A\frac{\delta F(\mu)}{\delta\mu(\cdot)}\rangle+\frac{\gamma}{2}\iint\frac{\delta^{2}F(\mu)}{\delta\mu(x)\delta\mu(y)}Q(\mu;dX, dy)$$+ \int_{\mathcal{M}_{F}}[F(\mu+g(\mu,\eta))-F(\mu)-\langle g(\mu, \eta), \frac{\delta F(\mu)}{\delta\mu(\cdot)}\rangle]n(\mu;d\eta)$

where$Q(\mu;dx, dy)=\mu(d_{X})\delta_{x}(dy)-\mu(dX)\mu(dy)$$\in \mathcal{M}_{F}$$g( \mu, \eta)=\frac{\mu+\eta}{1+\langle\eta,1\rangle}-\mu=\frac{\langle\eta,1\rangle}{1+\langle\eta,1\rangle}(\overline{\eta}-\mu)$and$n( \mu;d\eta)=\int\mu(d_{X})\int_{0}^{\infty}\nu(dv)\delta_{v}\tau_{x}\delta(d\eta)$

,

.

,

.

is

5
Here we give a formal calculation for general measrue-valued processes, i.e., Ito’s formulafor measure-valued processes.
For a set , let
be the class of finite singed measures on , i.e.,$\eta=\eta^{+}-\eta^{-};\eta^{\pm}\in \mathcal{M}_{F}(B)$ .
We denote $||\eta||=(\eta^{+}+\eta^{-})(B)$ . For simplicity,
if $B=S$ , then. Let $Q(\mu;dx, dy)$ :be measurable
such that $Q(\mu;dx,$ $d_{X)}\leq C\mu(d_{X)}$ for some $C>0$ . Let:and$n(\mu;d\eta)$ :be measurable such that
$B$

$B$

$\mathcal{M}_{F}^{\pm}(B)$$\eta\in \mathcal{M}_{F}^{\pm}(B)$$\Leftrightarrow$$\mathcal{M}_{F}^{\pm}=\mathcal{M}_{F}^{\pm}(S)$$\mathcal{M}_{F}arrow \mathcal{M}_{F}^{\pm}(S\cross S)$$g(\mu, \eta)$$\mathcal{M}_{F}\cross \mathcal{M}_{F}^{\pm}arrow \mathcal{M}_{F}^{\pm}$$\mathcal{M}_{F}arrow \mathcal{M}_{F}(\mathcal{M}_{F}^{\pm})$$\sup_{\mu\in \mathcal{M}_{F;\mu}(S)\leq K}\int \mathcal{M}^{\pm}F||g(\mu,\eta)||$In general, let$X_{t}$

be an$\mathcal{M}_{F}$-valued Markov process such that

$X_{t}=X_{0}+ \int_{0}^{t}A^{*}x_{s}dS+M_{t}^{c}+M_{t}^{d}$where
is defined by
is a pure discontinuous martingale;$A^{*}X_{s}$for every $K>1$ .

A $||g(\mu, \eta)||^{2}n(\mu;d\eta)<\infty$$\langle A^{*}X_{s}, f\rangle=\langle$ $X_{s},$A$f\rangle$,$M_{t}^{c}$,is a continuous martingale and$M_{t}^{d}$$M_{t}^{c}(dx)= \int_{0}^{t}M(ds, dx)$

$M_{t}^{d}(d_{X})= \int_{0}^{t}\int_{\mathcal{M}_{F}^{\pm}}g(x_{s-},\eta)(dX)\overline{N}(ds, d\eta)$with a continuous martingale measure $M(dS, dx)$ and a pure discontinuous martingale
measure. Suppose that the covariance of $M^{c}(dx)M^{C}(dy)$ is given as$\overline{N}(dS, d\eta)$$\langle\langle M^{C}(dx), Mc(dy)\rangle\rangle_{t}=\gamma\int_{0}^{t}Q(X_{s};dX, dy)d_{S}$and the compensator ofis $\overline{N}(dS, d\eta)=dsn(X_{s}, d\eta)$ (cf. for martingale measures, see
Walsh [10] .
Let $F(\mu)$ be a suitable functional ofand letbe given as in (2.1). Then,by a formal calcutation we have the following Ito’s formula:$\overline{N}$$)$$\mu\in \mathcal{M}_{F}$$F(X_{t})$$=$$\mathcal{L}F(\mu)$$F(X_{0})+ \int_{0}^{t}\mathcal{L}F(xs)ds+M_{t}^{c}(F)$$+ \int_{0}^{t}\int_{\lambda 4}\pm F[F(X_{s-}+g(X_{s-}, \eta))-F(X_{S-})]\overline{N}(ds, d\eta)$

,

where
$M_{t}^{c}(F)= \int_{0}^{t}\int\frac{\delta F(x_{s})}{\delta X_{s}(x)}M(dS, dx)$

is a martingale with quadratic variation$\langle\langle M^{c}(F)\rangle\rangle t=\gamma\int^{t}\mathrm{o}d_{S}\int\int\frac{\delta F(x_{s})}{\delta X_{s}(x)}\frac{\delta F(x_{s})}{\delta X_{s}(y)}Q(x_{s};dx, dy)$(for the stochastic integrals corresponding to the martingale measures, see Walsh [10] andalso Dawson [1] .
$)$We denote the space of pure atomic probability measures on by
We also denote the pure atomic part of a measure by , and for a process.
$S$$\eta$$(\mathrm{Y}_{t})_{a}$$\eta_{a}$$\mathcal{M}_{1,a}=\mathcal{M}_{1,a}(S)$$\mathrm{Y}_{t}$by$\mathrm{Y}_{t,a}$

.

or

6
We set$\mathrm{M}=\{\mathrm{m}=(m_{1}, m_{2}, \ldots);m_{1}\geq m_{2}\geq\cdots\geq 0, \sum m_{i}=1\}$$\overline{\mathrm{M}}=\{\mathrm{m}=(m_{1},m_{2}, \ldots);m_{1}\geq m_{2}\geq\cdots\geq 0, \sum m\leq 1\}$and let $M$ :masses of the atoms of$\mathcal{M}_{1}arrow\overline{\mathrm{M}};M(\eta)=M(\eta_{a})=(\mathrm{t}\mathrm{h}\mathrm{e}$$\eta_{a}$vector of descending order statistics of the).

Theorem 2. Let
starting fromfollowing $f_{or}m$ :be a jump-type Fleming-Viot process associated withis a bounded operator haning the. Suppose that$(\mathrm{Y}_{t}, \mathrm{P}_{\mu})$$(A,\gamma, \nu,\tau_{x})$$A$$\mu\in \mathcal{M}_{1}$$Af(x)=a(x) \int_{S}[f(y)-f(x)]B(x, dy)$ ,where $a(x)$ is a nonnegative bounded function on and $B(x, dy)$ is a nonnegative kemelsuch that $B(x, S)=1$ for all $x\in S$ . Also assume that $\gamma>0$ andfor every$x\in S$ . Thenis pure atomic for all $t>0a.s_{f}.i.e.$ ,$S$$T_{x}\delta\in \mathcal{M}_{1,a}$$\mathrm{Y}_{t}$$\mathrm{P}_{\mu}$($\mathrm{Y}_{t}\in \mathcal{M}_{1,a}$for all $t>0$ ) $=1$ .

Moreover if $a(\cdot)\equiv a(\geq 0),$for every $x\in S$ and $B(x, \cdot)$ has no atoms for every-martingale problem for theis a solution of the, theninfinite dimensional operator $(G, D(c))$ , where$T_{x}\delta=\delta_{x}$$x\in S$$\mathrm{D}([0, \infty),$$\{\mathrm{m}_{t}=M(\mathrm{Y}_{t})\}$$\mathrm{M})$$G \phi(\mathrm{m})=\frac{\gamma}{2}\sum i,jm_{i}(\delta ij-m_{j})\partial_{i}^{2}\phi j(\mathrm{m})-a\sum_{i}m_{i}\partial_{i}\phi(\mathrm{m})$$+$$\sum_{i}m_{i}\int_{0}^{\infty}[\phi((\frac{m_{j}+v\delta_{ij}}{1+v})_{j})-\phi(\mathrm{m})-\sum\frac{v}{1+v}(\sim j\delta_{ij}-m_{j})\partial_{j}\emptyset(\mathrm{m})]\nu(dv)$and $D(G)$ is the algebra generated byurith. That is,is the size ordered atom process for the infinitely
withmany neutral alleles (jump-type) model.
$\partial_{i}=\partial/\partial m_{i},$$\{1, \phi^{2}, \emptyset^{\mathrm{s}}, \ldots\}$$\partial_{ij}2=\partial^{2}/(\partial m_{i}\partial m_{j})$$\mu(\mathrm{m})=\sum_{i}m_{i}^{\beta}(\beta>1)$$\mathrm{Y}_{t}$Proof. It is enough to consider the case that $a(\cdot)\equiv a>0$ and$B(x, \cdot)$

Because if we set $\overline{S}=S\cross[0,1],$ $\overline{a}=\sup a(x)>0$ (note that the case ofhas no atoms.
is trivial),$\overline{a}=0$$\overline{B}(x, u, dydv)=\frac{a(x)}{\overline{a}}B(x, dy)dv+\frac{\overline a-a(x)}{\overline{a}}\delta_{x}(dy)dv$

and$\overline{A}f(x,u)=\overline{a}\int_{0}^{1}\int_{S}[f(y, v)-f(X, u)]\overline{B}(X,u, dydv)$,

of the martingale problem for , is the, with thethensolution of the martingale problem for .
, if, then set $F_{\beta}( \mu)=\phi_{\beta}(M(\mu))=\sum m_{i}^{\beta}(\beta>1)$ , andFor$\mathrm{Y}_{t}(\cdot):=\overline{\mathrm{Y}}_{t}(\cdot\cross[0,1])$$\overline{A}$$\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}\overline{\mathrm{Y}}_{t}}$$A$$\mu\in \mathcal{M}_{1}$$\mu_{a}=\sum_{i}m_{i}\delta_{x}\dot{.}$$F_{1+}( \mu)=\lim\beta\downarrow 1F_{\beta}(\mu)=\sum_{j}m_{j}(=\langle\mu_{a}, 1\rangle)$.

If $\beta>2$ , then$\frac{\delta F_{\beta}(\mu)}{\delta\mu(x)}=\sum_{i}\beta m_{i}^{\beta-}1_{x}(1):X$

and$\frac{\delta^{2}F_{\beta}(\mu)}{\delta\mu(x)\delta\mu(y)}=\sum_{i}\beta(\beta-1)m^{\beta}1.(x)i^{-2}x.1(x:y)$,7
We first give some formal calculations. For each fixedby. Since $B(x, \cdot)$ has no atoms, by (2.1) we have$x$

we denote all atoms of $\mu+T_{x}$$\{x_{\mathrm{j}}\}$$LF_{\beta}(\mu)$$=$$\sum_{i}[-a\beta m^{\beta}i+\frac{\gamma}{2}\beta(\beta-1)(m-m_{i}^{2})mi]\beta-2$$+ \int_{S}\mu(dX)\int_{0}^{\infty}[F_{\beta}(\frac{\mu+vT_{x}\delta}{1+v})-F_{\beta}(\mu)$.

$- \frac{v}{1+v}\sum_{j}(\tau_{x}\delta-\mu)(\{Xj\})\beta m_{j}\beta-1]\nu(dv)$$=$$-a \beta F_{\beta}(\mu)+\frac{\gamma}{2}\beta(\beta-1)(F_{\beta-}1(\mu)-F_{\beta(\mu}))$$+ \int_{S}\mu(dX)\int_{0}^{\infty}[F_{\beta}(\frac{\mu+v\tau_{x}\delta}{1+v})-F_{\beta}(\mu)$$- \frac{\beta v}{1+v}\sum_{j}(T_{x}\delta(\{x_{\mathrm{j}}\})-\mu(\{x_{j}\}))\mu(\{xj\})^{\beta-1}]\nu(dv)$Moreover if we set
formal Ito’s formula$F_{1}(\mu)\equiv 1$(2.2), then the above formula is still valid for $\beta=2$ . Hence by$M_{t}( \beta):=M_{t}(F_{\beta}(\mathrm{Y}t))=F_{\beta(}\mathrm{Y}t)-F\beta(\mathrm{Y}\mathrm{o})-\int_{0}^{t}\mathcal{L}F_{\beta}(\mathrm{Y}_{s})dS$

is an -martingale such thatwith quadratic variation$L^{2}$$M_{t}(\beta)=M_{t}^{C}’\beta+M_{t}d,\beta$, where$M_{t}^{c,\beta}$

is a continuous martingale$\langle\langle M^{c,\beta}\rangle\rangle_{t}=\beta^{2}\gamma\int_{0}^{t}[F_{2\beta-1}(\mathrm{Y}s)-F_{\beta}(\mathrm{Y}s)2]ds$

and$M_{t}^{d,\beta}= \int_{0}^{t}\int[F_{\beta}(\frac{\mathrm{Y}_{s-}+\eta}{1+\langle\eta,1\rangle})-F_{\beta}(\mathrm{Y}_{s-})]\overline{N}(dS, d\eta)$

is a pure discontinuous martingale with compensator$\overline{N}(dS, d\eta)=ds\int_{S}\mathrm{Y}(sd_{X})\int_{0}^{\infty}\nu(dv)\delta_{v}\tau_{x}\delta(d\eta)$.

To verify the above result we use an approximation method. Let
be a partition family for , i.e.,(a)
if $j\neq k$ ,(b),
for all(c) for each , there is such that
,
(d)
.
For
. For $\beta\geq 1$ , let $F_{\beta,n}(\mu)=$
, set
and
as $narrow\infty$ by the definition. Note that forof.
, and use Ito’s formula
For each, if we takefor, then by letting $karrow\infty$ and summing up on we can get the following: If$\{S_{j}^{n}, X_{j}^{n}, \rho n;n,j\in \mathrm{N}\}$$S$$S_{j}^{n}\subset S,$ $x_{j}^{n}\in S_{j}^{n},$$S_{j}^{n}\cap S_{k}^{n}=\emptyset$$\bigcup_{j}S_{j}^{n}=S$$n\in \mathrm{N}$

$k$

$j$

$S_{j}^{n+1}\subset S_{k}^{n}$$\rho_{n}:=\sup_{j}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(\mathrm{S}_{\mathrm{j}}\mathrm{n})arrow 0(narrow\infty)$$\mu\in \mathcal{M}_{1}$$\xi^{n}(\mu)=\sum_{j}\mu(S^{n}j)\delta x_{j}^{n}$$\sum_{j}\mu(s_{j}^{n})=\phi^{\beta}(\xi^{n}(\mu))$$\beta\geq 1,$$\xi_{t}^{n}=\xi^{n}(\mathrm{Y}_{t})$$F_{\beta,n}(\mu)arrow F_{\beta}(\mu)$$F_{\beta}(\mu)$$j,$ $n$$\langle \mathrm{Y}_{t}, f_{k}^{j,n}\rangle$$f_{k}^{j,n}\in D(A);0\leq f_{k}^{j,n}\uparrow 1_{S_{j}^{n}}(k\uparrow\infty)$

$j$

8

$\beta\geq 2$, then$M_{n,t}( \beta):=M_{t}(F_{\beta,n}(\mathrm{Y}_{t}))=F_{\beta,n}(\mathrm{Y}_{t})-F_{\beta,n}(\mathrm{Y}_{0)}-\int_{0}^{t}LF_{\beta},(n\mathrm{Y}_{S})d_{S}$is a bounded (uniformly in )$n$

$LF_{\beta,n}(\mu)$$L^{2}$-martingale, where$= \sum_{i}[\frac{\gamma}{2}\beta(\beta-1)(\mu(s_{i}^{n})-\mu(s_{i}^{n})^{2})\mu(s_{i}n)^{\beta 2}-$$+a\beta[\langle\mu, B(\cdot, S_{i}n)\rangle-\mu(s_{i}^{n})]\mu(S_{i}^{n})^{\beta-}1]$$+ \int_{S}\mu(dx)\int_{0}\infty\sum_{j}[(\frac{(\mu+vT_{x}\delta)(S^{n}j)}{1+v}\mathrm{I}^{\beta}-\mu(s_{j}^{n})^{\beta}$$- \frac{v}{1+v}(T_{x}\delta-\mu)(Sn)j\beta\mu(s_{j}^{n})^{\beta-1}]\nu(dv)$.

Note that if we set $p=1/(1+v),$ $q=v/(1+v)$ and(2.3)$h(a, b):=(pa+qb)^{\beta}-a^{\beta}-\beta q(b-a)a^{\beta-1}\geq 0$

for $0\leq a,$ $b\leq 1$ , then by$h(a, b)$$\beta\geq 2$$= \int_{0}^{1}dS\int^{s}0-\beta(\beta-1)(tq(ba)+a)^{\beta-2}q2(b-a)^{2}dt$$\leq\beta(\beta-1)q^{2}(a^{2}+b^{2})$.

(boundedSince
has no atoms, we can see that as
pointwisely), and hence( , in). Moreover the limit process
is an -martingale withas mentioned above (note that thisdecomposition can be shown by using the uniqueness of special martingales; cf. Theorem
6.1.3 in [1] .
By mean zero in (2.2) and by taking a limit ofcarefully, we haveas$B(x, \cdot)$$narrow\infty \mathcal{L}F_{\beta,n}(\mu)arrow \mathcal{L}F_{\beta}(\mu)$$M_{n,t}(\beta)arrow M_{t}(\beta)$$M_{t}(\beta)=M_{t}c,\beta+M_{t}d,\beta$$L^{2}$$\{M_{t}(\beta)\}$$L^{2}$$\mathrm{a}.\mathrm{s}.$$)$$\mathcal{L}F_{\beta}(\mu)$$\beta\downarrow 2$$0= \lim_{\beta\downarrow 2}\mathrm{E}_{\mu}[M_{t}(\beta)-Mt(2)]=\mathrm{E}_{\mu}[\gamma\int_{0}^{t}(1-F_{1+}(\mathrm{Y}s))ds](\geq 0)$.

By $\gamma>0$ this implies$\langle(\mathrm{Y}_{t})_{a}, 1\rangle=F_{1+}(\mathrm{Y}_{t})=1$

,

i.e.,

$\mathrm{Y}_{t}\in \mathcal{M}_{1,a}$for a.a.t $>0,$

In order to show that
, we mention thatfor all $t>0,$-martingales for $\beta\in(1,2)$ . In fact, tothis, let$\mathrm{Y}_{t}\in \mathcal{M}_{1,a}$$\mathrm{a}.\mathrm{s}.$$L^{2}$

with.

$M_{t}^{c,\beta},$$M_{t}^{d,\beta}$$F_{\beta,\epsilon}(\mu)=\phi_{\beta,\epsilon}(M(\mu))$$\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{i}\mathfrak{h}r$$\phi_{\beta,\epsilon}(\mathrm{m})=\sum_{i}\psi_{\epsilon}(m_{i})$$\mathrm{a}.\mathrm{s}$$\psi_{\epsilon}(m)=(m+\epsilon)^{\beta}-\epsilon^{\beta}-\beta\epsilon^{\beta}-1m$

are still, where.

Apply Ito’s formula toand take the limit, then the corresponding martingale
parts
converge to. Moreover for $M_{t}(\beta)=$respectively in, the formula (2.2) also holds. These can be checked as follows: First notethat for $1<\beta<2$ ,$\epsilon\downarrow 0$$F_{\beta,\epsilon}(\mathrm{Y}_{t})$$M_{t}^{c,\beta,\epsilon},$$M_{t}^{d,\beta,\epsilon}$$M_{t}^{c,\beta},$$M_{t}^{d,\beta}$$L^{2}$$M_{t}^{c,\beta}+M_{t}^{d,\beta}$$\psi_{\epsilon}(m)=\int_{0}^{1}ds\int_{0}^{s}\beta(\beta-1)(tm+\epsilon)^{\beta-22}mdt\uparrow m^{\beta}(\beta\downarrow 1)$,

9$\psi_{\epsilon}(m)$is convex in $0\leq m\leq 1$ and$\frac{\delta F_{\beta,\epsilon}(\mu)}{\delta\mu(x)}=\sum_{i}\beta\{(m_{i}+\epsilon)^{\beta 1}--\epsilon)\beta-11_{x}(:)X$,. $+\epsilon)^{\beta 2}-1_{x}(:)X1_{x:}(y)$$\frac{\delta^{2}F_{\beta,\epsilon}(\mu)}{\delta\mu(x)\delta\mu(y)}=\sum_{i}\beta(\beta-1)(m_{[]}$

.

Hence
$\mathcal{L}F_{\beta,\epsilon}(\mu)$$= \sum_{i}[-a\beta\{(m_{i}+\epsilon)^{\beta-}1-\epsilon^{\beta 1}-\}m_{i}+\frac{\gamma}{2}\beta(\beta-1)(mi-m_{i})2(mi+\epsilon)\beta-2]$$+ \int_{S}\mu(dX)\int_{0}^{\infty}[F_{\beta,\epsilon}(\frac{\mu+vT_{x}\delta}{1+v})-F_{\beta,\epsilon}(\mu)$$- \frac{\beta v}{1+v}\sum_{j}(\tau_{x}\delta-\mu)(\{Xj\})\{(mi+\epsilon)^{\beta 1}--\epsilon-1\}\beta]\nu(dv)$converges to$\mathcal{L}F_{\beta}(\mu)$as$\epsilon\downarrow 0$

by using$(m+ \epsilon)^{\beta-1}-\epsilon^{\beta}-1=(\beta-1)m\int_{0}^{1}(tm+\epsilon)^{\beta-2}dt\uparrow m^{\beta-1}$$(\epsilon\downarrow 1)$

and monotone convergence theorem. By the same way we also have$\int_{0}^{t}\mathcal{L}F_{\beta,\epsilon}(\mathrm{Y})sd_{S}arrow\int_{0}^{t}LF_{\beta}(\mathrm{Y}s)d_{S}$If we set$( \mathrm{Y}_{t})_{a}=\sum_{j}m_{\mathrm{j}}(t)\delta_{x}(t)$$=$$\langle\langle M^{c,\beta,\epsilon}\rangle\rangle_{t}$, then as$\epsilon\downarrow 0$.

,$\gamma\beta^{2}\int_{0}^{t}[\sum_{j}\{(m_{j}(S)+\epsilon)^{\beta 1}--\epsilon^{\beta}-1\}^{2}m_{j}(_{S)}$$- \{\sum_{j}((m_{j}(_{S})+\epsilon)^{\beta-1}-\epsilon-1)\beta mj(S)\}2]ds$$arrow\gamma\beta^{2}\int_{0}^{t}\sum j\{m_{j}(S)2\beta-1-(\sum_{j}m_{\mathrm{j}}(S)\beta)^{2}\}ds$$=$$\langle\langle M^{c,\beta}\rangle\rangle_{t}$

.

Hence
$M_{t}^{c,\beta,\epsilon}=B(\langle M^{\mathrm{c}},\rho,\epsilon\rangle\rangle tarrow M_{t}^{c,\beta}=B\langle(M\mathrm{c},\rho\rangle\rangle_{t}$

’

where
is a one-dimensional Brownian motion. The
the folowing:$\{B_{t}\}$$\langle\langle M^{c,\beta,\epsilon}-M^{c}’\beta\rangle\rangle_{t}$$=$$L^{2}$

convergence can be shown by$\gamma\beta^{2}\int_{0}^{t}[\sum_{j}\{(m_{j(_{S)+}}\epsilon)^{\beta}-1-\epsilon^{\beta}-1-m_{j(}s)^{\beta 1}-\}2m_{j}(s)$$- \{\sum_{j}((m_{j}(S)+\epsilon)^{\beta 1}--\epsilon^{\beta 1}--mj(S)\beta-1)mj(s)\}2]ds$$arrow$$0$in$L^{1}$.

10

For the discontinuous parts, we can see that$\int_{0}^{t\infty}ds\int \mathrm{Y}S(d_{X)\int_{0})}\nu(dv|F_{\beta},\epsilon(\frac{\mathrm{Y}_{s}+vT_{x}\delta}{1+v})-F_{\beta,\epsilon}(\mathrm{Y}_{s})$$- \{F_{\beta}(\frac{\mathrm{Y}_{s}+vT_{x}\delta}{1+v})-F_{\beta}(\mathrm{Y}_{s})\}|2arrow 0$

in$L^{1}$

in . Moreover by taking aby Lebesgue’s convergence theorem. Hence. convergence. Therefore (2.2) is also valid.
suitable subsequence we get the,Form the above results as$M_{t}^{d,\beta,\epsilon}arrow M_{t}^{d,\beta}$$L^{2}$$\mathrm{a}.\mathrm{s}$$\beta\downarrow 1$$\mathrm{E}_{\mu}[(M_{t}^{c,\beta})^{2}]=\mathrm{E}_{\mu}[\langle\langle M^{C}’\beta\rangle\rangle_{t}]arrow\gamma \mathrm{E}_{\mu}[\int_{0}^{t}F_{1+}(\mathrm{Y}_{s})(1-F_{1+}(\mathrm{Y}_{s}))dS]=0$$\mathrm{E}_{\mu}[(M_{t}^{d,\beta})2]$by$T_{x}\in \mathcal{M}_{1,a}$,$=$$\mathrm{E}_{\mu}[\int_{0}^{t}ds\int \mathrm{Y}(sdx)\int_{0}^{\infty}\nu(dv)|F_{\beta}(\frac{\mathrm{Y}_{s}+vT_{x}\delta}{1+v})-F_{\beta}(\mathrm{Y}_{s})|^{2}]$$arrow$$\mathrm{E}_{\mu}[\int_{0}^{t}ds\int \mathrm{Y}s(dx)\int 0v\nu(d)\infty(\frac{v}{1+v})^{2}\langle(T_{x}\delta-\mathrm{Y}_{S})a’ 1\rangle 2]=0$

for all $x\in S$ and$\mathrm{Y}_{t}\in \mathcal{M}_{1,a}$$\lim_{\beta\downarrow 1}\mathrm{E}_{\mu}[M_{t}(\beta)^{2}]$$\leq$

for$\mathrm{a}.\mathrm{a}$. $t>0,$$\mathrm{a}.\mathrm{s}$. Furthermore$2 \lim_{\beta\downarrow 1}(\mathrm{E}_{\mu}[(M_{t}^{c,\beta})^{2}]+\mathrm{E}_{\mu}[(M_{t}^{d,\beta})2])$$=0$ .

Hence by Doob’s maximal inequality and by taking a sequence$\mathrm{a}.\mathrm{s}$$\sup_{t\leq T}|M_{t}(\beta_{n})|arrow 0$$\{\beta_{n}\};\beta_{n}\downarrow 1$,. for each $T>0$ .

Note that$H^{\beta}(\mu)$$:= \int_{S}\mu(dx)\int_{0}^{\infty}[F_{\beta}(\frac{\mu+v\tau_{x}\delta}{1+v})-F_{\beta}(\mu)$$- \frac{\beta v}{1+v}(\sum_{j}T_{x}\delta(\{x_{j}\})\mu(\{Xj\})\beta-1-F_{\beta}(\mu)\mathrm{I}]\nu(dv)\geq 0$

and that for$\mu\in \mathcal{M}_{1}$,$\lim_{\beta\downarrow}\sup_{1}LF\beta(\mu)=-aF_{1+}(\mu)+\lim_{\beta\downarrow}\sup_{1}[\frac{\gamma}{2}\beta(\beta-1)F\beta-1(\mu)+H^{\beta}(\mu)]$.

Thus if we set$R_{t}= \lim\sup\int_{0}^{t}narrow\infty[\frac{\gamma}{2}\beta_{n}(\beta_{n}-1)F_{\beta 1}(\mathrm{Y}_{s})n^{-}+H^{\beta_{n}}(\mathrm{Y}_{s})]ds$(which is nondecreasing in ), then by (2.2) and the above result we have$t$

$F_{1+}( \mathrm{Y}_{t})-F_{1+}(\mathrm{Y}_{0})+a\int_{0}^{t}F_{1}+(\mathrm{Y}_{S})d_{S}-Rt=0$

By$F_{1+}(\mathrm{Y}_{t})=1$

for a.a.t $>0,$$\mathrm{a}.\mathrm{s}.$for all $0<t\leq T,$,$F_{1+}(\mathrm{Y}_{t})-F_{1+}(\mathrm{Y}_{0})=R_{t}-at$

for all $t\geq 0,$$\mathrm{a}.\mathrm{s}$.

$\mathrm{a}.\mathrm{s}$

.

11
and
$0=F_{1+}(\mathrm{Y}_{t})-F1+(\mathrm{Y}_{s})=R_{t}-R_{s}-a(t-S)$for$\mathrm{a}.\mathrm{a}$. $t>s>0,$$\mathrm{a}.\mathrm{s}.$,that is,for$R_{t}-R_{s}=a(t-s)$$\mathrm{a}.\mathrm{a}$. $t>s>0,$$\mathrm{a}.\mathrm{s}$.

The left hand side is nondecreasing in $t>s$ . Hence it is easy to see that
$R_{t}=at$This implies thateasy to check thatfor all $t\geq 0,$$\mathrm{a}.\mathrm{s}$.

. Finally in case of $T_{x}\delta=\delta_{x}(x\in S)$ , it isfor all $t>0,$
is a solution of the martingale problem for $(G,D(G))$ .

$\mathrm{Y}_{t}\in \mathcal{M}_{1,a}$$\mathrm{a}.\mathrm{s}$$\square$$\{\mathrm{m}_{\mathrm{t}}\}$

Remark 2 (Pure jump case). In case of $\gamma=0$ , even if is bounded, it is not ensurefor. For instance, suppose that
thatis pure atomic for all. Also assume that $a(\cdot)\equiv a>0$ and $B(x, \cdot)$ has no
every $x\in S$ and,be defined as in the previous proof. If
atoms for each $x\in S$ . Let$A$$T_{x}\delta\in \mathcal{M}_{1,a}$$t>0\mathrm{P}_{\mu^{-\mathrm{a}}}.\mathrm{S}$$\mathrm{Y}_{t}$$\mathrm{Y}_{0}=\mu\in \mathcal{M}_{1},{}_{a}\mathrm{P}_{\mu}- \mathrm{a}.\mathrm{s}$$\int_{0}^{\infty}v\nu(dv)<\infty$$H^{\beta}(\mu)$then it is easy to see that$\lim_{\beta\downarrow 1}H^{\beta}(\mu)=0$.

In fact, for $\beta>1$ , let $h(a, b)\geq 0$ be in $(2.3)$ with $p=1/(1+v),q=v/(1+v)$ , then it.
holds that for $0\leq a,$ $b\leq 1$ ,$h(a, b)$$\leq pa^{\beta}+qb^{\beta}-a^{\beta}-\beta q(b-a)a^{\beta-1}$$=$$q(-a^{\beta}+ \oint-\beta(b-a)a^{\beta 1}-)$$\leq$$q(1+\beta)(a+b)$ ,$\{$$arrow$$0$$(\beta\downarrow 1)$.

Threfore we can apply Lebegue’s convergence theorem for $H(\eta)$ . Now by mean zero
$0$

$=$

$\lim_{\beta\downarrow 1}\mathrm{E}_{\mu}[Mt]d,\beta n$$=$$\mathrm{E}_{\mu}[F_{1+}(\mathrm{Y}t)-F1+(\mathrm{Y}0)+a\int_{0}^{t}F_{1+}(\mathrm{Y}s)ds]$

This implies that $\mathrm{E}_{\mu}[F_{1+}(\mathrm{Y}_{t})]=F_{1+}(\mu)e^{-a}t=e^{-at}<1$ , i.e.,(Note that if $a=0$ , then
. $t>0,$thus,
for
we have$\mathrm{a}.\mathrm{a}$$\mathrm{a}.\mathrm{s}$is not pure atomic.

for al $T>0$ .
, i.e.,. Moreover by the same way as in the previous proof$\mathrm{E}_{\mu}[F_{1+}(\mathrm{Y}_{t})]=1$$\mathrm{Y}_{t}\in \mathcal{M}_{1,a}$$\mathrm{Y}_{t}$$\mathrm{E}_{\mu}[\int_{0}^{\tau_{F_{1}(\mathrm{Y}_{t})t]}}+d=T$$F_{1+}(\mathrm{Y}_{t})-1=R_{t}-at=H_{t}-at=0$

for all $t>0,$That is,$\mathrm{P}_{\mu}$(.

$\mathrm{Y}_{t}\in \mathcal{M}_{1,a}$for all $0\leq t\leq T$ ) $=1.)$$\mathrm{a}.\mathrm{s}$

.

12
3.

$\mathrm{S}\mathrm{p}\mathrm{A}\mathrm{C}\mathrm{E}$-TIME INHOMOGENEOUS$\mathrm{F}\mathrm{L}\mathrm{E}\mathrm{M}\mathrm{I}\mathrm{N}\mathrm{G}-\mathrm{V}\mathrm{I}\mathrm{o}\mathrm{T}$PROCESSES

Next we would like to extend the Perkins result to the space(-time) inhomogeneouscase.
Let $c(x)\geq 0$ be a bounded function.
According to Dawson [1], we first give a characterization of the space inhomogeneous
binary branching process.
is an-valued process such that $\mathrm{P}_{\mu}(Z0=\mu)=1,$$\inf\{t\geq 0;\langle Z_{t}, 1\rangle=0\})$ and,$(Z_{t}, \mathrm{P}_{\mu})_{t}\geq 0(\mu\in \mathcal{M}_{F})$$(Z_{t}, \mathrm{P}_{\mu})_{t}\geq 0$$Z_{t}=Z_{t\wedge\tau \mathfrak{o}}(\tau_{0}=$$\mathcal{M}_{F}$$\mathrm{P}_{\mu}[e^{-\langle z_{l},f)}]=e^{-\mathrm{t}\mu,Vf)}l$

where$V_{t}f$

is a unique solution to the following equation$V_{t}f(x)=P_{t}f(X)- \frac{1}{2}\int_{0}^{t}d_{SP_{s}}(c(\cdot)(Vt-Sf)(\cdot)^{2})(x)$

,

or
,

$\partial_{t}V_{t}f(x)=AV_{t}f(X)-\frac{1}{2}c(X)(Vtf)^{2}(x)$Moreover$\langle Z_{t}, f\rangle(f\in i)(A))$$V0^{f(X)}=f(x)$ .

has the following semi-martingale representation:$\langle Z_{t}, f\rangle=\langle Z_{0}, f\rangle+\int_{0}^{t}\langle Z_{s}, Af\rangle d_{S}+M_{t}(f)$where $\{M_{t}(f)\}$ is a continuous$L^{2}$-martingale with quadratic variation$(t<\tau_{0})$$\langle\langle M(f)\rangle\rangle_{t}=\int_{0}^{t}\langle Z_{s}, Cf^{2}\rangle ds$

The generator of this process$\mathcal{L}^{Z}$$F(\eta)$of$\eta\in \mathcal{M}_{F}$A$f\rangle$.

$+ \frac{1}{2}\langle\mu, cf^{2})\rangle]e^{-\langle f\rangle}\mu,$.

, a derivative at $x\in S$ is defined by$\frac{\delta F(\eta)}{\delta\eta(x)}=\lim_{\epsilonarrow 0}\frac{1}{\epsilon}[F(\eta+\epsilon\delta_{x})-F(\eta)]$and higher order derivatives
generator can be also expressed as$\delta^{2}F(\eta)/(\delta\eta(x)\delta\eta(y)),$(3.1)

such that$\langle\langle M(f)\rangle\rangle_{t}$is given as$L^{z}e^{-(}.,f\rangle(\mu)=[-\langle$$\mu,$

For a functional,$\ldots$(if exists),

are defined similarly. Note that the$c^{z_{F(\eta)}}= \langle\eta, A\frac{\delta F(\eta)}{\delta\eta(\cdot)}\rangle+\frac{1}{2}\iint\frac{\delta^{2}F(\eta)}{\delta\eta(x)\delta\eta(y)}Q(\eta;dX, dy)$

where $Q(\eta;dx, dy)=c(x)\eta(d_{X})\delta_{x}(dy)$ .

Next in oreder to introduce space-time inhomogeneous Fleming-Viot processes we definea family of operators
for a fixed $g\in C_{r,+}$ as follows: For functionals
and $r\leq t<\tau_{g}$ ,$L^{g}=(c_{t}^{g})_{r}\leq t<\mathcal{T}_{g}$$\exp[-\langle\eta, f\rangle](\eta\in \mathcal{M}_{1}, f\in D(A))$$L_{t}^{g}e^{-(\cdot,f\rangle}(\eta)$$=$$\{-\langle\eta, Af\rangle-g(t)-1[\langle\eta, C\rangle\langle\eta, f\rangle-\langle\eta, cf\rangle]\}e^{-\langle\eta,f}\rangle$$+ \frac{1}{2g(t)}[\langle\eta, cf^{2}\rangle+\langle\eta, c\rangle\langle\eta, f\rangle 2-2\langle\eta, cf\rangle\langle\eta, f\rangle]e^{-\langle\eta,f}\rangle$

with a domain$D_{0}(\mathcal{L}^{g}):=\mathrm{l}\mathrm{i}\mathrm{n}$span$\{e^{-\langle\cdot,f\rangle}$; $f\in D(A),$ $f\geq 0\}$ .

.

13This operator can be also expressed as in (3.1) by$\mathcal{L}_{t}^{g}F(\eta)$$=$$\langle\eta, A\frac{\delta F(\eta)}{\delta\eta(\cdot)}\rangle+g(t)^{-1}[\langle\eta, c\rangle\langle\eta, \frac{\delta F(\eta)}{\delta\eta(\cdot)}\rangle-\langle\eta, c\frac{\delta F(\eta)}{\delta\eta(\cdot)}\rangle]$$+ \frac{1}{2}\int\int\frac{\delta^{2}F(\eta)}{\delta\eta(x)\delta\eta(y)}Qt(\eta;d_{X}, dy)$

,

where
$Qt( \eta;dX, dy)=\frac{1}{g(t)}[C(x)\eta(dx)\delta_{x}(dy)+(\langle\eta, c\rangle-c(_{X)(y))(d)}-C\eta X\eta(dy)]$.

We need the following condition.
Condition 3.1. $c\in C(S)$ satisfies $0 \leq\sup c(x)\leq 2$ inf $c(x)$ .
This condition is equivalent to $c(x)+c(y)\geq c(z)\geq 0$ for every $x,$ $y,$ $z\in S$ .
The following result gives the definition of the space-time inhomogeneous Fleming-Viotprocess
associated with $(A,g, c)$ starting fromat $t=r$ .
$(\mathrm{Y}_{t}, \mathrm{P}_{r,\mu}^{FV})$$\mu\in \mathcal{M}_{1}$$g\in C_{r,+}$ and $c\in C(S);C(X)\geq 0$ .
Theorem 3. Let
$:=$
For
, set
. Then, under Condition 3.1 on $c(x)$ , there is asolutiononto the martingale problem forsatisfying thefollowing:(i),(ii) For $f\in D(A)$ andhas thesemi-martingale representation:$\mu\in \mathcal{M}_{1},$$\mathrm{C}([r,\tau_{g})arrow \mathcal{M}_{1})$$\mathrm{P}_{r,\mu}^{FV}$$\mathrm{Y}_{r}=\mu,$$\omega\in \mathrm{C}_{r,\tau_{g}-}$$\mathrm{Y}_{t}(\omega)=\omega(t)$$(\mathcal{L}_{t’ 0}^{g}D(\mathcal{L}^{g}))_{t}\in_{1g}r,\mathcal{T})$$\mathrm{C}_{r,\tau \mathit{9}^{-}}$$\mathrm{P}_{r,\mu}^{FV}-a.s.$$r\leq t<\tau_{g_{f}}\langle \mathrm{Y}_{t}, f\rangle$$\langle \mathrm{Y}_{t}, f\rangle$$=$$f_{\mathit{0}\iota}\iota_{ov\dot{n}n}g$$\langle \mathrm{Y}_{r}, f\rangle+\int_{r}^{t}\{\langle \mathrm{Y}_{S}, Af\rangle+g(s)-1[\langle \mathrm{Y}s’ c\rangle\langle \mathrm{Y}s’ f\rangle-\langle \mathrm{Y}s’ cf\rangle]\}d_{S}$$+M_{r,t}(f)$such that $\{M_{r},t(f)\}$ is a continuous$L^{2}$-martingale with quadratic$va\dot{n}\sigma tion$$\langle\langle M(f)\rangle\rangle_{r,t}=\int_{r}^{t}g(_{S})-1[\langle \mathrm{Y}_{S}, Cf^{2}\rangle+\langle \mathrm{Y}\cdot, C\rangle s\langle \mathrm{Y}fs’\rangle^{2}-2\langle \mathrm{Y}_{S}, cf\rangle\langle \mathrm{Y}f\rangle]s’ ds$Moreover if $A=0$ , then the solution$(\mathrm{Y}_{t}, \mathrm{P}_{\mu}^{FV})$.

is unique.

By using the same argument as in Perkins [8] and the author [6] it is possible to obtainthe following result. Recall.
$\overline{\mu}=\mu/\langle\mu, 1\rangle$

Corollary 1. Let
and set. For a fixed $r\geq 0$ , letbea space-inhomogeneous binary branching process with $A=0$ onsuch that$Z_{r}=\mu$ . Set
and $\tau_{0}=\inf\{t\geq r;x_{t}=0\}$ . If
, then$\mu\in \mathcal{M}_{F}\backslash \{0\}$$y=\langle\mu, 1\rangle$$(Z_{t}, \mathrm{P}_{\mu})$$\mathrm{C}([r, \infty),$ $\mathcal{M}_{F})$$x_{t}=\langle Z_{t}, 1\rangle$$\mathrm{P}_{\mu}(\overline{Z.}|_{[r,\mathcal{T}_{\mathrm{O}}})\in B|$

where$(0, g, c)$$Q_{y}=\mathrm{P}_{\mu}\mathrm{o}(\{x_{t}\}_{t\in[r},\tau 0))^{-}1$$\langle$Z.,$1\rangle$,$=g(\cdot)|_{1}r,\tau)g)=\mathrm{P}_{r}^{F_{\frac{V}{\mu}}},(\mathrm{Y}|_{1)}r,\tau_{g}\in B)$$Q_{y^{-}}a.a$. $g\in C_{r,+}$ ,is a space-time inhomogeneous Fleming-Viot process associated withstarting from.

$( \mathrm{Y}_{t}, \mathrm{P}_{r}^{F},\frac{V}{\mu})$$\mathrm{Y}_{r}=\overline{\mu}$Note thathas a decomposition $x_{t}=\langle\mu, f\rangle+m$ , wheremartingale starting from with quadratic variation,$x_{t}=\langle Z_{t}, 1\rangle$$0$$\langle\langle m\rangle\rangle_{t}=\int_{r}^{t}\langle Zs’ c\rangle ds$$(r\leq t<\tau_{0})$.

$\{m_{t}\}$is a continuous

14
Proof. For simplicity of the notations, as in [6] we set$x_{t}.$$Z_{t}(f)=\langle Z_{t}, f\rangle,$$|Z_{t}|=\langle Z_{t}, 1\rangle=$

RecaU that$dZ_{t}(f)=Z_{t}(Af)dt+dM_{t}(f)$ ,$Z_{0}(f)=\langle\mu, f\rangle$,-martingale with quadratic variation
where $\{M_{t}(f)\}$ is a continuous
Thus by using Ito’s formula we have$L^{2}$$d\langle\langle M(f)\rangle\rangle_{t}=$$\langle Z_{t},cf^{2}\rangle dt$$d(1/|Z_{t}|)=-d|z_{t}|/|Z_{t}|^{2}+d\langle\langle M(1)\rangle\rangle_{t}/|Z_{t}|^{3}$

and, noting that$d\langle\langle Z(f), (1/|Z|)\rangle\rangle_{t}=-d\langle\langle M(f), M(1)\rangle\rangle t/|Z_{t}|^{2}=-[\langle Zt, Cf\rangle/|Z_{t}|^{2}]dt$,

we ako have,$d\overline{Z_{t}}(f)=\overline{Z_{t}}(Af)dt+dU_{t}(f)+\ulcorner Z_{t}(C)\overline{Z}t(f)-\overline{Z}_{t}(Cf)]/|Z_{t}\mathrm{t}dt$

where$dU_{t}(f)=dM_{t}(f)/|Z_{t}|-\ulcorner Z_{t}(f)/|Z_{t}|]dMt(1)$is a continuous$d\langle\langle U(f)\rangle\rangle_{t}$$L^{2}$-martingale with quadratic variation$=$$\frac{d\langle\langle M(f)\rangle\rangle_{t}}{|Z_{t}|^{2}}+[\frac{\overline Z_{t}(f)}{|Z_{t}|}]^{2}d\langle\langle M(1)\rangle\rangle_{t}-2[\frac{\overline Z_{t}(f)}{|Z_{t}|^{2}}]d\langle\langle M(f), M(1)\rangle\rangle_{t}$$=$$[\overline{z_{t}}(Cf2)+\overline{Z_{t(f)^{2}\overline{Z}(_{C)}}}t-2\overline{Zt}(f)\overline{Z_{t(cf)]}}]|Z_{t}|^{-1}dt$

Hence by the same way as in [8] we can show thatthe spaoe inhomogeneous Fleming-Viot process.

Proof of Theorem 3.
Let $c(x)\geq 0$ be in $C(S)$ and$c^{g}(t;X):=C(X)/g(t)(t\leq\tau_{g})$ ,$\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{S}6^{r}$$\{\overline{Z}_{t}\}$.

under the condition $|Z.|=g(\cdot)$ is$\square$Condition 3.1. Fix $g\in C_{r,+}$ and set $d(x)=$$d(x, y, z)=c(gt;x,y, z):= \frac{1}{2}[C^{g}(t, X)+Cg(t, y)-C^{g}(t, Z)](\geq 0)$ .

for
It is enough to consider the uniqueness and the existence ineach fixed $r<T<\tau_{g}$ .
. Let
We ffist show the existence of the solution. Fix $n\geq 1$ . Let
be the independent particle system associated with the motion process$(w(t), P)x$ starting ffom
be the Markov particle system starti. Let
ising ffom $X_{r}=\mu^{(n)}$ such that the Laplaoe functional$\mathrm{C}_{r,T}:=\mathrm{C}([r, \tau]arrow \mathcal{M}_{1})$$\mu^{(n)}=\sum_{k=1k}^{n}\delta_{x}$$(X_{t}^{0}, \mathrm{Q}_{r,\mu}^{0}(n))$$X_{r}^{0}=\mu^{(n)}$$(X_{t}, \mathrm{Q}_{r,\mu}(n))$$L_{r,t}(\mu^{\mathrm{t}^{n)}})=\mathrm{Q}_{r,\mu^{(n)}}[\exp-\langle X_{t}, f\rangle]$

15
the unique solution to the following equation:$L_{r,t}( \mu)(n)=\frac{1}{n}\sum \mathrm{Q}_{f}^{0},\mathrm{t}n)k=1n\mu[\exp(-\sum_{m}c^{g}(r, t)-\sum_{ji\neq}C^{g}im,j,k(r,t)-\langle X_{t}0, f\rangle \mathrm{I}$$+$$\sum_{m}\int_{r}^{t}dSC(gms)\exp(-c_{m}^{g}(r, S)-,\sum_{m\neq m}c_{m}^{g},(r,t)-\sum_{ji\neq}C^{g}k(i,j,\mathrm{I}r,t)$$L_{s,t}(X_{s}^{0}-\delta wm(s)+\delta w_{k}(s))$$+$$\sum_{i\neq j}\int_{r}^{t}dSC_{i},(_{S}g)j,k\exp(c_{i,\mathrm{j},k}^{g}(r, S)-\sum C_{m}(g)-,\sum_{;(,j)\neq(i,j)}C_{i},,(gr,t)j’,k\mathrm{I}mr,ti^{l}\neq \mathrm{j}\prime i’$$L_{s,t}(X^{0_{-}}\delta(s)+w\delta(s))siwj]$,

where $c_{m}^{g}(t):=c^{g}(t;w_{m}(t)),$ $c_{m}(gr,t):= \int_{r}^{t}c_{m}(gs)ds$ and $C_{i,j,k}^{g}(t):=c^{g}(t;wi(t),w_{j}(t),wk(t))$ ,$c_{i,j,k}^{g}(r,t):= \int_{r}^{t}c_{i,j,k}^{g}(s)d_{S}(i\neq j)$.

This particle system can be constructed directly as follows: first -particles $\{w_{i}(t);i=$$1,2,$) is-th particle, }move independently and one particle (e.g.
selected with probability $1/n$ at the starting time $t=r$. Let$1,2,$, and $m,i,j,$ $k\in\{1, \ldots,n\};i\neq j)$ be independent random variables such thatand $P_{\mathrm{w}}(\tau_{i,j,k}(p)>t)=\exp[-c_{i,j,k}g(r, t)]$ , where $\mathrm{w}=\{w_{i}(t);i=$$1,2,$
andand theis uniquely defined by$\ldots,n(1)(\},$-th particle. If, then after the random time
particle is selectedjumps to the location of theand at the same time
particlei-thwith probability $1/n$ . Ifl’ then after the random timeisparticleparticle and at the same time
jumps to the location of the anotherselected with probability $1/n$ . Again these particles move independently according to thesame law. For these particles, we use the same notations $\{w_{i}(t)\}$ . Next the random times. Then according todefined as above by usingorparticles moves similarly to above. These operations are continued.
is called the space-time dependent Moran particle
This particle system
at $t=r$ associated with the motion process $(w(t), P_{r},x)_{f}$ samplingsystem starting fromrate function $c(t;x)$ .
We denote the generator of independent particle system $\{w_{k}(t)\}$ by , which is definedas$n$$\ldots$$n$$(\subset D_{r})$$k_{1}$$w_{k_{1}}$$\sigma_{m}^{(p)}=\sigma_{m}^{(\mathrm{p})},$$\tau_{i,\mathrm{j},k}^{(p)}(p=$$\ldots$$P_{\mathrm{w}}(\sigma_{m}^{(_{\mathrm{P}}})>t)=\exp[-C^{g}(mr, t)]$$\min_{m}\sigma_{m}^{(1)}=\sigma_{m_{1}}^{(1)}$$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{X}m_{1}\mathrm{l})$$\min_{i\neq j^{\mathcal{T}_{i}}j,k_{1}},=\tau_{i_{1},j_{1}},k_{1}$$(\mathrm{l}\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r})(i_{1},j_{1})$$\sigma_{m_{1}}^{(1)},$$\sigma_{m_{1}}^{(1)}<\tau_{i_{1},j_{1},k_{1}}$$m_{1}$$k_{2^{-}}\mathrm{t}\mathrm{h}$$k_{1^{-}}\mathrm{t}\mathrm{h}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{C}(1)1\mathrm{e}$$(1)$$\sigma_{m_{1}}^{(1)}>\tau_{i_{1},j_{1}},k_{1}$$\mathcal{T}$$i_{1},j_{1},k_{1}’$$k_{2^{-}}\mathrm{t}\mathrm{h}$$j_{1^{-}}\mathrm{t}\mathrm{h}$$\{\sigma_{m}^{(2)}\},$$\sigma_{m_{2}}^{(2)},$$\sigma_{m_{2}}^{(2)}>\tau_{i_{2},j_{2},k_{2}}^{(2)}$$\{\tau_{i,j,k_{2}}^{(2)}\}$$\tau_{i}\mathrm{a}(2)2,\mathrm{j}2,k2(2)\mathrm{r}\mathrm{e}$$\sigma_{m_{2}}^{(2)}<\tau_{i_{2},j2,k_{2}}$$(X_{t}, \mathrm{Q}_{r,\mu}(n))$$\mu^{(n)}$$\mathcal{G}^{0}$$\mathcal{G}^{0}e^{-\mathrm{t}\cdot,f}\rangle(\eta)=-\langle\eta, eAf(1-e^{-}f)\rangle e^{-}\langle\eta,f\rangle$

.

16
Let

$\mu^{(n)}=\Sigma_{i=1}^{n}\delta_{x}\dot{.}$$\mathcal{G}_{t}e^{-\langle\cdot,f\rangle}(\mu^{(n}))$. The generator$=$$\mathcal{G}_{t}$ofis given as$(X_{t}, \mathrm{Q}_{r,\mu^{()}}\#)$$\frac{1}{n}\sum_{k=1}^{n}[\mathcal{G}^{0}e^{-\mathrm{t}}.,(f\rangle)+\sum d(X_{m})(e^{-}-1)\mu^{(n)}m1em[f(xk)-f(x)-(\mu,f(n))$$+ \sum_{i\neq j}d(x_{ij}, X, xk)(e-[f\mathrm{t}x_{j})-f\mathrm{t}x:)1-1)e^{-}\langle\mu)(n,f\rangle]$$=$$\{-\langle\eta, e^{f}A(1-e-f)\rangle+\frac{1}{n}\sum^{n}k=1[\sum_{m}d(Xm)(e^{-1}-f\mathrm{t}xmf(x_{k}))]-1)$$+ \sum_{i\neq j}d(x_{i,j}X, xk)(e-[f(x\mathrm{j})-f(x.\cdot)]-1)]\}e^{-\langle\mu^{(n)},f\rangle}$

Set the domain of$\mathcal{G}=(\mathcal{G}_{t})$$D_{0}(\mathcal{G}):=\mathrm{l}\mathrm{i}\mathrm{n}$

.

by

span

$\{e^{-(\cdot,f\rangle}$; $f=-\log(1-h),$ $0\leq h<1,$ $h\in D(A)\}$ .

Then it is easy to see thatis a Markov process with sample paths inand the unique solution to the martingale problem foron.
Now we consider the scaled Moran particle system, wherewith$\mu_{n}=\mu^{(n)}/n$ andis its probability law. We also denote the generator by. We
shall show that ifin, then the scaling limit
exists as a space-timeinhomogeneous Fleming-Viot process associated with $(A, 1, c^{g})=(A, g, c)$ and has thefollowing generator ; for $r\leq t<\tau_{g}$ ,$(x_{t}, \mathrm{Q}_{r,\mu^{(}}n))$$\mathrm{D}([r, \infty)arrow \mathcal{M}_{F}(S))$$(\mathcal{G}_{t}, D_{0}(\mathcal{G}))$$\mathrm{D}([r, \infty)arrow \mathcal{M}_{F}(S))$$\mathrm{Y}_{n,t}=X_{t}/n$$(\mathrm{Y}_{n},, {}_{t}\mathrm{P}_{r,\mu_{*}}^{(}n).)$$\mathrm{P}_{r,\mu n}^{(n)}$$\mathcal{L}_{n,t}^{g}$$\mu_{n}arrow\mu$$\mathcal{M}_{1}$$(\mathrm{Y}_{t}, \mathrm{P}_{r,\mu}^{FV})$$\mathcal{L}_{t}^{g}$$\mathcal{L}_{t}^{g}e^{-\mathrm{t}}.,(f))\eta$,$=-\langle\eta, Af\rangle e^{-_{\mathrm{t}f\rangle}}\eta$$- \frac{1}{g(t)}[\langle\eta, c\rangle\langle\eta, f\rangle-\langle\eta,cf\rangle]e^{-}(\eta,f)$$+ \frac{1}{2g(t)}[\langle\eta, Cf^{2}\rangle+\langle\eta, c\rangle\langle\eta, f\rangle^{2}-2\langle\eta, Cf\rangle\langle\eta, f\rangle]e^{-(\eta,f})$.

For $f\in D(A)$ ,$\langle \mathrm{Y}_{n,t}, f\rangle-\langle \mathrm{Y}fn,r’\rangle-\int_{r}^{t}\langle \mathrm{Y}_{n,s}, Af\rangle dS$

is a$\mathrm{P}_{r,\mu_{n}}^{(n)}$-martingale and$\sup_{n}\mathrm{P}_{r,\mu_{n}}^{()}n[\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{p}r\leq t\leq T|\langle \mathrm{Y}t, Afn,\rangle|]\leq||Af||$.

Hence by Th. 9.4 in Chap. 3
of [3]
is relatively compact, i.e., tight in$T],$(becauseis Polish). Moreover since is compact and $D(A)$ isdense in $C(S)$ and closed under addition, by Th. 3.7.1 in [1]is tight, i.e., relatively
compact in. Therefore there exist a subsequence
and a limit
pointsuch thatconverges weakly toin.
For each integer , letbe a family of counting measures onofthe form. Moreover let $f_{n}=-n\log(1-f/n)$ for $f\in D(A)$ such that$(\mathrm{p}145)$$\mathrm{D}([r,T], \mathrm{R})$$\mathrm{D}([r,$$\{\langle \mathrm{Y}_{n,t}, f\rangle\}$$S$$\mathrm{R})$$\{\mathrm{Y}_{n,t}\}$$\mathrm{D}([r, \tau],\mathcal{M}1)$$(\mathrm{Y}_{t}, \mathrm{P}_{r,\mu})$$\{(\mathrm{Y}_{n_{k}},,{}_{t}\mathrm{P}_{r,\mu_{n}}^{(n}))\}$$\{\mathrm{Y}_{n_{k},t}\}$$n$$\eta_{n}=\sum_{k=1k}^{n}\delta_{x}/n$$\mathcal{M}_{1}^{(n)}=\mathcal{M}_{1}^{(n)}(s)$$\{\mathrm{Y}_{t}\}$$\mathrm{D}([r, T], \mathcal{M}1)$

$S$

17
$||f||<1$ and inf $f>0$ .

(3.2)

It is possible to show that for each $r<T<\tau_{g}$ ,$\lim_{narrow\infty}\sup_{r\leq t\leq\tau\eta\in}\sup_{\mathcal{M}1}|\mathcal{L}g\cdot,f_{n}\rangle(n,te^{-\mathrm{t}}\eta)-(n)c_{t}^{g}e-\mathrm{t}\cdot,f\rangle(\eta)|=0$(we shall show this at the end). Hence it is easy to that the limit point
is asolution to the martingale problem forin. We need to show thecontinuity and the semi-martingale representation of. However this can be shown by
the same way as in the proof of Th. 6.1.3 of [1].
$(\mathrm{Y}_{t,\mu}\mathrm{P}_{r},)$$(\mathcal{L}_{t}^{g}, D\mathrm{o}(\mathcal{L}))$$\mathrm{D}([r, T],\mathcal{M}1)$$\{\mathrm{Y}_{t}\}$Next in case of $A=0$ the uniqueness can be shown by the same way as in [9]. In fact,
for $r\leq t\leq T$ andlet$\eta\in \mathcal{M}_{1}$$a_{t}(\eta, f)=g(t)^{-}1[\langle\eta, c\rangle\langle\eta, f\rangle-\langle\eta, cf\rangle]$and.
We consider the following stochastic differential equation: for $r\leq t\leq T$ ,$b_{t}(\eta, f)=g(t)^{-1}[\langle\eta, cf^{2}\rangle+\langle\eta, c\rangle\langle\eta, f\rangle^{2}-2\langle\eta, Cf\rangle\langle\eta, f\rangle]$(3.3)$d\langle \mathrm{Y}_{t}, f\rangle=at(\mathrm{Y}_{t}, f)dt+\sqrt{b_{t}(\mathrm{Y}_{t},f)}dB_{t}$,$\mathrm{Y}_{r}=\mu$

,

where
is a one-dimensional standard Brownian motion. To show the (law) uniquenessof
it is enough to show the pathwise uniqueness of the solution to the above equationin(see [7]). However the pathwise uniqueness can be easily checked. Let
be solutions for the equation (3.3) defined on th same probability space (we denote theprobability measure as). By using the following inequality (let $g_{*}= \inf_{r\leq t\leq Tg(t}$ )$(B_{t})$$(\mathrm{Y}_{t})$$\mathrm{C}_{r,T}$$(\overline{\mathrm{Y}}_{t})$$(\mathrm{Y}_{t}),$$)$$\mathrm{P}_{\mu}$$|a_{t}(\eta, f)-a\iota(\overline{\eta}, f)|\leq Ng^{-1}*(||c||\mathrm{v}||f||))(|\langle\eta-\overline{\eta}, C\rangle|+|\langle\eta-\overline{\eta}, f\rangle|)$,$|bt(\eta, f)-bt(\overline{\eta}, f)|\leq Ng^{-1}*(||cf||\vee||f||^{2})(|\langle\eta-\overline{\eta}, c\rangle|+|\langle\eta-\overline{\eta}, f\rangle|)$,where $N$ is an appropriate number, and we have$\mathrm{E}_{\mu}[|\langle \mathrm{Y}_{t}-\overline{\mathrm{Y}}_{l}, f\rangle|]\leq C\int_{r}^{t}\mathrm{E}_{\mu}[|\langle \mathrm{Y}_{S}-\overline{\mathrm{Y}}C\rangle s’|+|\langle \mathrm{Y}_{s}-\overline{\mathrm{Y}}S’ f\rangle|]ds$,

where $C>0$ is a constant depending only on $(g_{*}, ||c||, ||f||)$ . Thus we get theuniqueness ofand of. Hence the law of uniqueness holds.
$A=0$
Therefore if, then the limit process
uniquely exists in.

$\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{W}^{\backslash }\mathrm{i}_{\mathrm{S}}\mathrm{e}$$\{\langle \mathrm{Y}_{t}, f\rangle\}(f\in C(S))$$\{\langle \mathrm{Y}_{t}, c\rangle\}$$(\mathrm{Y}_{t}, \mathrm{P}_{r,\mu}^{FV})_{r}\leq t\leq\tau$Finally we show (3.2). Note that for$c_{n,t}^{g}e^{-\langle}.,f_{n}\rangle(\eta_{n})$$\eta_{n}=\sum_{j}\delta_{x_{j}}/n\in \mathcal{M}_{1}^{(n)}$$=$$\mathcal{G}_{t}e^{-\langle\cdot,f_{n}}(/n\rangle\eta_{n}n)$$=$$- \langle\eta_{n}, \frac{Af}{1-f/n}\rangle e^{-}\langle\eta_{\#},fn)$$\mathrm{C}_{r,T}$,$+ \frac{1}{n}\sum_{1k,,m=}c_{t}(ngX_{m})(em-x_{k}))/n-(f_{n}(x)f_{n}(1)e-\langle\eta n’ f_{n}\rangle$$+ \frac{1}{n}\sum_{k=1i}\sum_{j\neq}d(Xi, xj, x_{k})(e)/n-n(f_{n}(x\dot{.})-fn(x_{j})1)e-\langle\eta_{\mathrm{B}},f_{n}\rangle$

It is easy to see that$\frac{1}{n^{2}}\sum_{k,m}C(x_{m})(f(X_{m})-f(x_{k}))=-(\langle\eta n’ c\rangle\langle\eta_{n}, f\rangle-\langle\eta n’ cf\rangle)$.

18
and that by symmetry of $d(x_{i},x_{j,k}X)$ in $(i,j)$$\sum_{i\neq j}d(x_{i}, Xj,’ Xk)[f(xi)-f(X_{j})]--0$.

Moreover$\frac{1}{n^{2}}\sum_{i\neq j}(f(X_{i})-f(x_{j}))^{2}$$=$$\frac{1}{n^{2}}\sum_{i,j}(f(_{X_{i}})-f(x_{j}))^{2}$$=2(\langle\eta_{n}, f^{2}\rangle-\langle\eta_{n}, f\rangle^{2})$and$\frac{1}{n^{2}}\sum C(x_{i})(f(_{X}i)-f(_{X_{j}}))i\neq j2$$=$$\frac{1}{n^{2}}\sum_{i,j}c(X_{i})(f(Xi)-f(_{X_{j}}))2$$=$$\langle\eta_{n}, cf^{2}\rangle+\langle\eta_{n}, C\rangle\langle\eta n’ f2\rangle-2\langle\eta_{n}, Cf\rangle\langle\eta n’ f\rangle$.

Hence by $d(x_{i}, x_{j’ k}X)=[d(x_{i})+d(x_{j})-d(x_{k})]/2$ we have$\frac{1}{n}\sum_{=k1i}^{n}\sum_{j\neq}d(_{X_{i,j}}X, x_{k})\frac{(f(x_{i})-f(x_{j}))^{2}}{2n^{2}}$$= \frac{1}{2g(t)}[\langle\eta_{n}, Cf2\rangle+\langle\eta n’ c\rangle\langle\eta n’ f\rangle^{2}-2\langle\eta_{n}, Cf\rangle\langle\eta n’ f\rangle]$.

Therefore by using Taylor’s expansion the equation (3.2) can be easily checked.

$\square$is pure atomic, i.e.,Remark 3. If $A=0$ and the starting measure
’
given
is
is also pure atomic and the corresponding generator
then clearly the processas$\mu=\sum m_{ix^{0}}^{0}\delta:$$\mathrm{Y}_{r}=\mu$$\mathrm{Y}_{t}$$c\psi(\mathrm{m})$$=$$\frac{1}{2}\sum_{i,j}[m_{i}c(x_{i})0\delta ij+mim_{j}\{\sum_{k}m_{k}C(X_{k}^{0})-c(X_{i})0-C(_{X^{0}}j)\}]\partial_{i}2\phi j(\mathrm{m})$$+ \sum_{i}b_{i}(\mathrm{m})\partial i\emptyset(\mathrm{m})$,where $b_{i}( \mathrm{m})=(\sum_{j}c(X_{j}^{0})mj-C(x_{i}^{0}))m_{i}$ . However in this case our result is contained
to Shiga’s result [9] in 1987. He showed the result under more general conditions on;$c(x)$ andand for some matrix
such that let$\beta_{i}=c(X_{i}^{0}),$$b_{i}(\mathrm{m})$$q_{ij}\geq 0,$ $\sup_{j}\sum q_{ij}<\infty,$$(q_{ij})$$\beta_{i}\geq 0;\sup_{i}\beta_{i}<\infty$$|bi(\mathrm{m})-bi(\mathrm{m}’)|\leq\Sigma_{j}q_{ij}|m_{j}-m_{j}|J$.

By using the same argument as in the proof of Theorem 3 we can see the following
results:

ing

Theorem 4. In Theorem 3, if the quadratic martingale part$\dot{?}S$changed to the follow-$\langle\langle M(f)\rangle\rangle_{r,t}=\int_{r}^{t}g(s)^{-}1[\langle \mathrm{Y}s’\rangle Cf^{2}+\langle \mathrm{Y}_{s}, C\rangle\langle \mathrm{Y}_{S}, f^{2}\rangle-2\langle \mathrm{Y}_{s}, Cf\rangle\langle \mathrm{Y}_{S}, f\rangle]ds$,

then the same claim holds for all bounded functions $c\in C(S);C(X)\geq 0$ wiffiout Condition 3.1. Moreover it is possible to construct the processes without the drift term$g(t)^{-1}[\langle\eta,c\rangle\langle\eta, f\rangle-\langle\eta, cf\rangle]$.

19. However it isenough to change $d(x, y, z)$ to $d(x,y)=c^{g}(t;X,y):=[c^{g}(t, X)+c^{g}(t,y)]/2$ , thus
to $c_{i,j}^{g}(t):=d(t;w_{i}(t),w_{j(t)})$ . Moreover for the processes without drift term, it enoughand $d(x_{m})$ .
to delete the terms corresponding to

Proof. We have to consider the approximating particle systems$\{X_{t}\}$$C_{i,j,k}^{g}(t)$$\square$$c_{m}^{g}(s)$REFERENCES[1] DAWSON, D. A. (1993) Measure-valued Markov processes. Lect. Notes in Math. 1541, Springer,$1-2\alpha]$.

[2] ETHIER, S. N. and KURTZ, T. G. (1981) The infinitely many neutral alleles diffusion modeL Adv.
Appl. Prob. 13, 429-452.
and Convergence.
[3] ETHIER, S. N. and KURTZ, T. G. (1986) Markov Processes:
Wiley, New York.
[4] ETHIER, S. N. and KURTZ, T. G. (1987) The infinitely many alleles model with selection as ameasure-valued diffusion. Lect. Notes in Biomath. 70, 72-86.
[5] FLEMING, W. H. and VIOT, M. (1979) Some measure-valued Markov processes in population genetics theory. Indiana Univ. Math. J. 28, 817-843.
Fleming-Viot processes. Preprint (submitted to journals of Applied[6] HIRABA, S. (1999)
Trust),
Prob.
1-19.
[7] IKEDA, N. and WATANABE, S. (1989) Stochastic Differential Equations and Diffiasion Processes.
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2nd ed.
[8] PERKINS, E. A. (1992) Conditional Dawson-Watanabe processes and Fleming-Viot processes. Seminar on Stochastic Processes 1991, Birkhauser, 142-155.
[9] SHIGA, T. (1987) A certain class of infinite dimensional diffusion processes arising in population
genetics. J. Math. Soc. Japan 39, 17-25.
tial equations. em Lect. Notes in[10] WALSH, J. B. (1986) An introduction to stochastic partial
Math. 1180, 265-439.
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