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Set up and main results

ドキュメント内 ランダム行列に関する無限粒子系の確率解析 (ページ 118-124)

This paper is organized as follows. In Section 6.2, we set up Dirichlet forms and state main results. There, two types of convergence of unlabeled dynamics are shown. In Section 6.4, we prove the first theorem. The second theorem, which is more convenient than first one, is proved in Section 6.5. In Section 6.6, we give examples of dynamical universality.

A probability measure µ on S is called a random point field (point process). We set L2(µ) =L2(S, µ). We define the bilinear form (E,Dµ) on L2(µ) as

E(f, g) =

S

D[f, g](s)dµ,

Dµ={f ∈ D∩L2(µ) ;E(f, f)<∞}. (6.8) For eachr, m∈Nwe set the bilinear form (Erm,Dµ) onL2(µ) as

Erm(f, g) =

S

Dmr [f, g](s)dµ. (6.9)

We make an assumption:

(A1) (Erm,Dµ) is closable onL2(µ) for each r, m∈N.

SetBrb={f;f is bounded andσ[πr]-measurable}. For functionsf, g∈ Bbr∩ Dµbeing constant outside the subset Smr ={sS;s(Sr) =m} we have

Dmr [f, g](s) =D[f, g](s) for all sSmr .

Here σmr is the density function ofµonSmr with respect to the Lebesgue measure onSrm, that is,σmr is the symmetric function such that

1 m!

Srm

fˇSrσmr dxm=

Smr

f dµ (6.10)

for any bounded σ[πr]-measurable functions f, where πr=πSr. From this we see that Erm(f, g) =

Smr

D[f, g]σrmdxm forf, g∈ Brb∩ Dµ. (6.11) This obvious identity is one of the key points of the argument in [44]. In the following, we quote a sequence of results from [44].

Lemma 6.1 ([44, Lemma 2.2]). Assume (A1). Then the following holds:

(1) (E,Dµ ∩ Brb) is closable onL2(µ).

(2) (Er,Dµ) is closable onL2(µ), where we set Er =∑

m=1Erm. We write (E1,D1)(E2,D2) if

D1 ⊃ D2 and E1(f, f)≤ E2(f, f) for any f ∈ D2, and (E1,D1)(E2,D2) if

D1 ⊂ D2 and E1(f, f)≥ E2(f, f) for any f ∈ D1.

For a sequence{(En,Dn)}n∈N of positive definite, symmetric bilinear forms onL2(µ), we say {(En,Dn)} is increasing if (En,Dn) (En+1,Dn+1) for anyn N, and decreasing if (En,Dn)(En+1,Dn+1) for any n∈N.

By Lemma 6.1 (E,Dµ ∩ Brb) and (Er,Dµ) are closable on L2(µ). Then we denote the closures by (Er,Dr) and (Er,Dr), respectively.

Lemma 6.2 ([44, Lemma 2.2]). Assume (A1). Then the following hold.

(1){(Er,Dr)}r∈N is decreasing.

(2){(Er,Dr)}r∈N is increasing.

By definition the largest closable part (( ˜E)reg,( ˜D)reg) of a given positive symmetric form ( ˜E,D˜) with a dense domain is a closable form such that (( ˜E)reg,( ˜D)reg)( ˜E,D˜) and that (( ˜E)reg,( ˜D)reg) is the largest element of closable forms dominated by ( ˜E,D˜). Such a form exists uniquely.

Because (E,Dµ) is closable on L2(µ) under(A1), we define (E,D) as the closure. Let (E,D) be the symmetric form such that

E(f, f) = lim

r→∞Er(f, f) with the domainD=∪

r∈NDr. Then the closure of the largest closable part ((E)reg,(D))reg) of (E,D) corresponds to (E,D) [44].

Let (E,D) be the closed symmetric form such that E(f, f) = lim

r→∞Er(f, f) with the domain D={f

r=1Dr; limr→∞Er(f, f)<∞}. Summarizing above we obtain the next lemma.

Lemma 6.3. Assume (A1). Then the following hold.

(1) (E,D) is the strong resolvent limit of{(Er,Dr)}r∈N asr→ ∞. (2) (E,D) is the strong resolvent limit of{(Er,Dr)}r∈N asr→ ∞. (3) (E,D)(E,D).

Proof. The first two statements follow from Lemma 6.2 and the general theory of the monotone convergence theorem of closed forms. The third follows from (Er,Dr)(Er,Dr) for anyr, and thus we have (E,D)(E,D) by the monotone convergence of these forms given by Lemma 6.2.

(A2) The random point field µsatisfies

r=1

mµ(Smr )<∞ for each m∈N.

We refer to [42] for the quasi-regularity and the locality of Dirichlet forms and related notions. The importance of the quasi-regularity and the locality is that they guarantee the existence of diffusion associated with the Dirichlet form.

We obtain an unlabeled diffusion from [44]. The next result is one of the main theorems in [44]. To be more precise, boundedness of density functions was assumed in addition to (A2) in [44], this was removed in [31].

Proposition 6.4 ([44, Theorem 1, Corollary 1]). Assume (A1) and (A2). Then (E,D) is a local quasi-regular Dirichlet form on L2(µ). In particular, there exists an S-valued, µ-reversible diffusionX associated with (E,D).

Let (E,D) and (E,D) be as in Lemma 6.3. We assume:

(A3) (E,D) = (E,D).

Remark 6.5. From Proposition 6.4 and (A3) we deduce that (E,D) is a quasi-regular Dirichlet form, and there exists the associated S-valued diffusion. This diffusion is the same as that of the diffusion associated with (E,D).

6.2.2 Finite particle approximation for a random point field and main result:

convergence of unlabeled dynamics.

In Section 6.2.1 we introduced two schemes of finite volume approximations related to bounded domains Sr and we take r → ∞. In the present section, we introduce another approximation consisting of Dirichlet forms describingN-particles. Note that the particles in the present section move in whole S, but the number of particles at the each stage of approximating dynamics is N N, and we letN go to infinity.

Let N} be a sequence of random point fields such that µN(s(S) =N) = 1 for any N N and limN→∞µN =µweakly. For r, m∈N, let σN,mr be the m-particles density of µN on Sr with respect to the Lebesgue measure. We set

Er,kN (f) =

k m=1

Srm

D[f]σN,mr dxm.

Hereafter, E(f) and D[f] denote E(f, f) and D[f, f], respectively. We remark that, if f ∈ Bbr∩ Dµ, then by (6.9) and (6.11) we have

Er,kN(f) =

k m=1

S

Dmr [f](s)dµN =

k m=1

ErN,m(f).

FromµN(s(S) =N) = 1 we have D =DµN, whereDµN is defined by (6.8) with µN. Recall that there exists a diffusion associated with a local, regular Dirichlet form. We refer to [16] for the definition of regular Dirichlet forms and related notions. To guarantee the existence ofN-particles dynamics, we assume:

(M1)For anyN N, (EN,D) is closable onL2N). Furthermore, the closure (EN,DN) of (EN,D) is a regular Dirichlet form on L2N).

Let XN and X be the diffusions associated with the Dirichlet space (EN,DN, L2N)) and (E,D, L2(µ)), respectively. We assume the initial distributions satisfy:

(M2) The distributions of XN0 and X0 have densities ξN L2N) and ξ L2(µ) with respect toµN and µ, respectively, and satisfy

Nlim→∞ξN =ξ strongly in the sense of Definition 6.15.

We assume density functions σrN,m and σmr of µN and µdefined in (6.10) satisfy:

(M3)For each r, m∈N

Nlim→∞

σrN,m

σrm 1

Smr = 0. (6.12)

Here ∥ · ∥Smr denotes theL(Srm, dx)-norm.

Theorem 6.6. Assume(A1)–(A3). Assume(M1)–(M3). Then we have

Nlim→∞XN =Xin distribution in C([0,∞);S). (6.13) The densityσmr in (6.12) may vanish in general. Then we introduce the condition

Cap ( ∪

m,r=1

{sSmr ;σrm(s) = 0})

= 0. (6.14)

Here, σmr = σrm(s1, . . . , sm) is regarded as a function on Smr = {s S;s(Sr) = m} such that σrm(s) = σmr (s1, . . . , sm) fors(· ∩Sr) =∑m

i=1δsi, and Cap is the capacity associated with (E,D) on L2(µ). See [16, 66p] for the definition of capacity.

We now relax the assumption (M3) as below. We shall use (M3’) when we present examples in Section 6.6.

(M3’) For eachr, m∈N,

Nlim→∞

σrN,m−σrm

Smr = 0. (6.15)

Furthermore, (6.14) holds.

Theorem 6.7. Assume (A1)–(A3). Assume (M1)–(M2) and (M3’). Then (6.13) holds.

A symmetric and locally integrable function ρn : Sn [0,) is called the n-point correlation function of a random point fieldµonS with respect to the Lebesgue measure ifρn satisfies

Ak11×···×Akmm

ρn(x1, . . . , xn)dx1· · ·dxn=

S

m i=1

s(Ai)!

(s(Ai)−ki)!

for any sequence of disjoint bounded measurable setsA1, . . . , Am∈ B(S) and a sequence of natural numbersk1, . . . , km satisfying k1+· · ·+km =n. If correlation functions converge compact uniformly, (6.15) is satisfied. In fact, the following relation between correlation functions and density functions hold. If for each r N there exist constantsc39 and c40 satisfying c39>0 and c40<1 such that

sup

xnSrn

ρn(xn)≤cn39nc40n,

then

σrm(xm) =

j=0

(1)j j!

Srm

ρm+j(xm,yj)m(dyj).

Let ρN,n be the n-correlation function of µN. We shall obtain (6.15) from uniform convergence ofρN,m toρm on Srm.

(M3”)Correlation functionsρN,n and ρn satisfy

Nlim→∞ρN,m−ρm

Srm = 0 for each r, m∈N, (6.16) sup

N∈N sup

xnSrn

ρN,n(xn)≤cn39nc40n. (6.17) Furthermore, (6.14) is satisfied.

Theorem 6.8. Assume (A1)–(A3). Assume (M1)–(M2) and (M3”). Then (6.13) holds.

Remark 6.9. (1) If σrm are bounded, then (6.12) implies (6.15).

(2) Clearly, (6.16) and (6.17) imply (6.15).

(3) Because of the variational formula of capacity, one can obtain (6.14) easily from esti-mates of correlation functions.

6.2.3 Convergence of labeled dynamics (SDE) and proof of Theorem 6.10 In this section, we consider labeled dynamics and formulate convergence of finite-dimensional SDEs to the limit ISDE.

Let u :SN S be the unlabeling map given byu(s) =∑

iδsi, where s = (si)i∈N. We assume the following:

(A4) Each particle is non-explosion and non-collision.

Because of(A4), we can construct the labeled dynamicsX= (Xi)i∈N∈C([0,∞);SN) such that Xt =∑

i∈NδXi

t with initial label l(X0) = X0. Next theorem proves dynamical convergence of labeled dynamics.

Theorem 6.10. Make the same assumptions as Theorem 6.6 or Theorem 6.7 or Theo-rem 6.8. Assume(A4) and that the initial distributions of the labeled dynamicsXN and X satisfy for eachm∈N,

Nlim→∞µN(lN,1, . . . ,lN,m)1 =µ◦(l1, . . . ,lm)1 (6.18) weakly. Then for each m∈N,

Nlim→∞(XN,1, . . . , XN,m) = (X1, . . . , Xm) (6.19) in distribution inC([0,∞);Sm).

Proof. From(A4)we can construct the labeled dynamicsXN andXsuch thatXN0 =lN(s) and X0=l(s). Note that the initial distribution is inLpN) for some 1< p. Then using Lyons-Zheng decomposition, we see the tightness of {(XN,i)mi=1}N∈N inC([0,∞);Sm) for each m.

The convergence of the finite-dimensional distributions of XN follows from the weak convergence of the unlabeled processes XN and the convergence of the labeled initial distributions (6.18).

Collecting these we obtain Theorem 6.10.

We next present the ISDE representation of the limit labeled dynamics.

We write f ∈Lploc[1]) if f ∈Lp(Sr×S, µ[1]) for all r N. Let C0(S)⊗ D be the algebraic tensor product ofC0(S) and D, that is,

C0(S)⊗ D ={

N i=1

fi(x)gi(y) ;fi ∈C0(S), gi ∈ D, N N}.

Definition 6.11 ([47]). An Rd-valued function dµ L1loc[1])d is called the logarithmic derivative ofµ if, for allf ∈C0(S)⊗ {D∩L(µ)},

S×S

dµ(x,y)f(x,y)µ[1](dxdy) =

S×S

xf(x,y)µ[1](dxdy).

Lemma 6.12 ([47]). Assume (A1)–(A4). Assume the logarithmic derivatives dµ of µ exists. Then, the following ISDE has a solution.

dXti =dBti+ 1

2dµ(Xti, Xti,)dt (iN). (6.20) Here Xti, denotes ∑

j̸=iδxj

t

Assume the logarithmic derivativedµN ofµN exists. Then the finite particle dynamics XN = (XN,1, . . . , XN,N)∈C([0,∞) ;SN) are solutions of SDEs such that

dXtN,i=dBN,it +1

2dµN(XtN,i, XtN,i)dt (i= 1, . . . , N). (6.21) Combining Lemma 6.12 with Theorem 6.10, we obtain convergence in distribution of solutions of SDEs (6.21) to a solution of the ISDE (6.20).

Remark 6.13. If we assume that the ISDE (6.20) has a unique solution in distribution, then the condition(A3) holds [31].

ドキュメント内 ランダム行列に関する無限粒子系の確率解析 (ページ 118-124)

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