数理解析研究所講究録
第 1672 巻 2010 年 39-44

39

A universal framework of the homogenization
problem of infinite dimensional diffusions
Sergio ALBEVERIO 1

Michael R\"OCKNER 2

Minoru W. YOSHIDA 3

andAbstract

By generalizing the ccncrete formulations in [ABRYl,2], A universal frame workof the homogenization pioblem of infinite dimensional diffusions is proposed. The
corresponding general stiucture is considered.

Probability space
efficients are defined

1$(\Theta, \overline{\mathcal{B}}, \overline{\mu})$on which the random co-Suppose that we are given the following:$\{(\Theta_{k}, \mathcal{B}_{k}, \lambda_{k})\}_{k\in Z^{d}}$:a system of coniplete probability spaces, where$d$

is a given naturalnumber.
:the probability space that is tlie completion of
the completion of the direct product probability space.
$(\Theta, \overline{\mathcal{B}}, \overline{\lambda})$$( \prod_{k}\Theta_{k}, \otimes_{k}\mathcal{B}_{k}, \prod_{k}\lambda_{k})$, i.e.,

a complete $pro\supset ability$ space (corresponding to a Gibbs state) defined asfollows: forand for any bounded measurable function defined onwith somesatisfies$(\Theta,\overline{\mathcal{B}}, \mu)$:$\forall D\subset\subset Z^{d}$$\prod_{k\in D},$$\varphi$$\forall D’\subset\subset Z^{d},$$\Theta_{k}$$\mu$$(E^{D}\varphi.l^{\iota})=(\varphi, \mu)$,where$(E^{D}\varphi)(\theta)$$\equiv$$\int_{\Theta}E^{D}((l\theta’|\theta)$$\equiv$$\int_{\Theta}\varphi(\theta_{D}’.

\theta_{D^{r}})nz_{D}(\theta_{D}’\cdot\theta_{D^{\Gamma}})\overline{\lambda}(d\theta’)$,with$m_{D}( \theta_{D}’.

,\theta_{D^{c}}.)\equiv\frac{1}{Z_{D}(\theta_{D^{r}})}e^{-U_{D}(\theta_{D}’\cdot\theta_{D^{\ulcorner}})},$$U_{D} \equiv\sum_{k\in D}U_{k}$,$\Theta\ni\theta\theta_{D}\in\prod_{k\in D}\Theta_{k}$lInst.

Angewandte Matheniatik, Universit\"at Bonn, Wegelerstr. 6,D-53ll5 Bonn (Germany),
et turti SI, Mendrisio; Ist. Mathematica, Universit\‘a di$SFB611|$ BiBoS; CERFIM, Locarno; $A(($ .

$\bigwedge_{1t}\cdot 1\iota it$$l\uparrow$

TYento2Dept. Math., Univ. Bielefeld

3e-mail wyoshida@ipcku.kansai-u ac.jp fax $+816$ 63303770. Kansai Univ., Dept. Mathematics, 564-8680 Yamate-Tyou 3-3-35 Suita Osaka(Japan)

40

is the natural projection,$\theta_{D}’$.

$\theta_{D^{C}}$is the element$\theta_{D}’’=\theta_{D}’$,such that$\theta’’\in\Theta$$\theta_{D^{\Gamma}}’’=\theta_{D^{r}}$,also, for each $k\in Z^{d},$is a given bounded measurable function of which support is in(the range of interactions) does not depend on , and, where the number
is the normalizing constant.
$U_{k}$$L$$\prod_{|k-k|<L}\Theta_{k’}$$k$$Z_{D}(\theta_{D^{c}}\overline{)}$2

The ergodic flow

(which is also a map on
we are given an ergodic flowa measure preserving map on it) as follows:
Suppose thatOn$T_{x}$$(\Theta,\overline{\mathcal{B}}, \overline{\lambda})$$\exists M_{1}<\infty$

For eachthe mapi

$x\in\prod_{k}\mathbb{R}^{d_{k}}$$T_{x}$onandthere exists a$\forall k\in Z^{d}$such that $x=(x^{k})_{k\in Z^{d}}$ withis defined bysuch that$d_{k}$$(\Theta, \overline{\mathcal{B}}, \mu)$, but is not.

(2.1)

$d_{k}<M_{1}$$x^{k}=(x_{1}^{k}, \ldots , x_{d_{k}}^{k})$$(\Theta, \overline{\mathcal{B}}, \overline{\lambda})$$)$

T.,$:\Thetaarrow\Theta$that is a measure preserving transformation with respect to the measureii)$T_{0}=$ the identity,

for $x,$$x’\in x\in\prod_{k\in Z^{d}}\mathbb{R}^{d_{k}}$$T_{x+x’}=T_{x}\circ T_{x’}$$\overline{\lambda}$

;

,

where
$x+x’\equiv(x^{k}+x^{\prime k})_{k\in Z^{d}}$,with$x^{k}+x^{;k}=(x_{1}^{k}+\backslash \tau_{1}^{;k}\ldots., x_{d^{k}}^{k}+x_{d^{k}}^{\prime k})$,for
$x=(x^{k})_{k\in Z^{ri}}$$x’=(x^{k})_{k\in Z^{d}}$.

,$x^{k}=(x_{1}^{k}, \ldots, x_{d_{k}}^{k})$,$x^{\prime k}=(x_{1)}^{\prime k}\ldots, x_{d_{k}}^{\prime k})$,and$0\equiv(0^{k})_{k\in Z^{d}}$,$0^{k}=(0, \ldots, 0)\in \mathbb{R}^{d_{k}}$;iii)$( x.\theta)\in(\prod_{k\in Z^{d}}\mathbb{R}^{d_{k}})x\thetaarrow T_{x}(\theta)\in\Theta$is$\mathcal{B}(\prod_{k\in Z^{d}}\mathbb{R}^{d_{k}})\cross\overline{\mathcal{B}}/\overline{\mathcal{B}}$-measurable, where,with the direct product $topc_{I}logy$ ;$\prod_{k\in Z^{d}}\mathbb{R}^{d_{k}}$is assumed to be the topological space41iv)

A function which isvFor$)$$D\subset Z^{d}$invariant for all$T_{x}$

is a constant function;$x\in\prod_{k\in Z^{d}}\mathbb{R}^{d_{k}}$, let$\prod_{k\in Z^{d}}\mathbb{R}^{d_{k}}\ni x\mapsto x_{D}\in\prod_{k\in D}\mathbb{R}^{d_{k}}$be the natural$p_{\Gamma O}J^{ecbion}$. If, then$x_{D^{c}}=0_{D^{\Gamma}}$$(T_{x}(\theta)|_{D^{r}}=\theta_{D^{r}}$,$\forall\theta\in\Theta$,$\forall D\subset\subset Z^{d}$.
$\square$

The core3

We assume that an existence of a core . Namely, there exists which is a dense subsetof both $L^{2}(P)$ and $L^{1}(P)$ , andsatisfies$mea_{\sim^{S}}urable$
is a bounded
funct.ion having only a finite number of variables,for some$\mathcal{D}$$\mathcal{D}$$\forall\varphi\in \mathcal{D}$$(\mathcal{D}- 1)$$\theta_{D}$$\varphi$$D\subset\subset Z^{d}$$(\mathcal{D}- 2)$,$\varphi(T_{\mathfrak{c}_{L}}(\theta))\in C^{\infty}(\prod_{k\in D}\mathbb{R}^{d_{k}}arrow \mathbb{R})$$\forall\theta\in\Theta$,(cf. v) in the previous section) where we identify
with anof which projection tois,
in, all t,he partial derivatives of all orders of the function
for each$x_{D}\in\prod_{k\in D}\mathbb{R}^{d_{k}}$$\prod_{k\in D}\mathbb{R}^{d_{k}}$$(\mathcal{D}- 3)$$(\mathcal{D}- 2)$$\varphi(T.(\theta))$$x_{D}$$\theta\in\Theta$(with the variables$\forall\varphi\in \mathcal{D},$$x\in(\prod_{k\in Z^{d}}\mathbb{R}^{d_{k}})$$x_{D}$$\exists M<\infty$;) are bounded and$|\nabla_{k}\varphi(T_{x}(\theta))|<\Lambda I$,$\forall\theta\in\Theta,$$\forall x$, $Vk$$\in Z^{d}$

(3.1)

,

where
$\nabla_{k}=(\frac{\partial}{x_{1}^{k}}\ldots., \frac{\partial}{x_{d_{k}}^{k}})$.
$\square$4

Probability spaceon which the infinite
dimensional diffusions are defined$((l, \mathcal{F}, P;\mathcal{F}_{t})$Suppose that we are given a system of family of functions
such that for each $k\in Z^{d}$ and each $1\leq i,$ $j\leq d_{k}$ ,onand there exists
and$(\Theta, \overline{\mathcal{B}}, \overline{\mu})$$\Theta_{k}$$a_{ij}^{k},$$a_{ij}^{k}$on
is a measurable function$k\in Z^{d},$$1\leq i,$ $j\leq d_{k}$$\Lambda f_{2}\in(0, \infty)$$M_{2}^{-1} \leq\sum_{1\leq i,j\leq d_{k}}a_{ij}^{k}(\theta_{k})x_{j}x_{j}\leq\lrcorner lI_{2}$,$\forall k\in Z^{d}$.

$\forall\theta_{k}\in\Theta_{k},$$\forall(x_{1}, \ldots, x_{d_{k}})\in \mathbb{R}^{d_{k}}$We assume that$U_{k}$.

$a_{ij}^{k}\in \mathcal{D}$.

$k\in Z^{d,}.1\leq i,\cdot j\leq d_{k}$.

.

(4.1)

42

by which the evaluation (3.1) holds
Also, we assume that there exists a common.
andfor all,Finally, suppose that we are given a $c\cdot omplete$ probability space
suppose that there exists a system. Onwith a filtorationof independent l-dimensional -adapted Brownian motion processes$\Lambda I<\infty$$U_{k}$$a_{ij}^{k}$$(\Omega,\mathcal{F}, P;\mathcal{F}_{t})$$(t\in \mathbb{R}_{+})$$(f1, \mathcal{F}, P;\mathcal{F}_{t})$$\mathcal{F}_{t}$$\mathcal{F}_{t}$$\{(B^{k}’{}^{t}(t))_{t\geq 0}\}_{k\in Z^{d},1\leq i\leq d_{k}}$Now, for each$\theta\in\Theta$.

, let

X

$\theta_{\equiv\{(X^{\theta,k,i}(t))_{t\geq 0}\}_{k\in Z^{d},1\leq i\leq d_{k}}}$.

be the unique solution ofX$\theta k)’{}^{t}(t)=X^{\theta,k}’{}^{t}(0)+\int_{0}^{t}\sum_{1\leq j\leq d_{k}}\{\frac{\partial}{\partial x_{j}^{k}}a_{ij}^{k}(T_{X^{\theta,k}(s)}(\theta))$$-a_{ij}^{k}(T_{X^{\theta,k}(s)}( \theta))(\frac{\partial}{\partial x_{j}^{k}}(\sum_{|k-k’|\leq L}U_{k’}(T_{X^{\theta}(s)}(\theta))))\}ds$,$+ \int_{0}^{t}\sum_{1\leq j\leq d_{k}}\sigma_{ij}^{k}(T_{X^{\theta,k}(s)}(\theta))dB^{k_{1}j}(s)$$t\geq 0$

(4.2)

,

where
$(\sigma_{ij}^{k})=(2a_{\mathfrak{i}j}^{k})^{\frac{1}{2}}$,and$X^{\theta.k}(t)=(X^{\theta,k,1}(t), \ldots, X^{\theta,k,d_{k}}(t))$

also, by$X^{\theta}(t)$we denote the vector$(X^{\theta,k}(t))_{k\in Z^{d}} \in\prod_{k\in Z^{d}}\mathbb{R}^{d_{k}}$Then, the random variable on$(\Omega, \mathcal{F}, P;\mathcal{F}_{t})$$(X^{\theta}(t))_{t\geq 0}$

we define a$\ominus valne(1$Proposition 4.1 The following hold:P–a.s.
,i
Ifthen$T_{X^{\theta}(0)}(\theta)=T_{X^{\theta’}(0)}(\theta’)$.

$(v\in\Omega$.

$(X^{\theta}(t)-X^{\theta}(0))_{t\geq 0}=(X^{\theta’}(t)-X^{\theta’}(0))_{t\geq 0_{T}}$ii).

process such that$\{T_{X^{\theta}(t)}(\theta)\}_{t\geq 0}$$)$.

is the one taking values in$C([0. \infty)arrow\prod_{k\in Z^{d}}\mathbb{R}^{d_{k}})$Through,For$\theta’=T_{\lambda^{\theta}(0)}(\theta)$,.

$P-a.s$ .

$\omega\in\Omega$

,

(4.3)

43

$(T_{X^{\theta}(t)}(\theta))_{t\geq 0}=(T_{X_{0}^{\theta’}(t)}(\theta’))_{t\geq 0}$

where
$it$

$X_{0}^{\theta’}(t)(\theta’)$.

By$(4\cdot 4)$

suchis the diffusion (lefined by$(T_{X^{\theta}(t)}(\theta))_{t\geq 0}$$P-a.s$ .

,$(4\cdot 2)$with$X_{0}^{\theta’}(0)=0$

(4.4)

,

$\omega\in\Omega$and replacing$\theta$by$\theta’$inare represmted by$(Y_{\theta’}(t))_{t\geq 0}\equiv(T_{X_{0}^{\theta’}(t)}(\theta’))_{t\geq 0}$

The processiii$)$$(Y_{\theta’}(t))_{t\geq 0})sati_{\iota}sf\grave{]}_{\theta.9}Y_{\theta’}(O)=\theta’$and is a Markov process.
$\square$Definition 4.1 By Propositzon 4.1. we define Markovt,an semi-groups corresponding toand $(Y(t))_{t\geq 0}$ :,For bounded measurablewith some dounded$(X^{\theta}(t))_{t\geq 0}$$D\subset\subset Z^{d}$$f \in\iota_{\vee}^{\cap}(\prod_{k\in D}\mathbb{R}^{d_{k}}arrow \mathbb{R})$$(p_{t}^{X,\theta}f)(x)\equiv E[f(X^{\theta}(t))|X^{\theta}(0)=x]$For,$x\in\prod_{k}\mathbb{R}^{d_{k}}$;$\varphi\in \mathcal{D}$$(p_{t}^{Y}\varphi)(y)\equiv E[\varphi(T_{X^{\theta}(t)}(\theta))|T_{X^{\theta}(0)}(\theta)=y]$,$y\in\Theta$.
$\square$5

Key assumption and the result

As was done in [ABRYl,2], we assuine the following:
There exist $K<\infty$ and $\gamma>0$ such that$\sup_{y\in\ominus}|(p_{t}^{Y}\varphi)(y)-<\varphi\}\mu>|\leq Ke^{-\gamma t}(\Vert\nabla\varphi\Vert_{L^{\infty}}+\Vert\varphi\Vert_{L^{\infty}})$,,$\forall\theta\in\Theta$$\forall\varphi\in \mathcal{D}$

,

(5.1)

where
$\Vert\nabla\varphi\Vert_{L^{\infty}}=\sup_{x.\theta}|\nabla\varphi(T_{x}(\theta))|$.

$\Vert\varphi\Vert_{L^{\infty}}=\sup_{\theta}|\varphi(\theta)|$.
$\square$Definition 5.1 For eachsuch that

and$k\in Z^{d}$$i=1\ldots$ . , $d_{k}$ ,$(D^{k_{1}i} \varphi)(y)\equiv\frac{\partial}{\partial x_{i}^{k}}\varphi(T_{x}(y))|_{x=0}$,define an operator$\varphi\in \mathcal{D}$,$y\in\Theta$$D^{k,i}$:$\mathcal{D}arrow \mathcal{D}$.
$\square$Proposition 5.1 For each$k\in Z^{d}$.

$i=1\ldots..d_{k}$let$b_{i}^{k}( y)\equiv\sum_{1\leq j\leq d_{k}}\{(D^{k,j}a_{vj}^{k})(y)-a_{jj}^{k}(y)(D^{k,j}(\sum_{|k-k’|\leq L}U_{k’})(y))\}$

then.

$\dagger_{i}^{\theta.k}(y)\equiv.1_{0}^{\infty}(p_{t}^{\}}.b_{r}^{k})(y)dt\in L^{2}(\mu)$,$y\in\Theta$,

44

口Through the same discussions performed in [ABRYl,2], for the present general framework we are also able to show the following (cf. [ABRYI,2] for the exact statement andthe terminologies)

Theorem 5.1 By taking a subsequence of the scaling process with the scaling parameter, for.
such thatit converges weakly to a Gaussian$\epsilon>0$$\{\epsilon X^{\theta}(\frac{t}{\epsilon^{2}})\}_{t\geq 0}$$l^{x-0.\epsilon’}$$\theta\in\Theta$$k\in Z^{d},$ $1\leq i,j\leq d_{k}$ ,process with a constant covan qnce matrix, characterized by, as
andis the sequence of the parameter corresponding to.
the subsequence of$\sigma_{ij}^{k},$$\theta\in\Theta$$\lambda_{i}^{\theta,k},$$\epsilon_{n}\downarrow 0wher\epsilon\{\epsilon_{n}\}_{n\in N}$$\{\epsilon X^{\theta}(\epsilon\tau^{t})\}:\geq 0$

口References[ABRYI] S. Albeverio, M.S. Bernabei, M. R\"ockner, M.W. Yoshida:
HomogenizaGibb.q
measure.s for periodic
tion with respect todiffusions on lattices.
C.R.Acad.Sci.Paris, Ser.I, vol. 341, 675-678 (2005).
$d\uparrow\eta.fl$[ABRY2] S. Albeverio, M.S. Bernabei, M. R\"ockner, M.W.Yoshida: Homogenization of, coefficients, applying a loganthmic
with periodic
Diffusions on the lattice
Sobolev inequality or a weak Poincare inequality. Stochastic Analysis and Applications (The Abel Sympo. 2005 Oslo) pp. 53-72, Springer Berlin Heidelberg (2007).
$Z^{d}$$d\uparrow\eta fl$