Content-Type: multipart/mixed; boundary="-------------0805030243107" This is a multi-part message in MIME format. ---------------0805030243107 Content-Type: text/plain; name="08-88.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="08-88.keywords" central limit theorem, linear systems, binary contact path process, diffusive behavior, delocalization. ---------------0805030243107 Content-Type: application/x-tex; name="lclt" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="lclt" %%%%%%%%%%%%%%%%%% \documentclass[11pt]{article} %%%%%%%%%%%%%%%%%%% show label, keys, .. %% \usepackage{showkeys} \usepackage{graphicx} \usepackage{epsfig} \addtolength{\topmargin}{-10ex} \addtolength{\topskip}{0pt} \setlength{\oddsidemargin}{0pt} %\setlength{\evensidemargin}{15pt} \addtolength{\textwidth}{80pt} \addtolength{\textheight}{130pt} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % ====== List of New Commands ====== % % % 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\newcommand{\ovZ}{\ov{Z}} \newcommand{\ovn}{\ov{N}} \newcommand{\ovN}{\ov{N}} \makeatletter \def\section{\@startsection{section}{1}{\z@}{-3.5ex plus -1ex minus -.2ex}{2.3ex plus .2ex}{\bf}} \def\subsection{\@startsection{subsection}{2}{\z@}{-3.25ex plus -1ex minus -.2ex}{1.5ex plus .2ex}{\bf}} \makeatother \newcommand{\cvlaw}{\stackrel{\rm{ law}}{\longrightarrow}} \newcommand{\eqlaw}{\stackrel{\rm{ law}}{=}} %%%%%%%%%%%%%%% added n0v 19 \newcommand{\bz}{\bar \zeta} \newcommand{\md}{\mu^{(2)}_\8} \newcommand{\IW}{{\mathbb{W}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % BEGINNING OF TEXT % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \bcenter \large{\bf Central Limit Theorem for a Class of Linear Systems}\footnote{\today} \vvs \normalsize \noindent Yukio NAGAHATA and Nobuo YOSHIDA\footnote{ Supported in part by JSPS Grant-in-Aid for Scientific Research, Kiban (C) 17540112}\\ \ecenter \begin{abstract} We consider a class of interacting particle systems with values in $[0,\8)^{\zd}$, of which the binary contact path process is an example. For $d \ge 3$ and under a certain square integrability condition on the total number of the particles, we prove a central limit theorem for the density of the particles, together with upper bounds for the density of the most populated site and the replica overlap. \end{abstract} \small \noindent Abbreviated Title: CLT for Linear Systems.\\ AMS 2000 subject classification : Primary 60K35; secondary 60F05, 60J25. \\ Key words and phrases: central limit theorem, linear systems, binary contact path process, diffusive behavior, delocalization. %%%%%%%%%%% \tableofcontents %%%%% \normalsize %%%%%% \SSC{Introduction} %%%%%% We write $\N=\{0,1,2,...\}$, $\N^*=\{1,2,...\}$ and $\Z=\{ \pm x \; ; \; x \in \N \}$. For $x=(x_1,..,x_d) \in \rd$, $|x|$ stands for the $\ell^1$-norm: $|x|=\sum_{i=1}^d|x_i|$. For $\h=(\h_x)_{x \in \zd} \in \R^{\zd}$, $|\h |=\sum_{x \in \zd}|\h_x|$. Let $(\W, \cF, P)$ be a probability space. We write $P[X]=\int X \; dP$ and $P[X:A]=\int_A X \; dP$ for a r.v.(random variable) $X$ and an event $A$. %%%%%%%%% \subsection{The binary contact path process (BCPP)} %%%%%%%%%%%%%% We start with a motivating simple example. Let $\h_t =(\h_{t,x})_{x \in \zd} \in \N^{\zd}$, $t \ge 0$ be binary contact path process (BCPP for short) with parameter $\lm>0$. Roughly speaking, the BCPP is an extended version of the basic contact process, in which not only the presence/absence of the particles at each site, but also their number is considered. The BCPP was originally introduced by D. Griffeath \cite{Gri83}. Here, we explain the process following the formulation in the book of T. Liggett \cite[Chapter IX]{Lig85}. Let $N^z=(N^z_t)_{t \ge 0}$, ($z \in \zd$) be independent rate one Poisson processes. We suppose that the process $(\h_t)$ starts from a deterministic configuration $\h_0=(\h_{0,x})_{x \in \zd}\in \N^{\zd}$ with $|\h_0|<\8$. At the $i$-th jump time $t=T^{z,i}$ of $N^z$, $\h_{t-}$ is replaced by $\h_t$ randomly as follows: for each $e \in \zd$ with $|e|=1$, $$ \h_{t,x}=\lef\{ \barray{ll} \h_{t-,x}+\h_{t-,z} & \mbox{if $x = z+e$,}\\ \h_{t-,x} & \mbox{if otherwise} \\ \earray \ri.\; \; \mbox{with probability ${\lm \over 2d\lm +1}$,} $$ (all the particles at site $z$ are duplicated and added to those on the site $z=x+e$), and $$ \h_{t,x}=\lef\{ \barray{ll} 0 & \mbox{if $x=z$}, \\ \h_{t-,x} & \mbox{if $x \neq z$} \earray \ri.\; \; \mbox{with probability ${1 \over 2d\lm +1}$} $$ (all the particles at site $z$ disappear). The replacement occurs independently for different $(z,i)$ and independently from the Poisson processes. A motivation to study the BCPP comes from the fact that the projected process $$ \lef( \h_{t,x} \wedge 1\ri)_{x \in \zd},\; \; \; t \ge 0 $$ is the basic contact process \cite{Gri83}. Let $$ \kp_p={2d\lm -1 \over 2d\lm +1}\; \; \; \mbox{and}\; \; \; \ov{\h}_t=(\exp(-\kp_1t)\h_{t,x})_{x \in \zd}. $$ Then, $(|\ov{\h}_t|)_{t \ge 0}$ is a martingale and therefore, the following limit exists almost surely: $$ |\ov{\h}_\8|\st{\rm def}{=} \lim_t|\ov{\h}_t|. $$ Moreover, $P (|\ov{\h}_\8| >0)>0$ if \bdnl{|h_8|>0} \mbox{$d \ge 3$ and $\lm > {1 \over 2d (1-2\pi_d)}$,} \edn where $\pi_d$ is the return probability for the simple random walk on $\zd$ \cite{Gri83}. It is known that $\pi_d \le \pi_3=0.3405...$ for $d \ge 3$ \cite[page 103]{Spi76}. %%%%%%%%%%%%%%%%%%%%%%% %We now look at the density: %$$ %\rh_{t,x}={\h_{t,x} \over |\h_t|}. %$$ %%%%%% We denote the density of the particles by: \bdnl{rh} \rh_{t,x}=\frac{\h_{t,x}}{|\h_t|} =\frac{\ov{\h}_{t,x}}{|\ov{\h}_t|}, \; \; t >0, x \in \zd. \edn Interesting objects related to the density would be \bdnl{rh^*} \rh^*_t=\max_{x \in \zd}\rh_{t,x}, \; \; \mbox{and}\; \; \cR_t=\sum_{x \in \zd}\rh_{t,x}^{2}. \edn $\rh^*_t$ is the density at the most populated site, while $\cR_t$ is the probability that a given pair of particles at time $t$ are at the same site. We call $\cR_t$ the {\it replica overlap}, in analogy with the spin glass theory. Clearly, $(\rh^*_t)^{2} \le \cR_t \le \rh^*_t$. These quantities convey information on localization/delocalization of the particles. Roughly speaking, large values of $\rh^*_t$ or $\cR_t$ indicates that the most of the particles are concentrated on small numbers of ``favorite sites" ({\it localization}), whereas small values of them implies that the particles are spread out over large number of sites ({\it delocalization}). As a special case of \Cor{CLT} below, we have the following result, which shows the diffusive behavior and the delocalization of the BCPP under the condition (\ref{|h_8|>0}): %%%%%%%%%% \Theorem{BCPP} %%%%%%% Suppose (\ref{|h_8|>0}). Then, for any $f \in C_{\rm b}(\rd)$, $$ \lim_{t \ra \8} \sum_{x \in \zd} f\lef(x/\sqrt{t}\ri)\rh_{t,x} =\int_{\rd}fd\n\; \; \; \mbox{in $P(\; \cdot \; | |\ov{\h}_\8|>0)$-probability,} $$ where $C_{\rm b} (\rd)$ stands for the set of bounded continuous functions on $\rd$, and $\n$ is the Gaussian measure with $$ \int_{\rd}x_id\n (x)=0, \; \; \; \int_{\rd}x_ix_jd\n (x)={\lm \over 2d\lm +1}\del_{ij},\; \; \; i,j=1,..,d. $$ Furthermore, $$ \cR_t=\cO (t^{-d/2})\; \; \; \mbox{as $t \nearrow \8$ in $P(\; \cdot \; | |\ov{\h}_\8|>0)$-probability}. $$ %%%%%%% \end{theorem} %%%%% \subsection{The results} %%%%%% We generalize \Thm{BCPP} to a certain class of linear interacting particle systems with values in $[0,\8)^{\zd}$\cite[Chapter IX]{Lig85}. Recall that the particles in BCPP either die, or make binary branching. To describe more general ``branching mechanism", we introduce a random vector $K=(K_x)_{x \in \zd }$ which is bounded and of finite range in the sense that \bdnl{K_x} K_x \lef\{\barray{l} \in [0,b_K] \; \; \mbox{a.s. for all $x \in \zd$,}\\ =0, \; \; \mbox{a.s. if $|x| >r_K$} \earray \rig.\; \; \mbox{for some non-random $b_K,r_K \in [0,\8)$.} \edn Let $N^z=(N^z_t)_{t \ge 0}$, ($z \in \zd$) be independent rate one Poisson processes, and let $K^{z,i}=(K_x^{z,i})_{x \in \zd }$ ($z \in \zd$, $i \in \N^*$) be i.i.d. random vectors with the same distributions as $K$, independent of $\{N^z\}_{z \in \zd}$. We suppose that the process $(\h_t)_{t \ge 0}$ starts from a deterministic configuration $\h_0=(\h_{0,x})_{x \in \zd} \in [0,\8)^{\zd}$ with $|\h_0|<\8$. At the $i$-th jump time $t=T^{z,i}$ of $N^z$, $\h_{t-}$ is replaced by $\h_t$, where \bdnl{h_(x,t)} \h_{t,x}=\lef\{ \barray{ll} K^{z,i}_0\h_{t-,z} & \mbox{if $x=z$}, \\ \h_{t-,x}+K^{z,i}_{x-z}\h_{t-,z} & \mbox{if $x \neq z$}. \earray \ri. \edn The BCPP is a special case of this set-up, in which \bdnl{binK} K= \lef\{ \barray{ll} 0 & \mbox{with probability ${1 \over 2d\lm +1}$} \\ \lef(\del_{x,0}+\del_{x,e} \rig)_{x \in \zd} & \mbox{with probability ${\lm \over 2d\lm +1}$, for each $2d$ neighbour $e$ of 0.} \earray \rig. \edn %%%%%%%% %As compared with BCPP where only binary branching occurs, %we allow more general, possibly random ``branching". %%%%%%% It is convenient to introduce the following matrix representation of the linear transformation $\h_{t-} \mapsto \h_t$: $$ \h_{t,x}=\sum_{y \in \zd}A^{z,i}_{x,y}\h_{t-,y}, $$ where \bdnl{Miz} A^{z,i}_{x,y}=\lef\{ \barray{ll} 1+\del_{x,z}(K^{z,i}_0-1) & \mbox{if $x=y$}, \\ \del_{y,z}K^{z,i}_{x-y} & \mbox{if $x \neq y$}. \earray \ri. \edn %%%%%%%%%% %Letting %$$ %\tl{A}^{z,s}_{x,y}=\lef\{ \barray{ll} %A^{z,i}_{x,y} & \mbox{if $s=T^{z,i}$}, \\ %\del_{x,y} & %\mbox{if $s \neq T^{z,i}$}, %\earray \ri. %\; \; \; s \ge 0, %$$ %%%%%%% The precise definition of the process $(\h_t)_{t \ge 0}$ is then given by the following stochastic differential equation: \bdnl{sde} \h_{t,x}=\h_{0,x} +\sum_{z \in \zd}\int^t_0 \lef( \sum_{y \in \zd}\tl{A}^{z,s}_{x,y}\h_{s-,y}-\h_{s-,x}\ri)d N^z_s, \edn where $\tl{A}^{z,s}_{x,y}=A^{z,N^z_s}_{x,y}$. By (\ref{K_x}), it is standard to see that (\ref{sde}) defines a unique process $\h_t=(\h_{t,x})$, ($t \ge 0$) and that $(\h_t)$ is Markovian. We set \bdmn \kp_1&=&\sum_{x \in \zd}P[(K_x-\del_{x,0})^p] \; \; \; p=1,2,\label{kp12} \\ \ov{\h}_t &=&(\exp (-\kp_1t )\h_{t,x})_{x \in \zd}. \label{ovh_t} \edmn We will see that $(|\ov{\h}_t|)_{t \ge 0}$ is a martingale (\Lem{FK1} below) and therefore, the following limit exists almost surely: \bdnl{ovh_8} |\ov{\h}_\8|\st{\rm def}{=} \lim_t|\ov{\h}_t|. \edn To state \Thm{CLT}, We define \bdnl{G(x)} G(x)=\int^\8_0P_R^0(R_t=x)dt, \edn where $((R_t)_{t \ge 0}, P_R^x)$ is the continuous-time random walk on $\zd$ starting from $x \in \zd$, with the generator \bdnl{L_R} L_Rf (x)=\half \sum_{y \in \zd}\lef( P[K_{x-y}]+P[K_{y-x}] \ri) \lef( f(y)-f(x)\rig). \edn As before, $C_{\rm b} (\rd)$ stands for the set of bounded continuous functions on $\rd$. %%%%%%%%%%%% \Theorem{CLT} %%%%%% Suppose (\ref{K_x}) and that \bdmn & & \mbox{the set $\{x \in \zd\; ;\; P[K_x]\neq 0\}$ contains a linear basis of $\rd$,} \label{K1}\\ %%%%%%%%%%%%%%%%%%%%% %& & P[K_0K_x]=P[K_x] \; \; \mbox{for $x \in \zd \bsh \{ 0\}$,}\label{K3}\\ %%%%%%%%%%%%%%%%%%%%%% & & P[(K_x-\del_{x,0})(K_y-\del_{y,0})]=0 \; \; \mbox{for $x,y \in \zd$ with $x \neq y$.} \label{K4} \edmn Then, referring to (\ref{kp12})--(\ref{L_R}), the following are equivalent: \bds \item[(a)] ${\kp_2 \over 2}G(0)<1$, \item[(b)] ${\dps \sup_{t \ge 0} P[|\ov{\h}_t|^2]<\8}$, \item[(c)] ${\dps \lim_{t \ra \8} \sum_{x \in \zd} f\lef((x -mt)/\sqrt{t}\ri)\ov{\h}_{t,x}=|\ov{\h}_\8|\int_{\rd}fd\n}$ in $\bL^2 (P)$ for all $f \in C_{\rm b} (\rd)$, \eds where $ m=\sum_{x \in \zd}xP[K_x] \in \rd $ and $\n$ is the Gaussian measure with \bdnl{nu} \int_{\rd}x_id\n (x)=0, \; \; \; \int_{\rd}x_ix_jd\n (x) =\sum_{x \in \zd}(x_i-m_i)(x_j-m_j)P[K_x],\; \; \; i,j=1,..,d. \edn Moreover, if ${\kp_2 \over 2}G(0)<1$, then, there exists $C \in (0,\8)$ such that \bdnl{RepDec} \sum_{x, \tl{x}\in \zd}f(x-\tl{x})P[\ov{\h}_{x,t}\ov{\h}_{\tl{x},t}] \le Ct^{-d/2}|\h_0|^2\sum_{x \in \zd}f(x) \edn for all $t>0$ and $f :\zd \ra [0,\8)$ with $\sum_{x \in \zd}f(x)<\8$. %%%%%%%% \end{theorem} %%%%%%% The proof of \Thm{CLT} will be presented in section \ref{pCLT}. \vvs \noindent {\bf Remarks:} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\bf 1)} The condition (\ref{K1}) guarantees a reasonable non-degeneracy for the transition mechanism (\ref{h_(x,t)}). On the other hand, (\ref{K4}) amounts to saying that the transition mechanism (\ref{h_(x,t)}) updates the configuration by ``at most one coordinate at a time". Here is an explanation for the relevance of the condition (\ref{K4}). To prove \Thm{CLT}, we use a certain Markov chain on $\zd \times \zd$, which is introduced in \Lem{FK2} below. Thanks to (\ref{K4}), the Markov chain is stationary with respect to the counting measure on $\zd \times \zd$. The stationarity plays an important role in the proof of \Thm{CLT}-- see \Lem{FK3} below. %%%%%%%%%%%%%%%%%%%%%% \vs \noindent{\bf 2)} %%%%%%%%%%%%%%%%%%%5 Because of (\ref{K1}), the random walk $(R_t)$ is recurrent for $d=1,2$ and transient for $d \ge 3$. Therefore, ${\kp_2 \over 2}G(0)<1$ is possible only if $d \ge 3$. As will be explained in the proof, ${\kp_2 \over 2}G(0)<1$ is equivalent to $$ P_R^0\lef[\exp \lef( {\kp_2 \over 2}\int^\8_0 \del_0 (R_t)dt\ri) \ri]<\8. $$ %%%%%%%%%%%%%%%%%%%%%%5 \noindent{\bf 3)} If, in particular, \bdnl{SRW} P[K_x] = \lef\{ \barray{ll} c >0 & \mbox{for $|x|=1$,} \\ 0 & \mbox{for $|x| \ge 2$,} \earray \rig. \edn then, $(R_t)_{t \ge 0} \st{\mbox{\scriptsize law}}{=} (\widehat{R}_{2dct})_{t \ge 0}$, where $(\widehat{R}_\cdot)$ is the simple random walk. Therefore, the condition (a) becomes \bdnl{Khas2} {\kp_2 \over 4dc(1-\pi_d)}<1. \edn By (\ref{binK}), the BCPP satisfies (\ref{K1})--(\ref{K4}). Furthermore, $\kp_2=1$ and we have (\ref{SRW}) with $c={\lm \over 2d\lm +1}$. Therefore, (\ref{Khas2}) is equivalent to (\ref{|h_8|>0}). \vs %%%%%%%%%%%%%%%%%%% \noindent{\bf 4)} %%%%%%%%%%%%%%%%%% The proof of \Thm{CLT} is roughly divided into two steps; \\ \bds \item[(i)] to represent the two-point function $P[\h_{t,x}\h_{t,\tl{x}}]$ in terms of a continuous-time Markov chain on $\zd \times \zd$ via the Feynman-Kac formula (\Lem{FK2} and \Lem{FK3} below), \item[(ii)] to show the central limit theorem for the ``weighted" Markov chain, where the weight comes from the additive functional due to the Feynman-Kac formula (\Lem{perturb2} below). \eds The above strategy was adopted earlier by one of the authors for branching random walk in random environment \cite{Yos08}. There, the Markov chain alluded to above is simply the product of simple random walks on $\zd$, so that the central limit theorem with the Feynman-Kac weight is relatively easy. Since the Markov chain in the present paper is no longer a random walk, it requires more work. However, the good news here is that the Markov chain we have to work on is ``close" to a random walk. In fact, we get the central limit theorem by perturbation from that for a random walk case. \vs %%%%%%%%%%%%%%%%%%% \noindent{\bf 5)} %%%%%%%%%%%%%%%%%% In connection to 4) above, D. Griffeath discussed a Feynman-Kac formula for $$ \sum_{y \in \zd}P[\h_{t,x}\h_{t,\tl{x}+y}] $$ in the case of BCPP (cf. proof of Theorem 1 in \cite{Gri83}). In the present paper, we have the formula for each summand of the above summation, so that finer information is available. \vs %%%%%%%%%%%%%%%%%%% \noindent{\bf 6)} %%%%%%%%%%%%%%%%%% The dual process of $(\h_t)$ above (in the sense of \cite[page 432]{Lig85}) is given by replacing (\ref{h_(x,t)}) by: \bdnl{h_(x,t)*} \h_{t,x}=\lef\{ \barray{ll} \sum_{y \in \zd}K^{z,i}_{y-x}\h_{t-,y} & \mbox{if $x=z$}, \\ \h_{t-,x} & \mbox{if $x \neq z$}. \earray \ri. \edn This amounts to replacing the matrix $(A^{z,i}_{x,y})_{x,y}$ by its transpose. As can be seen from the proofs, all the results in this paper remain true for the dual process. %%%%%%%%%%%%%%%%%%%%% \vvs We define the density and the replica overlap in the same way as (\ref{rh})--(\ref{rh^*}). Then, as an immediate consequence of \Thm{CLT}, we have the following %%%%%%%%%% \Corollary{CLT} %%%%%%% Suppose (\ref{K_x}), (\ref{K1})--(\ref{K4}) and that ${\kp_2 \over 2}G(0)<1$. Then, $P (|\ov{\h}_\8|>0)>0$ and for all $f \in C_{\rm b} (\rd)$, $$ \lim_{t \ra \8} \sum_{x \in \zd} f\lef((x -mt)/ \sqrt{t}\ri)\rh_{t,x} =\int_{\rd}fd\n\; \; \; \mbox{in $P(\; \cdot \; | |\ov{\h}_\8|>0)$-probability,} $$ where $ m=\sum_{x \in \zd}xP[K_x] \in \rd $ and $\n$ is the same Gaussian measure defined by (\ref{nu}). Furthermore, $$ \cR_t=\cO (t^{-d/2})\; \; \; \mbox{as $t \nearrow \8$ in $P(\; \cdot \; | |\ov{\h}_\8|>0)$-probability}. $$ %%%%%%% \end{corollary} %%%% Proof: The first statement is immediate from \Thm{CLT}(c). Taking $f(x)=\del_{x,0}$ in (\ref{RepDec}), we see that $$ P[\sum_{x \in \zd}\ov{\h}_{x,t}^2 ]\le Ct^{-d/2}|\h_0|^2\; \; \; \mbox{for $t >0$}. $$ This implies the second statement. \hfill $\Box$ %%%%%%%%%%%%%%%% \vvs For $a \in \zd$, let $\h_t^a$ be the process starting from $\h_0 =(\del_{a,x})_{x \in \zd}$. As a by-product of \Thm{CLT}, we have the following formula for the covariance of $(|\ov{\h}_\8^a|)_{a \in \zd}$. For BCPP, this formula was obtained by D. Griffeath as the main result of \cite{Gri83} (cf. Theorem 1 in that paper). %%%%%%%%%%%% %We define %$\ov{\h}_t^a$ and $|\ov{\h}_\8^a|$ exactly in the same way as %(\ref{ovh_t}) and (\ref{ovh_8}) ($|\ov{\h}_\8^a|$ exists, %since $(|\ov{\h}_t^a|)_{t \ge 0}$ is a martingale). %%%%%%%%%%%%% \Theorem{cov} %%%%%%%%% Suppose (\ref{K_x}), (\ref{K1})--(\ref{K4}) and that ${\kp_2 \over 2}G(0)<1$. Then, $$ P[|\ov{\h}_\8^a||\ov{\h}_\8^b|]=1+{\kp_2G(a-b) \over 2-\kp_2G(0)}, \; \; \; a,b \in \zd. $$ %%%%%%%%% \end{theorem} %%%%%%%%%%% The proof of \Thm{cov} will be presented in section \ref{pcov}. %%%%%%%%%%%%%%%%% %Proof: The first statement is immediate from \Thm{CLT}(c). Taking %$f(x)=\del_{x,0}$ in (\ref{RepDec}), we see that %$$ %P[\sum_{x \in \zd}\ov{\h}_{x,t}^2 ]\le Ct^{-d/2}|\h_0|^2\; \; \; %\mbox{for $t >0$}. %$$ %This implies the second statement. %\hfill $\Box$ %%%%%%%%%%%%%%%% %%%%%%%%% \SSC{Lemmas} %%%%%%%%% \subsection{Markov chain representations for the point functions} %%%%%%%%%%%%% We assume (\ref{K_x}) throughout, but not (\ref{K1})--(\ref{K4}) for the moment. We start with the Feynman-Kac formula for one-point function: %%%%%%%%%%%%% \Lemma{FK1} %%%%%%%%% \bdnl{FK1} P[\h_{t,x}]=e^{\kp_1t}P^x[\h_{0,S_t}], \; \; (t,x)\in [0,\8) \times \zd, \edn where $\kp_1$ is defined by (\ref{kp12}) and $((S_t)_{t \ge 0}, P^x)$ is the continuous-time random walk on $\zd$ with the generator $$ L_S f (x)= \sum_{y \in \zd}P[K_{x-y}] \lef( f(y)-f(x)\rig). $$ As a consequence, $(|\ov{\h}_t|)_{t \ge 0}$ (cf. (\ref{ovh_t})) is a martingale. %%%%%%%%%% \end{lemma} %%%%%%% Proof: It is easy to see that, \bds \item[(1)] \hspace{1cm} ${\dps \gm_{x,y} \st{\rm def}{=} \sum_{z \in \zd}\lef( P[A^{z,i}_{x,y}]-\del_{x,y}\ri)= P[K_{x-y}]-\del_{x,y},}$ for $x,y\in \zd$, \item[(2)] \hspace{1cm} ${\dps \kp_1=\sum_{x \in \zd}P[K_x]-1=\sum_{y \in \zd}\gm_{x,y}}$, (cf. (\ref{kp12})). \eds We show that $u(t,x)\st{\rm def}{=}P[\h_{t,x}]$ solves the integral equation \bds \item[(3)] \hspace{1cm} ${\dps u(t,x)-u(0,x)=\int^t_0( L_S +\kp_1)u(s,x)ds,}$ \eds and that \bds \item[(4)] \hspace{1cm} ${\dps \sup_{t \in [0,T]}\sup_{x \in \zd}|u(t,x)|<\8}$ for any $T \in (0,\8)$. \eds To show (3), we average (\ref{sde}): \bdnn u(t,x)-u(0,x) &=&\sum_{z \in \zd}\int^t_0 \lef( \sum_{y \in \zd}P[A^{z,1}_{x,y}]u(s,y)-u(s,x)\ri)ds \\ &=& \int^t_0 \sum_{y \in \zd}\sum_{z \in \zd} \lef(P[A^{z,1}_{x,y}]-\del_{x,y}\ri)u(s,y)ds \\ &\st{\mbox{\scriptsize (1)}}{=}& \int^t_0 \sum_{y \in \zd}\gm_{xy}u(s,y)ds \\ &\st{\mbox{\scriptsize (2)}}{=}& \lef( \sum_{y \in \zd}\gm_{xy}(u(s,y)-u(s,x))+\kp_1 u(s,x) \ri)ds =\mbox{RHS of (3)}. \ednn On the other hand, we have by (\ref{K_x}) and (\ref{sde}) that, for any $p \in \N^*$, there exists $C_1 \in (0,\8)$ such that $$ P[\h_{t,x}^p] \le C_1\sum_{y:|x-y| \le r_K}\int^t_0 P[\h_{s,y}^p]ds, \; \; \; t \ge 0. $$ By iteration, we see that there exists $C_2 \in (0,\8)$ such that \bdnl{sde2} P[\h_{t,x}^p] \le e^{C_2t}\sum_{y \in \zd}e^{-|x-y|}(1+\h_{0,y}^p), \; \; \; t \ge 0, \edn which implies (4). \\ The solution to (3) subject to (4) is unique, for each given $\h_0$. This can be seen by using Gronwall's inequality with respect to the norm $\| u\|=\sum_{x \in \zd}e^{-|x|}|u(x)|$. Moreover, the RHS of (\ref{FK1}) is a solution to (3) subject to the bound (4). This can be seen by adapting the argument in \cite[page 5,Theorem 1.1]{Szn98}. Therefore, we get (\ref{FK1}). Now, $$ P[|\ov{\h}_t|]\st{\mbox{\scriptsize (\ref{FK1})}}{=} \sum_{x \in \zd}P^x[\h_{0,S_t}] =\sum_{x,y \in \zd}\h_{0,y}P^x[S_t=y] =\sum_{y \in \zd}\h_{0,y}=|\h_0|. $$ Then, from this and the Markov property of $(\h_t)$, we see that $|\ov{\h}_t|$ is a martingale. \hfill $\Box$ %%%%%%%%%%% \vvs To prove the Feynman-Kac formula for two-point function, we prepare: %%%%%%%%%%%%% \Lemma{Gm} %%%%%%%%% For $x,y,\tl{x},\tl{y} \in \zd$, \bdmn \Gm_{x,\tl{x},y,\tl{y}} & \st{\rm def}{=}& \sum_{z \in \zd}\lef( P[A^{z,i}_{x,y}A^{z,i}_{\tl{x},\tl{y}}] -\del_{x,y}\del_{\tl{x},\tl{y}}\ri) \nn \\ &=& \lef\{ \barray{ll} P[K_{x-y}K_{\tl{x}-\tl{y}}]\del_{y,\tl{y}} & \mbox{if $x \neq y$ and $\tl{x} \neq \tl{y}$,}\\ P[K_{\tl{x}-\tl{y}}]+ P[(K_0-1)K_{\tl{x}-\tl{y}}]\del_{y,\tl{y}} & \mbox{if $x=y$ and $\tl{x} \neq \tl{y}$,}\\ P[K_{x-y}]+ P[(K_0-1)K_{x-y}]\del_{y,\tl{y}} & \mbox{if $x \neq y$ and $\tl{x} =\tl{y}$,}\\ 2P[K_0-1]+\del_{x,\tl{x}}P[(K_0-1)^2] & \mbox{if $x=y$ and $\tl{x}=\tl{y}$.} \earray \rig. \label{Gm2} \\ V(x,\tl{x}) & \st{\rm def}{=}& \sum_{y,\tl{y} \in \zd}\Gm_{x,\tl{x},y,\tl{y}} =2\kp_1 + \sum_{y \in \zd }P[(K_{x-y}-\del_{x,y})(K_{\tl{x}-y}-\del_{\tl{x},y})], \label{C_Gm2} \edmn %%%%%%%% \end{lemma} %%%%%%%%% Proof: Direct computations. \hfill $\Box$ %%%%%%%%%%% \vvs We have the Feynman-Kac formula for two-point function: %%%%%%%%%%%%% \Lemma{FK2} %%%%%%%%% Let $(X,\tl{X})=((X_t, \tl{X}_t)_{t \ge 0}, P_{X,\tl{X}}^{x,\tl{x}})$ be the continuous-time Markov chain on $\zd \times \zd$ starting from $(x,\tl{x})$, with the generator $$ L_{X,\tl{X}}f (x,\tl{x})=\sum_{y,\tl{y} \in \zd} \Gm_{x,\tl{x},y,\tl{y}} \lef( f(y,\tl{y})-f(x,\tl{x})\rig), $$ where $\Gm_{x,\tl{x},y,\tl{y}}$ is defined by (\ref{Gm2}). Then, for $(t,x,\tl{x})\in [0,\8) \times \zd \times \zd$, %%%% %\bdnl{FK2} %%%% $$ P[\h_{t,x}\h_{t,\tl{x}}]=P^{x,\tl{x}}_{X,\tl{X}} \lef[ \exp \lef( \int^t_0 V(X_s,\tl{X}_s)ds\ri) \h_{0,X_t}\h_{0,\tl{X}_t} \ri], $$ where $V$ is defined by (\ref{C_Gm2}). %%%%%%%%%% \end{lemma} %%%%%%% Proof: We show that $u(t,x,\tl{x})\st{\rm def}{=}P[\h_{t,x}\h_{t,\tl{x}}]$ solves the integral equation \bds \item[(1)] \hspace{1cm} ${\dps u(t,x,\tl{x})-u(0,x,\tl{x})= \int^t_0( L_{X,\tl{X}}+V(x,\tl{x}))u(s,x,\tl{x})ds.}$ \eds Then, the lemma can be obtained in the same way as in \Lem{FK1}. By (\ref{sde}), we have $$ \h_{t,x}\h_{t,\tl{x}}-\h_{0,x}\h_{0,\tl{x}} =\sum_{z \in \zd}\int^t_0 \lef( \sum_{y, \tl{y}\in \zd} \tl{A}^{z,s}_{x,y}\tl{A}^{z,s}_{\tl{x},\tl{y}}\h_{s-,y}\h_{s-,\tl{y}} -\h_{s-,x}\h_{s-,\tl{x}}\ri)d N^z_s. $$ Therefore, \bdnn u(t,x,\tl{x})-u(0,x,\tl{x}) &=& \sum_{z \in \zd}\int^t_0 \lef( \sum_{y, \tl{y}\in \zd} P[A^{z,1}_{x,y}A^{z,1}_{\tl{x},\tl{y}}]u(s,y,\tl{y}) -u(s,x,\tl{x})\ri)ds \\ &=& \int^t_0 \sum_{y, \tl{y}\in \zd} \sum_{z \in \zd}\lef( P[A^{z,1}_{x,y}A^{z,1}_{\tl{x},\tl{y}}] -\del_{x,y}\del_{\tl{x},\tl{y}}\ri) u(s,y,\tl{y})ds \\ &=& \int^t_0 \sum_{y, \tl{y}\in \zd}\Gm_{x,\tl{x},y,\tl{y}} u(s,y,\tl{y})ds\\ &=& \int^t_0 \lef(\sum_{y, \tl{y}\in \zd}\Gm_{x,\tl{x},y,\tl{y}} ( u(s,y,\tl{y})-u(s,x,\tl{x}) )+V(x,\tl{x})u(s,x,\tl{x})\ri) ds \\ &=& \int^t_0( L_{X,\tl{X}}+V(x,\tl{x}))u(s,x,\tl{x})ds. \ednn \hfill $\Box$ %%%%%%%%%%%%%%%%%% \vvs We assume (\ref{K1})--(\ref{K4}) from here on. Then, (\ref{Gm2})--(\ref{C_Gm2}) are reduced to: \bdmn \Gm_{x,\tl{x},y,\tl{y}} &=& \lef\{ \barray{ll} P[K_{x-y}^2]\del_{x,\tl{x}} \del_{y,\tl{y}} & \mbox{if $x \neq y$ and $\tl{x} \neq \tl{y}$,}\\ P[K_{\tl{x}-\tl{y}}] & \mbox{if $x=y$ and $\tl{x} \neq \tl{y}$,}\\ P[K_{x-y}] & \mbox{if $x \neq y$ and $\tl{x} =\tl{y}$,}\\ 2P[K_0-1]+\del_{x,\tl{x}}P[(K_0-1)^2] & \mbox{if $x=y$ and $\tl{x}=\tl{y}$.} \earray \rig. \label{Gm} \\ V (x,\tl{x}) &=& 2\kp_1 +\kp_2 \del_{x,\tl{x}}, \; \; \; \mbox{(cf.(\ref{kp12})).} \label{C_Gm} \edmn It is easy to see from (\ref{Gm})--(\ref{C_Gm}) that \bdnl{stat} \sum_{y,\tl{y} \in \zd}\Gm_{x,\tl{x},y,\tl{y}} =\sum_{y,\tl{y} \in \zd}\Gm_{y,\tl{y},x,\tl{x}}=2\kp_1 +\kp_2 \del_{x,\tl{x}}, \edn which implies that $(X,\tl{X})$ is stationary with respect to the counting measure on $\zd \times \zd$. We denote the dual process of $(X,\tl{X})$ by $(Y,\tl{Y})=((Y_t,\tl{Y}_t)_{t \ge 0}, P_{Y,\tl{Y}}^{x,\tl{x}})$, that is, the continuous time Markov chain on $\zd \times \zd$ with the generator \bdnl{L_Y*} L_{Y,\tl{Y}}f (x,\tl{x})=\sum_{y,\tl{y} \in \zd} \Gm_{y,\tl{y},x,\tl{x}} \lef( f(y,\tl{y})-f(x,\tl{x})\rig). \edn Thanks to (\ref{stat}), $L_{X,\tl{X}}$ and $L_{Y,\tl{Y}}$ are dual operators on $\ell^2 (\zd \times \zd)$. %%%%%%%% %where $\Gm_{x,\tl{x},y,\tl{y}}$ is defined by (\ref{Gm2}). %\sum_{z \in \zd}\Gm_{x,0,y+z,z}&=&2P[K_{x-y}],\; \; \; %\mbox{if $x \neq y$.} %\label{Gm^(-)} %%%%%%%%%%%% %In particular, $\Gm_{x,\tl{x},y,\tl{y}}=\Gm_{y,\tl{y},x,\tl{x}}$ %and hence that the Markov chain $((X_t, \tl{X}_t)_{t \ge 0}, P^{x,\tl{x}})$ %is symmetric in this case. %%%%%%%%%%% \vvs \noindent {\bf Remark:} If we additionally suppose that $P[K_x^p]=P[K_{-x}^p]$ for $p=1,2$ and $x \in \zd$, then, $\Gm_{x,\tl{x},y,\tl{y}}=\Gm_{y,\tl{y},x,\tl{x}}$ for all $x,\tl{x},y,\tl{y} \in \zd$. Thus, $(X,\tl{X})$ and $(Y,\tl{Y})$ are the same in this case. \vvs The relative motion $Y_t-\tl{Y}_t$ of the components of $(Y,\tl{Y})$ is nicely identified by: %%%%%%%%%%%%% \Lemma{Y-Y} %%%%%%%%% $((Y_t-\tl{Y}_t)_{t \ge 0}, P_{Y,\tl{Y}}^{x,\tl{x}})$ and $((R_{2t})_{t \ge 0}, P_R^{x-\tl{x}})$ (cf. (\ref{L_R})) have the same law. %%%%%%%%%% \end{lemma} %%%%%%% Proof: Since $(Y,\tl{Y})$ is shift invariant, in the sense that $\Gm_{x+v,\tl{x}+v,y+v,\tl{y}+v} =\Gm_{x,\tl{x},y,\tl{y}} $ for all $v \in \zd$, $((Y_t-\tl{Y}_t)_{t \ge 0}, P_{Y,\tl{Y}}^{x,\tl{x}})$ is a Markov chain. Moreover, its jump rate is computed as follows. For $x \neq y$, \bdnn \sum_{z \in \zd}\Gm_{y+z,z,x,0} & = & P[K_{x-y}]+P[K_{y-x}] +\del_{x,0}\sum_{z \in \zd} P[(K_{y+z}-\del_{y,z})(K_{z}-\del_{0,z})] \\ & \st{\mbox{\scriptsize (\ref{K4})}}{=} & P[K_{x-y}]+P[K_{y-x}]. \ednn \hfill $\Box$ %%%%%%%%%% \vvs To prove \Thm{CLT}, the use of \Lem{FK2} is made not in itself, but via the following lemma. It is the proof of this lemma, where the duality of $(X,\tl{X})$ and $(Y,\tl{Y})$ plays its role. %%%%%%%%%%%%% \Lemma{FK3} %%%%%%%%% For $g:\zd \times \zd \ra [0,\8)$, \bdmn \lefteqn{\sum_{x,\tl{x} \in \zd} P[\ov{\h}_{t,x}\ov{\h}_{t,\tl{x}}]g(x,\tl{x})} \nn \\ &=&\sum_{x,\tl{x} \in \zd}\h_{0,x}\h_{0,\tl{x}} P_{Y,\tl{Y}}^{x,\tl{x}} \lef[ \exp \lef( \kp_2\int^t_0 \del_0(Y_s-\tl{Y}_s)ds\ri) g(Y_t,\tl{Y}_t) \ri].\label{ell_2} \edmn In particular, for $f:\zd \ra [0,\8)$, \bdnl{FK3} \sum_{x,\tl{x} \in \zd}P[\ov{\h}_{t,x}\ov{\h}_{t,\tl{x}}]f(x-\tl{x}) =\sum_{x,\tl{x} \in \zd}\h_{0,x}\h_{0,\tl{x}} P_R^{x-\tl{x}}\lef[ \exp \lef( {\kp_2 \over 2}\int^{2t}_0 \del_0(R_u)du\ri) f(R_{2t})\ri]. \edn %%%%%%%%%% \end{lemma} %%%%%%% Proof: It follows from \Lem{FK2} and (\ref{C_Gm}) that \bds \item[(1)] \hspace{1cm} ${\dps \mbox{LHS of (\ref{ell_2})}=\sum_{x,\tl{x} \in \zd} P_{X,\tl{X}}^{x,\tl{x}} \lef[ \exp \lef( \kp_2\int^t_0 \del_0(X_s-\tl{X}_s)ds\ri) \h_{0,X_t}\h_{0,\tl{X}_t} \ri]g(x,\tl{x}).}$ \eds We now observe that the operators \bdnn f (x,\tl{x}) & \mapsto & P_{X,\tl{X}}^{x,\tl{x}} \lef[ \exp \lef( \kp_2\int^t_0 \del_0(X_s-\tl{X}_s)ds\ri) f(X_t, \tl{X}_t) \ri], \\ f (x,\tl{x}) & \mapsto & P_{Y,\tl{Y}}^{x,\tl{x}} \lef[ \exp \lef( \kp_2\int^t_0 \del_0(Y_s-\tl{Y}_s)ds\ri) f(Y_t, \tl{Y}_t) \ri] \ednn are dual to each other with respect to the counting measure on $\zd \times \zd $. Therefore, $$ \mbox{RHS of (1)}=\mbox{RHS of (\ref{ell_2})}. $$ Taking $g (x,\tl{x})=f(x-\tl{x})$ in particular, we have by (\ref{ell_2}) and \Lem{Y-Y} that \bdnn \mbox{LHS of (\ref{FK3})} &=&\sum_{x,\tl{x} \in \zd}\h_{0,x}\h_{0,\tl{x}}P^{x,\tl{x}}_{Y,\tl{Y}} \lef[ \exp \lef( \kp_2\int^t_0 \del_0(Y_s-\tl{Y}_s)ds\ri) f(Y_t-\tl{Y}_t)\ri] \\ &=& \sum_{x,\tl{x} \in \zd}\h_{0,x}\h_{0,\tl{x}} P_R^{x-\tl{x}}\lef[ \exp \lef( \kp_2\int^t_0 \del_0(R_{2u})du\ri) f(R_{2t}) \ri] =\mbox{RHS of (\ref{FK3})}. \ednn \hfill $\Box$ %%%%%%%%%%%%%%%% \subsection{Central limit theorems for Markov chains} %%%%%%%%%%%%%%% We prepare central limit theorems for Markov chains, which is obtained by perturbation of random walks. %%%%%%%% \Lemma{perturb1} %%%%%%% Let $((Z_t)_{t \ge 0}, P^x)$ be a continuous-time random walk on $\zd$ with the generator $$ L_Z f(x)=\sum_{y \in \zd}a_{y-x}(f(y)-f(x)), $$ where we assume that $$ \sum_{x \in \zd}|x|^2a_x<\8. $$ Then, for any $B \in \s [Z_u\; ; \; u \in [0,\8)]$, $x \in \zd$, and $f \in C_{\rm b}(\rd)$, $$ \lim_{t \ra \8}P^x [f((Z_t -mt)/\sqrt{t}):B] =P^x (B)\int_{\rd}fd\n, $$ where $m=\sum_{x \in \zd}xa_x$ and $\n$ is the Gaussian measure with \bdnl{nua} \int_{\rd}x_id\n (x)=0, \; \; \; \int_{\rd}x_ix_jd\n (x) =\sum_{x \in \zd}(x_i-m_i)(x_j-m_j)a_x,\; \; \; i,j=1,..,d. \edn %%%%%% \end{lemma} %%%%%%% Proof: By subtracting a constant, we may assume that $\int_{\rd}fd\n =0$. We first consider the case that $B \in \cF_s \st{\rm def}{=}\s [Z_u\; ; \; u \in [0,s]]$ for some $s \in (0,\8)$. It is easy to see from the central limit theorem for $(Z_t)$ that for any $x \in \zd$, $$ \lim_{t \ra \8}P^x [f((Z_{t-s} -mt)/\sqrt{t})]=0. $$ With this and the bounded convergence theorem, we have $$ P^x[f((Z_t -mt)/\sqrt{t}):B] =P^x[P^{Z_s}[f((Z_{t-s} -mt)/\sqrt{t})]:B] \lra 0\; \; \mbox{as $t \nearrow \8$}. $$ Next, we take $B \in \s [Z_u\; ; \; u \in [0,\8)]$. For any $\e>0$, there exist $s \in (0,\8)$ and $\tl{B} \in \cF_s$ such that $P^x[|{\bf 1}_B-{\bf 1}_{\tl{B}}|] <\e$. Then, by what we already have seen, $$ \suplim_{t \ra \8} P^x[f((Z_t -mt)/\sqrt{t}):B] \le \suplim_{t \ra \8}P^x[f((Z_t -mt)/\sqrt{t}):\tl{B}] +\| f \|\e = \| f \|\e, $$ %%%%%% %\le (P^x (B)+\e)\int_{\rd}fd\n +\| f \|\e, %%%%%% where $\| f\|$ is the sup norm of $f$. Similarly, $$ \inflim_{t \ra \8} P^x[f((Z_t -mt)/\sqrt{t}):B] \ge -\| f \|\e. $$ Since $\e>0$ is arbitrary, we are done. \hfill $\Box$ %%%%%%%% \Lemma{perturb2} %%%%%%% Let $Z=((Z_t)_{t \ge 0}, P^x)$ be as in \Lem{perturb1} and and $D \sub \zd$ be transient for $Z$. On the other hand, let $\tl{Z}=((\tl{Z}_t)_{t \ge 0}, \tl{P}^x)$ be the continuous-time Markov chain on $\zd$ with the generator $$ L_{\tl{Z}} f(x)=\sum_{y \in \zd}\tl{a}_{x,y}(f(y)-f(x)), $$ where we assume that $\tl{a}_{x,y}=a_{y-x}$ if $x \not\in D$ and that $D$ is also transient for $\tl{Z}$. Furthermore, we assume that a function $v:\zd \ra \R$ satisfies \bdnn & & \mbox{$v \equiv 0$ outside $D$,}\\ & & \tl{P}^z\lef[\exp \lef( \int_0^\8 |v(\tl{Z}_t)|dt\ri)\ri]<\8\; \; \mbox{for some $z \in \zd$}. \ednn Then, for $f \in C_{\rm b}(\rd)$, $$ \lim_{t \ra \8}\tl{P}^z \lef[\exp \lef( \int_0^t v(\tl{Z}_u)du\ri)f((\tl{Z}_t-mt)/\sqrt{t})\ri] =\tl{P}^z\lef[\exp \lef( \int_0^\8 v(\tl{Z}_t)dt\ri)\ri]\int_{\rd}fd\n, $$ where $\n$ is the Gaussian measure such that (\ref{nua}) holds. %%%%%% \end{lemma} %%%%%%% Proof: Define \bdnn H_D(\tl{Z})&=&\inf \{ t \ge 0\; ; \; \tl{Z}_t \in D\}, \; \; T_D(\tl{Z})=\sup \{ t \ge 0\; ; \; \tl{Z}_t \in D\},\\ e_t&=&\exp \lef( \int_0^t v(\tl{Z}_s)ds \ri). \ednn Then, \bdmn \lefteqn{\tl{P}^z\lef[e_tf((\tl{Z}_t-mt)/\sqrt{t})\ri]} \nn \\ &=&\tl{P}^z\lef[e_tf((\tl{Z}_t-mt)/\sqrt{t}):T_D(\tl{Z})0$, we have by \Lem{perturb1} that $$ \lim_{t \ra \8}\tl{P}^x\lef[f((\tl{Z}_{t-s}-mt)/\sqrt{t}):H_D(\tl{Z})=\8 \ri] = \tl{P}^x[H_D(\tl{Z})=\8] \int_{\rd}fd\n. $$ Therefore, \bdnn \lefteqn{\lim_{t \ra \8} \tl{P}^z\lef[e_s{\bf 1}_{\tl{Z}_s \not\in D} \tl{P}^{\tl{Z}_s}\lef[f((\tl{Z}_{t-s}-mt)/\sqrt{t}):H_D(\tl{Z})=\8 \ri]\ri]}\\ &=&\tl{P}^z\lef[e_s{\bf 1}_{\tl{Z}_s \not\in D} \tl{P}^{\tl{Z}_s}[H_D(\tl{Z})=\8] \rig]\int_{\rd}fd\n \\ &=& \tl{P}^z\lef[e_s :T_D (\tl{Z})0$ and $f :\zd \ra [0,\8)$ with $\sum_{x \in \zd}f(x)<\8$. %%%%%%%%%%%% \end{lemma} %%%%%%%%% Proof: We adapt the argument in \cite[Lemma 3.1.3]{CY04}. For a bounded function $f:\zd \ra \R$, we introduce \bdnn (T_tf)(x)&=&P^x \lef[ \exp \lef( \int^t_0v(Z_u)du\rig)f(Z_t)\rig], \; \; \; x \in \zd,\\ T_t^hf&=&\frac{1}{h}T_t [fh], \; \; \mbox{where ${\dps h(x)=P^x\lef[\exp \lef( \int_0^\8 v(Z_t)dt\ri)\ri]}$.} \ednn Then, $(T_t)_{t \ge 0}$ extends to a symmetric, strongly continuous semi-group on $\ell^2(\zd)$. We now consider the measure $\sum_{x \in \zd}h(x)^2\del_x$ on $\zd$, and denote by $(\ell^{p,h}(\zd), \| \; \cdot\; \|_{p,h})$ the associated $\bL^p$-space. Then, it is standard to see that $(T_t^h)_{t \ge 0}$ defines a symmetric strongly continuous semi-group on $\ell^{2,h}(\zd)$ and that for $f \in \ell^{2,h}(\zd)$, \bdnn \cE^h (f,f) & \st{\rm def.}{=} & \lim_{t \searrow 0}{1 \over t} \sum_{x\in \zd} f (x)(f-T_t^hf)(x)h(x)^2 \\ &=& \half \sum_{x,y \in \zd}a_{y-x}| f(y)-f(x)|^2h(x)h(y). \ednn %%%%%%% %where the bracket stands for the inner product of $\ell^{2,h}(\zd)$. %%%%%%%% By the assumptions on $(a_x)$, we have the Sobolev inequality: \bds \item[(1)] \hspace{1cm} ${\dps \sum_{x \in \zd}| f (x)|^{\frac{2d}{d-2}} \le c_1 \lef( \half\sum_{x,y \in \zd}a_{y-x}| f(y)-f(x)|^2 \rig)^{\frac{d}{d-2}} }$ for all $f \in \ell^2 (\zd)$, \eds where $c_1\in (0,\8)$ is independent of $f$. This can be seen via isoperimetric inequality \cite[page 40, (4.3)]{Woe00}. We have on the other hand that \bds \item[(2)] \hspace{1cm} $ 1/C_v\le h(x) \le C_v.$ \eds We see from (1) and (2) that $$ \sum_{x \in \zd}| f (x)|^{\frac{2d}{d-2}}h(x)^2\le c_2 \cE^h (f,f) ^{\frac{d}{d-2}} \; \; \; \mbox{for all $f \in \ell^{2,h} (\zd)$,} $$ where $c_2\in (0,\8)$ is independent of $f$. This implies that there is a constant $C$ such that $$ \| T^h_t \|_{2 \ra \8,h} \le Ct^{-d/4} \; \; \; \mbox{for all $t >0$}, $$ e.g.,\cite[page 75, Theorem 2.4.2]{Dav89}, where $\| \cdot \|_{p \ra q,h}$ denotes the operator norm from $\ell^{p,h}(\zd)$ to $\ell^{q,h}(\zd)$. Note that $\| T^h_t \|_{1 \ra 2,h} =\| T^h_t \|_{2 \ra \8,h}$ by duality. We therefore have via semi-group property that \bds \item[(3)] \hspace{1cm} $\| T^h_t \|_{1 \ra \8,h} \le \| T^h_{t/2} \|_{2 \ra \8,h}^2 \le C^2t^{-d/2}$ for all $t >0$. \eds Since $T_tf=h T^h_t[f/h]$, the desired bound (\ref{RepDec2}) follows from (2) and (3). \hfill $\Box$ %%%%%%%%% \SSC{Proof of \Thm{CLT} and \Thm{cov}} %%%%%%%%%%%%%%%%%% \subsection{Proof of \Thm{CLT}} \label{pCLT} %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%% (a) $\LRa$ (b): %%%%%%%%%%%%% Define $$ h (x)=P_R^x\lef[\exp \lef( {\kp_2 \over 2}\int^\8_0 \del_0 (R_t)dt\ri)\ri]. $$ Then, $\sup_{t \ge 0} P[|\ov{\h}_t|^2]=h(0)$ by (\ref{FK3}). Therefore, it is enough to show that (a) is equivalent to $h(0)<\8$. In fact, (a) implies that $h(0)<\8$, by Khas'minskii's lemma \cite[page 7]{Szn98}. Note that $\sup_{x \in \zd}G(x)=G(0)$ to use Khas'minskii's lemma. On the other hand, we have that %%%%%% %by differentiating %$\exp \lef( {\kp_2 \over 2}\int^t_0 \del_0 (R_s)ds\rig)$ %and then integrating, %%%%%%%%%%%% $$ \exp \lef( {\kp_2 \over 2}\int^t_0 \del_0 (R_s)ds\rig) =1+{\kp_2 \over 2}\int^t_0\del_{0,R_s} \exp \lef( {\kp_2 \over 2}\int^t_s \del_0 (R_u)du\rig)ds, $$ and hence that $h (x)=1+{\kp_2 \over 2} h(0)G(x)$. Thus, $h(0)<\8$ implies (a) and that \bdnl{h(x)} h(x)=1+{\kp_2G(x) \over 2-\kp_2G(0)}. \edn %%%%%%%% % the condition (a) is equivalent to %$\sup_{t \ge 0}P[|\ov{\h}_t|^2]<\8$, which in turn %is equivalent to the $\bL^2 (P)$-convergence of the martingale %$|\ov{\h}_t|$. %%%%%%%%%%%%%%% (a),(b) $\Ra$ (c): %%%%%%%%%%%% %By a standard approximation arguments, we may assume that %$f \in C_{b,u}(\rd)$, where $C_{b,u}(\rd)$ %denotes the set of bounded, uniformly continuous functions. %%%%%%%%%%%%%%%%% Since (b) implies that $\lim_{t \ra \8}|\ov{\h}_t|=|\ov{\h}_\8|$ in $\bL^2 (P)$, it is enough to prove that, $$ U_t\st{\rm def.}{=}\sum_{x \in \zd}\ov{\h}_{t,x}f \lef( (x-mt)/\sqrt{t}\rig) \lra 0\; \; \; \mbox{in $\bL^2 (P)$ as $t \nearrow \8$} $$ for $f \in C_{\rm b} (\rd)$ such that $\int_{\rd}fd\n =0$. We set $f_t(x, \tl{x}) =f ((x-m)/\sqrt{t})f ((\tl{x}-m)/\sqrt{t})$. Then, by \Lem{FK3}, $$ P[U_t^2] = \sum_{x,\tl{x} \in \zd} P[\ov{\h}_{t,x}\ov{\h}_{t,\tl{x}}]f_t(x, \tl{x}) = \sum_{x,\tl{x} \in \zd}\h_{0,x}\h_{0,\tl{x}} P_{Y,\tl{Y}}^{x,\tl{x}}\lef[ e_tf_t(Y_t, \tl{Y}_t) \ri], $$ where $e_t=\exp \lef( \kp_2\int^t_0 \del_0(Y_s-\tl{Y}_s)ds\ri)$. Note that by \Lem{Y-Y} and (a), \bds \item[(1)]\hspace{1cm} ${\dps P_{Y,\tl{Y}}^{x,\tl{x}}\lef[ e_\8\ri] =h(x-\tl{x}) \le h(0) <\8}$. \eds Since $|\h_0|<\8$, it is enough to prove that for each $x,\tl{x} \in \zd$ \bds \item[(2)]\hspace{1cm} ${\dps \lim_{t \ra \8} P_{Y,\tl{Y}}^{x,\tl{x}}\lef[ e_tf_t(Y_t, \tl{Y}_t) \ri]=0}$. \eds We now apply \Lem{perturb2} to the Markov chain $\tl{Z}_t\st{\rm def.}{=}(Y_t,\tl{Y}_t)$ and the random walk $(Z_t)$ on $\zd \times \zd$ with the genaretor $$ L_Zf (x,\tl{x}) =\sum_{y,\tl{y} \in \zd}a_{x,\tl{x},y,\tl{y}} \lef( f(y,\tl{y})-f(x,\tl{x})\rig)\; \; \; \mbox{with}\; \; a_{x,\tl{x},y,\tl{y}}=\lef\{\barray{ll} \Gm_{y,\tl{y},x,\tl{x}} & \mbox{if $x=\tl{x}$ or $y=\tl{y}$,} \\ 0 & \mbox{if otherwise}.\earray \rig. $$ Let $D=\{ (x,\tl{x}) \in \zd \times \zd\; ; \; x=\tl{x}\}$. Then, $a_{x,\tl{x},y,\tl{y}}=\Gm_{x,\tl{x},y,\tl{y}}$ if $(x,\tl{x}) \not\in D$. Moreover, $D$ is transient both for $(Z_t)$ and for $(\tl{Z}_t)$. Finally, the Gaussian measure $\n \otimes \n $ is the limit law in the central limit theorem for the random walk $(Z_t)$. Therefore, by (1),(2) and \Lem{perturb2}, $$ \lim_{t \ra \8} P_{Y,\tl{Y}}^{x,\tl{x}}\lef[ e_tf_t(Y_t, \tl{Y}_t) \ri] =P_{Y,\tl{Y}}^{x,\tl{x}}\lef[ e_\8\ri] \lef( \int_{\rd}fd\n \rig)^2=0. $$ (c) $\Ra$ (b): This can be seen by taking $f \equiv 1$.\\ (\ref{RepDec}):By (\ref{FK3}), $$ \sum_{x,\tl{x} \in \zd}P[\ov{\h}_{t,x}\ov{\h}_{t,\tl{x}}]f(x-\tl{x}) =\sum_{x,\tl{x} \in \zd}\h_{0,x}\h_{0,\tl{x}}P^{x-\tl{x}}_R\lef[ \exp \lef( {\kp_2 \over 2}\int^{2t}_0 \del_0(R_u)du\ri) f(R_{2t})\ri]. $$ We apply \Lem{Nash} to the right-hand-side to get (\ref{RepDec}). \hfill $\Box$ %%%%%%%%%%%%%%%%%% \subsection{Proof of \Thm{cov}} \label{pcov} %%%%%%%%%%%%%%%%%%%%%%%%%%%% By the shift-invariance, we may assume that $b=0$. We have by \Lem{FK3} that $$ P[\ov{\h}^a_{t,x}\ov{\h}^0_{t,\tl{x}}] =P_{Y,\tl{Y}}^{a,0} \lef[ \exp \lef( \kp_2 \int^{t}_0 \del_0(Y_u-\tl{Y}_u)du\ri) :(Y_t,\tl{Y}_t)=(x,\tl{x}) \ri], $$ and hence by \Lem{Y-Y} that $$ P[|\ov{\h}_t^a||\ov{\h}_t^0|] =P^{a,0}_{Y,\tl{Y}} \lef[ \exp \lef( \kp_2 \int^{2t}_0 \del_0(Y_u-\tl{Y}_u)du\ri)\ri] =P^{a}_R\lef[ \exp \lef( {\kp_2 \over 2}\int^{2t}_0 \del_0(R_u)du\ri)\ri]. $$ By \Thm{CLT}, both $|\ov{\h}_t^a|$ and $|\ov{\h}_t^0|$ are convergent in $\bL^2 (P)$ if $ {\kp_2 \over 2}G(0)<1$. Therefore, letting $t \nearrow \8$, we conclude that $$ P[|\ov{\h}_\8^a||\ov{\h}_\8^0|]= P^{a}_R\lef[ \exp \lef( {\kp_2 \over 2}\int^{\8}_0 \del_0(R_u)du\ri)\ri] \st{\mbox{\scriptsize (\ref{h(x)}) }}{=}1+{\kp_2G(a) \over 2-\kp_2G(0)}. $$ \hfill $\Box$ %%%%%%%%%%%%%% \small \vvs \noindent{\bf Acknowledgements:} The authors thank Shinzo Watanabe for the improvement of \Lem{perturb1}. %%%%%%%% %He also thanks Yukio Nagahata for discussions. %%%%%%%%%%% \begin{thebibliography}{99} %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bibitem{CY04} Comets, F., Yoshida, N.: Some New Results on Brownian Directed Polymers in Random Environment, RIMS Kokyuroku {\bf 1386}, 50--66, avilable at authors' web pages. (2004). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bibitem{Dav89} Davies, E. B.: ``Heat kernels and spectral theory", Cambridge University Press (1989). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bibitem{Gri83} Griffeath, D.: The Binary Contact Path Process, Ann. Probab. Volume 11, Number 3 (1983), 692-705. %%%%%%%%%%%%%%%%%%%%%%%%%% \bibitem{Lig85} Liggett, T. M. : ``Interacting Particle Systems", Springer Verlag, Berlin-Heidelberg-Tokyo (1985). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bibitem{Spi76} Spitzer, F.: ``Principles of Random Walks", Springer Verlag, New York, Heiderberg, Berlin (1976). %%%%%%%%%%%%%%%%%%% \bibitem{Szn98} Sznitman, A.-S., : Brownian Motion, Obstacles and Random Media, Springer monographs in mathematics, Springer (1998). %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% \bibitem{Woe00} Woess, W.: ``Random Walks on Infinite Graphs and Groups", Cambridge University Press. (2000). %%%%%%%%%%%%%%%%%%%%%% \bibitem{Yos08} Yoshida, N.: Central Limit Theorem for Branching Random Walk in Random Environment, to appear in Ann. Appl. Proba., arXiv:0712.0649 (2008). %%%%%%%%%%%%%%%%%%%%%%% %%%%%% \end{thebibliography} %%%%% \vvs \noindent Yukio Nagahata \\ Department of Mathematics, \\ Graduate School of Engineering Science \\ Osaka University,\\ Toyonaka 560-8531, Japan.\\ email: {\tt nagahata@sigmath.es.osaka-u.ac.jp}\\ URL: {\tt http://www.sigmath.osaka-u.ac.jp/}$\widetilde{}$ {\tt nagahata/} \vvs \noindent Nobuo Yoshida \\ Division of Mathematics \\ Graduate School of Science \\ Kyoto University,\\ Kyoto 606-8502, Japan.\\ email: {\tt nobuo@math.kyoto-u.ac.jp}\\ URL: {\tt http://www.math.kyoto-u.ac.jp/}$\widetilde{}$ {\tt nobuo/} %%%%%%%%%% \end{document} %%%%%%%% \Lemma{perturb1} %%%%%%% Let $((Y_t)_{t \ge 0}, P^x)$ be a continuous-time, mean-zero random walk on $\zd$ with the generator $$ L_Y f(x)=\sum_{y \in \zd}a_{y-x}(f(y)-f(x)), $$ where we assume $$ \sum_{x \in \zd}|x|^2a_x<\8. $$ Let $H_D=\inf \{t \ge 0\; ; \; Y_t \in D\}$ for a transient set $D \sub \zd$. Then, $x \not\in \zd$, $$ \lim_{t \ra \8}P^x (Y_t /\sqrt{t} \in \cdot |H_D=\8)=\n,\; \; \; \mbox{weakly,} $$ where $\n$ is the Gaussian measure with \bdnl{nua} \int_{\rd}x_id\n (x)=0, \; \; \; \int_{\rd}x_ix_jd\n (x)=\sum_{x \in \zd}x_ix_ja_x,\; \; \; i,j=1,..,d. \edn %%%%%% \end{lemma} %%%%%%% Proof: We have to show that for $f \in C_{\rm b}(\rd)$, \bds \item[(1)] ${\dps \lim_{t \ra \8}P^x [f(Y_t /\sqrt{t}):H_D=\8] =P^x [H_D=\8]\int_{\rd}fd\n}$. \eds We may assume that $f$ is Lipschitz continuous. Define $$ T_D =\sup \{ t \ge 0\; ; \; Y_t \in D\},\; \; e_{t,\a}=\exp \lef( -\a\int^t_0{\bf 1}_D(Y_s)ds \ri). $$ Since ${\dps \lim_{\a \ra \8}e_{\8,\a}={\bf 1}_{\{H_D=\8\}}}$, it is enough to prove that \bds \item[(2)] ${\dps \lim_{t \ra \8}P^x [e_{\8,\a}f(Y_t /\sqrt{t})] =P^x[e_{\8,\a}]\int_{\rd}fd\n}$, uniformly in $\a >0$. \eds We write $s=t^{1/3}$ and approximate the LHS of (2) by the RHS: \bdnn P^x[e_{\8,\a}f(Y_t /\sqrt{t})] &=&P^x[e_{\8,\a}f(Y_t /\sqrt{t}):T_D