A Support Problem for Superprocesses in Terms of Random Measure (Probability Symposium)
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(2) 109. §2.. Main result. $\mu$\in M_{F}(\mathbb{R}). For. \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}( $\mu$). the support of $\mu$ , say,. ,. is defined. \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}( $\mu$):=\{A\in B(\mathbb{R}): $\mu$(A^{c})=0\} While,. the. global support of. superprocess X. Gsupp(X) It is. by. (4). .. Gsupp(X). say,. :=\displaystyle\bigcup_{t\geq0}\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(X_{t}(dx). is defined. by. (5). .. key point that we relate the support Gsupp(X) of superprocess X_{t} in terms of locally finite measure L=L(dx) on \mathbb{R} to a nonlinear singular elliptic boundary problem. a. Let. d=1, a(x)>. differential operator. O.. We consider the associated. P=\displaystyle \frac{1}{2}a(x)\frac{d^{2} {dx^{2} +b(x)\frac{d}{dx}+c(x). boundary problem: for. ,. \left\{ begin{ar y}{l Pv= ^{2}(x)\frac{L(dx)}{dx},a_{1}<x a_{2}\ v(a_{1})=$\beta$_{1},v(a_{2})=$\beta$_{2}. \end{ar y}\right. When \infty ,. we. the. lution. denote the solution. problem (6). v(x). Moreover,. is. for. a. of(6) by v(x;$\beta$_{1}, $\beta$_{2}). possesses. continuous. a. unique solution. convex. (6). \exists\{$\beta$_{1}^{(n)}\}_{n}\nearrow\infty, \exists\{$\beta$_{2}^{(n)}\}_{n}\nearrow v(x;$\beta$_{1}^{(n)}, $\beta$_{2}^{(n)}) Note that the. ,. since. so‐. .. function defined. \forall a_{1}\leq x_{0}\leq x\leq a_{2}, v(x). a. on. the interval. I=[a_{1}, a_{2}].. satisfies. v(x)=v(x_{0})+$\Phi$_{0}(x_{0})(x-x_{0})+\displaystyle \int_{x0}^{x}$\Phi$_{1}(y)v(y)dy +\displaystyle \int_{x_{0} ^{x}dy\int_{x_{0} ^{y}$\Phi$_{2}(z)v(z)dz+\int_{x\mathrm{o} ^{x}dy\int_{x0}^{y}\frac{2v^{2}(z)}{a(z)}L(dz),\cdot. (7). where. $\Phi$_{0}(x)=v(x+)+\displaystyle \frac{2b(x)}{a(x)}, $\Phi$_{1}(x)=\frac{2b(x)}{a(x)}, $\Phi$_{2}(x)=\displaystyle \frac{2b(x)a'(x)-2b'(x)a(x)+2a(x)c(x)}{a(x)^{2} . Then. we can. obtain. an. explicit expression. C^{+}(\mathbb{R}) \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}( $\psi$)\subset(-K, K) ,. ,. $\theta$>0 , when. of the. we. approximate solution.. denote. by v_{K}(t, x; $\theta \psi$). For. $\psi$\in. the solution of. u(t, x)=0, x\in(-K, K)^{c}. u(t, x)= $\theta$\displaystyle \int_{0}^{t}\int_{-K}^{K}p_{K}(t-s, x, y) $\psi$(y)dyds -\displaystyle \int_{0}^{t}\int_{-K}^{K}p_{K}(t-s, x, y)u^{2}(s, y)L(dy)ds, x\in(-K, K). ,. (8).
(3) 110. then. a. simple fact v_{K}\geq 0 yields concurrently. to. v_{K}\nearrow \mathrm{i}\mathrm{n}t\hslash^{1} つ v_{K}\nearrow \mathrm{i}\mathrm{n} $\psi$. and. ,. furthermore it follows immediately that. v_{K}(t, x; $\theta \psi$)\displaystyle \leq\sup_{t,x}\int_{0}^{t}\int_{-K}^{K}p_{K}(t-s, x, y) $\theta \psi$(y)dyds<\infty. On the other tion. hand, v_{K}( $\theta$, t, x;a_{1}, a_{2}) replaced by $\psi$=1_{[a,a]^{c}}12^{\cdot}. For. simplicity,. b(x)=0, c(x)> perprocess. denote the solution of. henceforth that. we. assume. O.. We shall represent the. (8). with the test func‐. \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(X_{0})\subset[a_{1}, a_{2}]\subset(-K, K) positive support probability of. ,. su‐. X. by the solution of (6). The argument of Iscoe (1988) for occupation. time processes. \displaystyle\int_{0}^{t}X_{s}^{K}ds \displaystyle\int_{0}^{t}X_{s}ds E_{X_{0} ^{L}[\displaystyle \exp\{- $\theta$\int_{0}^{\infty}X_{S}^{K}([a_{1}, a_{2}]^{\mathrm{c} )ds\}] =\displaystyle \exp\{-\int_{-\infty}^{\infty}v_{K}( $\theta$, x;a_{1}, a_{2})X_{0} ( ) \} implies. or. that. (9). dx. holds. And besides. we. have. v_{K}( $\theta$, x;a_{1}, a_{2})=\displaystyle \lim_{t\rightar ow\infty}(\lim_{n\rightar ow\infty}v_{K}( $\theta \psi$_{n};t,x and v. is. we can a. deduce that. signed. v(x)\equiv v_{K}( $\theta$, x;a_{1}, a_{2}) satisfies x\in(-K, K). measure, and also that for. that its second derivative. ,. \displaystyle \frac{dv}{dx}(x\pm)=\int_{x_{0^{\backslash } ^{x\pm}\frac{2c(y)v(y)}{a(y)}dy+\int_{x}^{x\pm}0\frac{2v^{2}(y)}{a(y)}L(dy) -2$\theta$\displaystyle\int_{x_{0}^{x\pm}1_{[a_{1} a2]^{c}(y)dy+( ,. Thus the. representation. of. Constant). probability for the support. can. .. be derived.. P_{x0}^{L}(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(X_{t})\cap[a_{1}, a_{2}]^{c}=\emptyset, \foral t\geq 0). =\displaystyle \lim_{K\rightar ow\infty}P_{X_{0} ^{L}(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(X_{t}^{K})\cap[a_{1}, a_{2}]^{c}=\emptyset,\foral t\geq 0) \Leftar ow \mathrm{b}\mathrm{y}. virtue of the. right continuity of the path. X_{t}^{K}( $\omega$). =\displaystyle \lim_{K\rightar ow\infty}P_{X_{0} ^{L}(\int_{0}^{\infty}X_{s}^{K}([a_{1}, a_{2}]^{c})ds=0) \Leftar ow \mathrm{b}\mathrm{y}. the. expression of the occupation. time process. =\displaystyle \lim_{K\rightar ow\infty}\lim_{ $\theta$\rightar ow\infty}\exp\{-\int_{-\infty}^{\infty}v_{K}( $\theta$, x;a_{1}, a_{2})X_{0} ( ) \} =\displaystyle \lim_{n\rightar ow\infty}\exp\{-\int_{a1}^{a_{2} v(x;$\beta$_{1}^{(n)}, $\beta$_{2}^{(n)} X_{0} ( ) \}. (9). dx. dx. (10).
(4) 111. By. virtue of the above‐mentioned facts. get the following principal result,. we can. the theorem for compact support.. (Main Result). $\mu$\in M_{F}(\mathbb{R}) and \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}( $\mu$)\subset[a_{1}, a_{2}] Suppose that d=1, a(x)>0, b(x)=0, c(x)>0 For \forall $\epsilon$>0 ( $\epsilon$<<1 : sufficiently small), there exist proper real numbers \exists\underline{x}=\underline{x}( $\epsilon$)<a_{1}, \exists\overline{x}=\overline{x}( $\epsilon$)>a_{2} such that v is a nonnegative solution of (7) on the interval (\underline{x}, x i.e. v(x)\geq 0 for x\in(\underline{x}, x If v. Theorem 1.. Let. .. .. satisfies. the conditions. a\displaystyle \leq x\leq a\sup_{12}v(x)\leq $\epsilon$, \lim_{x\rightar ow\underline{x} v(x)=\lim_{x\rightar ow\overline{x} v(x)=\infty then the superprobess. §3.. X=\{X_{t}, t\geq 0\}. Formulation of superprocess Let. (11). ,. has the compact support.. by. admissible functional. by X=\{X_{t}, t\geq 0\} the measure‐valued branching process a locally finite random measure L\cdot and P_{ $\mu$}^{L} denotes the probability. denote. us. corresponding to. ,. law of the measure‐valued process X terms of random. measure. L is. given by. E_{ $\mu$}^{L}[e^{-\langle $\varphi$,X_{\mathrm{t} \rangle}]=e^{-\langle u(t), $\mu$)} Here the function. u(t, x). Then. .. satisfies the. the. a. measure‐valued process. following Laplace. (12). .. following Cauchy problem.. \left{\begin{ar y}{l \partil_{}u=P-\frac{L(dx)}{ u^{2},\ u(0,x)=$\varphi$\nC_{b}^+(\mathb{R}). \end{ar y}\right. Now, suggested by. a. (13). formulation. the above initial value we. by Dawson‐Fleischmann (1995), we shall consider problem as an integral equation. As a matter of fact, when. write the fundamental solution to the aforementioned. then. we. Cauchy problem by. This. means. shall. assume. that. [Assumption]. we. consider the mild solution. to the above. For any. method to. processes. by. .. Cauchy problem.. (14) We. henceforth:. c>0,. \displaystyle \int_{-\infty}^{\infty}e^{-cx^{2} L(dx)<\infty a. p,. have. u(t, x)=\displaystyle \int p(t, x, y) $\varphi$(y)dy-\int_{0}^{t}\int p(t-s, x, y)u^{2}(s, y)L(dy)ds. Recall. in. transition functional.. X_{0}= $\mu$\in M_{F}(\mathbb{R}). with. (X_{t}, P_{ $\mu$}^{L}). E.B.. ,. \mathrm{a} \mathrm{s} .. .. (15). apply admissible Brownian functional in the studies on super‐ Dynkin (1994). Roughly speaking, it is nothing but a special.
(5) 112. case. that the. branching. rate term $\gamma$ in the. super‐Brownian. Watanabe superprocess would be does not. \ell_{t,x}( $\omega$). always. the Dawson‐. or. additive functional which. changed general density. For a finite measure \tilde{L} on \mathbb{R} and a local time motion B_{s} we define the additive functional K_{t}^{[\overline{L}] ( $\omega$) by into. a. possess its. of Browninan. ,. K_{t}^{[\overline{L}] ( $\omega$) We shall impose the. :=\displaystyle \int P_{t,x}( $\omega$)\tilde{L} (. following admissible. [Dynkins Admissibility] (i) $\Pi$_{r,x}[K^{[\tilde{L}]}(r, t)]<\infty (ii) $\Pi$_{r,x}[K^{[\overline{L}]}(r, t)]\rightarrow 0 Theorem 2.. motion. For ,. a. for. Brownian motion. uniformly. in. (16). conditions.. \forall r<t,. (Dynkin, 1994) If. dx ).. (B_{t}, $\Pi$_{0,x}). ,. x. (r,t\rightarrow s). x. \forall s. junction \mathcal{P}(r, $\mu$;t, C)=P_{r, $\mu$}(X_{t}\in. the transition. C) satisfied the following two conditions, then the measure‐valued Markov process named ( $\xi$, K, $\psi$) ‐superprocess with parameters X=(X_{t}, P_{r, $\mu$}) can be determined.. \displaystyle \int \mathcal{P}(r, $\mu$;t, dv)e^{-\langle f, $\nu$\rangle}=\exp\{-\langle v(r) , $\mu$. (17). v(r, x)+$\Pi$_{r,x}\displaystyle \int_{r}^{t} $\psi$(s, v(s) ($\xi$_{s})dK_{s}=$\Pi$_{r,x}f($\xi$_{t}) §4.. Construction of sequence of. (\sim. In this section. sequence of finite. probability. we. This. space.. approximate measure‐valued. shall construct. measure. basic process. a. M_{F}(\mathbb{R}) ‐valued. provides. target superprocess. For each K\in \mathrm{N} ,. we. a. we. denote. by. \tilde{X}_{t}^{K}\equiv\tilde{X}_{t}^{K}(dx). construct this measure‐valued baisc process a. Markov process w_{K}. on. E_{K} starting. w_{K}(t):=(\{n\}, w(t)). at. for. ,. on. processes. limit of. the. increasing. common. basic our. put. a. \tilde{X}_{t}^{K}. (19). ,. M_{F}(E_{K}) ‐valued. an. a. proto‐type in the construction of. E_{K}:=\displaystyle \bigcup_{n=1}^{K}\{n\}\times(-n, n) and. as. processes realized. with. us. (18). .. process. We shall first of all. x\in(-n, n). in what follows. For. point (n, x). can. be defined. ,. as. 1\leq t\leq$\tau$_{n}. w_{K}($\tau$_{n}) :=(\displaystyle \{n+1\}, w($\tau$_{n})) , $\tau$_{n}=\inf\{t>0 : w(t)=\pm n\} where. w. is. a. w_{K} dies out we. P‐diffusion. finally. starting. at time $\tau$_{K}. .. at. a. Next. point. we. x. .. Notice that the stochastic process. consider. a. randam. measure. for. n\leq K.. define. L_{K}(\{n\}\times(a, b)):=L((-n, n)\cap(a, b. L_{K} In fact, ..
(6) 113. On this account,. making. use. we can. define the admissible additive functional. of this random. \mathcal{K}_{t}^{[L_{K}]}(w_{K}) where. \tilde{P}_{t,x}. is. a. :=\displaystyle \int\tilde{P}_{t,y}(w_{K})L_{K} ( dy). (20). given by. \displaystyle\tilde{\el}_{t,x}(w):=\lim_{$\epsilon$\downar ow0}\frac{1}{2$\epsilon$}\int_{0}^{t}1_{(a-$\epsilon$,a+\mathrm{e}) (w(s) ds Then. application. an. of the. previous Dynkins. this admissible additive functional. by. \tilde{X}_{t}^{K}=\tilde{X}_{t}^{K}(dx). .. \mathcal{K}_{t}^{[L_{K}]. (21). .. existence theorem. gives. us a. (Theorem 2). superprocess, which. we. That is to say,. (22). v(r, x)+$\Pi$_{r,x}^{P}^{\sim}\displaystyle \int_{r}^{t}v(s, w_{K}(s) ^{2}d\mathcal{K}_{t}^{[L_{K}] =$\Pi$_{r,x}^{P} $\varphi$(w_{K}(t) \sim we. shall construct. a new. approximate. by employing terization. Before constructing. sequence of. branching. the above‐mentioned process, and shall. processes. initial. the superprocess in. its initial value. We choose. measure as. candidate of the initial. for each subset B\subset \mathbb{R}. measure. we. for. our. a. question,. finite. we. measure. measure‐valued process. (23). .. measure‐valued. give. its charac‐. consider first the. $\mu$\in M_{F}(\mathbb{R}). \tilde{X}_{t}^{K}. .. For. process. it is the. \tilde{X}_{t}^{M}. case. of the number M\in \mathrm{N}. restricted to. of the process. \tilde{X}_{t}^{K}. .. a. In other. set. and. us now. we. denote. put E_{\infty}. by. a new. (24). M>K , the law of the is. equivalent. to the law. for. ,. \forall M>K.. P_{X_{0} ^{L,K} the probability law of the measure‐valued process \tilde{X}^{K},. :=\displaystyle \bigcup_{n=1}^{\infty}\{n\}\times(-n, n). and. \{P_{X_{0} ^{L,K}\}_{K}. limit induces the law of. M_{F}((-K,. n\geq 1,. words,. Then note that since the law. projective. satisfying. E_{K}=\displaystyle \bigcup_{n=1}^{K}\{n\}\times(-n, n). \mathcal{L}(\tilde{X}_{t}^{M}[E_{K})=\mathcal{L}(\tilde{X}_{t}^{K}) Let. as \mathrm{a}. define. \tilde{X}_{0}^{K}(\{n\}\times B) := $\mu$(B\cap\{[n-1, n)\cup(-n, -n+1 Then, if. with. denote. E_{r,x}^{(L_{K})}e^{-\langle $\varphi$,X_{t}^{-K})}=\exp\{-\{v(r) , $\mu$ Next. by. ,. random variable. positive. \mathcal{K}_{t}^{[L_{K}]}(w_{K}). L_{K} i.e.. measure. \tilde{X}^{\infty} of. denotes. \tilde{X}^{K}. an. becomes. M(E_{\infty}) ‐valued process. K)) ‐valued process. X_{t}^{K}. M(E_{\infty}) ‐valued a. consistent. \tilde{X}^{\infty} Hence, .. process.. family, if. we. its. define. as. X_{t}^{K}(B) :=\displaystyle \sum_{n=1}^{K}\tilde{X}_{t}^{\infty}(\{n\}\times B). ,. (25).
(7) 114. then. an. increasing. Proposition. X_{t}^{K}. Then. .. (Characterization). 3.. X_{t}^{K} satisfies. be. a. obtained.. log‐Laplace function of. following. the. with. ,. function u_{K}(t, x) satisfies uniquely. the. for $\varphi$\in C_{0}(\mathbb{R}). u_{K}(t, x). Let. E_{X_{0}^{K} [e^{-\langle $\varphi$,X_{t}^{K}\rangle}]=e^{-\langle u(t), $\mu$)}K Moreover,. \{X_{t}^{K}(B)\}_{K}\nearrow \mathrm{i}\mathrm{s}. sequence of stochastic processes. X_{0}^{K}(dx)= $\mu$(dx). (26). .. follwoing integral equation:. the. ,. u_{K}(t, x)=\displaystyle \int_{-K}^{K}p_{K}(t, x, y) $\varphi$(y)dy -\displaystyle \int_{0}^{t}\int_{-K}^{K}p_{K}(t-s, x, y)u_{K}^{2}(s, y)L ( E[X_{t}^{K}(B)]=\displaystyle \int_{-K}^{K}\int_{B}p_{K}(t, x, y) $\mu$(dx)dy. dy ) ds. (27). ,. (28). ,. where. p_{K}(t,x, y). is the. fundamental. solution. of the. Dirichlet. boundary. value. prob‐. lem:. (29). \partial_{t}u-Pu=0, u|_{\partial(-K,K)}=0. §5. Existence of Therefore. $\mu$\in M_{F}(\mathbb{R}). superprocess in terms of finite. M_{F}(\mathbb{R}) ‐valued. can. be defined. by. process. the. which represents. a. X=\{X_{t}, t\geq 0\}. following. X_{t}. with the initial. measure. limit. :=\displaystyle \lim_{K\rightar ow\infty}X_{t}^{K} ( dx ).. X_{t}(dx) We call this stochastic process. measure. a. random media.. (30). superprocess in terms of randam. Next. we. shall extend. p_{K}(t, \cdot, \cdot). measure. L. onto \mathbb{R}\times \mathbb{R}.. Namely,. p_{K}(t, x, y)=0 Then,. since. p_{K}(t, \cdot, \cdot)\nearrow p(t,. \cdot. ,. we. if. x or. may. y\not\in(-K, K). .. apply the monotone convergence theorem. to obtain. E[X_{t}(B)]=\displaystyle \int_{-\infty}^{\infty}\int_{B}p(t, x, y) $\mu$(dx)dy \foral B\in B(\mathbb{R}) On the other functions cesses. hand,. since. \{u_{K}(t, \cdot)\}_{K}. is also. have. \{X_{t}^{K}(\cdot)\}_{K}\nearrow \mathrm{i}\mathrm{n}K. .. again,. X_{t}(dx). ,. the sequence of. \log‐Laplace. associated with the sequence of those measure‐valued pro‐. increasing \nearrow \mathrm{i}\mathrm{n}K As. vergence theorem. limit process. we. (31). .. can. the. a. consequence,. \log‐Laplace. also be obtained. function. by using. u(t, x). the monotone. con‐. of the above‐mentioned. by. u(t, x)=\displaystyle \lim_{K\rightarrow\infty}u_{K}(t, x). .. (32).
(8) 115. Finally,. application of. an. the monotone convergence theorem. again. leads to the. following:. u(t, x)=\displaystyle \lim_{K\rightarrow\infty}u_{K}(t, x). =\displaystyle \lim_{K\rightar ow\infty}\int_{-K}^{K}p_{K}(t, x, y) $\varphi$(y)dy -\displaystyle \lim_{K\rightar ow\infty}\int_{0}^{t}\int_{-K_{\backslash } ^{K}p_{K}(t-s, x, y)u_{K}^{2}(s, y)L(dy)ds =\displaystyle \int_{-\infty}^{\infty}p(t, x, y) $\varphi$(y)dy-\int_{0}^{t}\int_{-\infty}^{\infty}p(t-s, x, y)u^{2}(s, y)L(dy)ds Remark. It is. interesting. to note that the above construction. local finiteness of the random. measure. L(dx). (33). .. requires. us. only. .. Acknowledgements. This work is supported in part by Japan MEXT Grant‐in‐ Aids SR(C) 24540114 and also by ISM Coop.Res. Program: 2016‐ISM‐CRP‐50II. References.. [1] Dawson, sions with. D. and. Fleischmann,. absolutely. continuous. K.. :. Super‐Brownian. measure. states.. higher. dimen‐. J. Theoret. Probab. 8. (1995),. motions in. 179‐206.. 2] Dôku,. I.. :. A certain class of. immigration. superprocesses and its limit theorem.. Appl. Stat. 6 (2006), 145‐205. [3] Dôku, I. : A limit theorem of superprocesses with non‐vanishing deterministic immigration. Sci. Math. Japn. 64 (2006), 563‐579. [4] Dôku, I. : A limit theorem of homogeneous superprocesses with spatially de‐ pendent parameters. Far East J. Math. Sci. 38 (2010), 1‐38. [5] Dôku, I. : Tumour.immunoreaction and environment‐dependent models. Trans. Japn. Soc. Indu. Appl. Math. 26 (2016), 213‐252. [6] Dynkin, E. B. : An Introduction to Branching Measure‐Valued Processes. Amer. Math. Soc., Providence, RI, 1994. [7] Iscoe, I. : On the supports of measure‐valued critical branching Brownian Adv.. motion. Ann. Probab. 16. (1988),. 200‐221.. Department of Mathematics Faculty of Education Saitama. University. 338‐8570 Saitama. JAPAN E‐‐mail:. [email protected]‐u.ac.jp. \mathrm{f}\S_{\mathrm{Q} \mathrm{f}\backslash$\lambda$^{\backslash}\mathrm{F}^{\backslash}\cdot\mathrm{a}_{\mathrm{s}-$\beta$\mathscr{X}^{\backslash}\not\equiv^{\backslash}\mathscr{X}\subsetneq }^{\mathrm{R}\frac{\backslash\backslash}{\neq}\mathrm{k} \supset. \grave{}\ovalbox{\t smal REJ CT} 工. J. \ovalbox{\t smal REJ CT}.
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