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
*L*^{2}(µ) =*L*^{2}(S, µ). We define the bilinear form (*E,D*_{◦}* ^{µ}*) on

*L*

^{2}(µ) as

*E*(f, g) =

∫

S

D[f, g](s)dµ,

*D*_{◦}* ^{µ}*=

*{f*

*∈ D*

*◦*

*∩L*

^{2}(µ) ;

*E*(f, f)

*<∞}.*(6.8) For each

*r, m∈*Nwe set the bilinear form (

*E*

*r*

^{m}*,D*

^{µ}*) on*

_{◦}*L*

^{2}(µ) as

*E**r** ^{m}*(f, g) =

∫

S

D^{m}*r* [f, g](s)dµ. (6.9)

We make an assumption:

**(A1)** (*E**r*^{m}*,D*_{◦}* ^{µ}*) is closable on

*L*

^{2}(µ) for each

*r, m∈*N.

Set*B**r** ^{b}*=

*{f*;

*f*is bounded and

*σ[π*

*r*]-measurable

*}*. For functions

*f, g∈ B*

^{b}*r*

*∩ D*

*◦*

*being constant outside the subset S*

^{µ}

^{m}*=*

_{r}*{*s

*∈*S;s(S

*) =*

_{r}*m}*we have

D^{m}*r* [f, g](s) =D[f, g](s) for all s*∈*S^{m}_{r}*.*

Here *σ*^{m}* _{r}* is the density function of

*µ*onS

^{m}*with respect to the Lebesgue measure on*

_{r}*S*

_{r}*, that is,*

^{m}*σ*

^{m}*is the symmetric function such that*

_{r}1
*m!*

∫

*S*_{r}^{m}

*f*ˇ*S**r**σ*^{m}_{r}*dx**m*=

∫

S^{m}_{r}

*f dµ* (6.10)

for any bounded *σ[π** _{r}*]-measurable functions

*f, where*

*π*

*=*

_{r}*π*

_{S}*. From this we see that*

_{r}*E*

_{r}*(f, g) =*

^{m}∫

*S*^{m}_{r}

D[f, g]σ_{r}^{m}*dx** _{m}* for

*f, g∈ B*

_{r}

^{b}*∩ D*

^{µ}

_{◦}*.*(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,*D*^{µ}_{◦}*∩ B*_{r}* ^{b}*) is closable on

*L*

^{2}(µ).

(2) (*E**r**,D**◦** ^{µ}*) is closable on

*L*

^{2}(µ), where we set

*E*

*r*=∑

_{∞}*m=1**E**r** ^{m}*.
We write (

*E*

^{1}

*,D*

^{1})

*≤*(

*E*

^{2}

*,D*

^{2}) if

*D*^{1} *⊃ D*^{2} and *E*^{1}(f, f)*≤ E*^{2}(f, f) for any *f* *∈ D*^{2}*,*
and (*E*^{1}*,D*^{1})*≥*(*E*^{2}*,D*^{2}) if

*D*^{1} *⊂ D*^{2} and *E*^{1}(f, f)*≥ E*^{2}(f, f) for any *f* *∈ D*^{1}*.*

For a sequence*{*(*E*^{n}*,D** ^{n}*)

*}*

*n*

*∈N*of positive definite, symmetric bilinear forms on

*L*

^{2}(µ), we say

*{*(

*E*

^{n}*,D*

*)*

^{n}*}*is increasing if (

*E*

^{n}*,D*

*)*

^{n}*≤*(

*E*

^{n+1}*,D*

*) for any*

^{n+1}*n*

*∈*N, and decreasing if (

*E*

^{n}*,D*

*)*

^{n}*≥*(

*E*

^{n+1}*,D*

*) for any*

^{n+1}*n∈*N.

By Lemma 6.1 (*E,D*^{µ}*◦* *∩ B**r** ^{b}*) and (

*E*

*r*

*,D*

^{µ}*◦*) are closable on

*L*

^{2}(µ). Then we denote the closures by (

*E*

*r*

*,D*

*r*) and (

*E*

*r*

*,D*

*r*), respectively.

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

(1)*{*(*E**r**,D**r*)*}**r**∈N* is decreasing.

(2)*{*(*E**r**,D**r*)*}**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,D**◦** ^{µ}*) is closable on

*L*

^{2}(µ) under

**(A1), we define (**

*E,D*) as the closure. Let (

*E*

_{∞}*,D*

*) be the symmetric form such that*

_{∞}*E**∞*(f, f) = lim

*r**→∞**E**r*(f, f)
with the domain*D** _{∞}*=∪

*r**∈N**D**r*. 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**→∞**E**r*(f, f)
with the domain *D*=*{f* *∈*∩_{∞}

*r=1**D**r*; lim_{r→∞}*E**r*(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*{*(*E**r**,D**r*)*}**r**∈N* as*r→ ∞*.
(2) (*E,D*) is the strong resolvent limit of*{*(*E**r**,D**r*)*}**r**∈N* as*r→ ∞*.
(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 (*E**r**,D**r*)*≤*(*E**r**,D**r*)
for any*r, 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 *µ*satisfies

∑*∞*
*r=1*

*mµ(S*^{m}* _{r}* )

*<∞*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 *L*^{2}(µ). In particular, there exists an S-valued,
*µ-reversible diffusion*X 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 *S**r* and we take *r* *→ ∞. In the present section, we introduce another*
approximation consisting of Dirichlet forms describing*N*-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 let*N* go to infinity.

Let *{µ*^{N}*}* be a sequence of random point fields such that *µ** ^{N}*(s(S) =

*N*) = 1 for any

*N*

*∈*N and lim

_{N}

_{→∞}*µ*

*=*

^{N}*µ*weakly. For

*r, m∈*N, let

*σ*

^{N,m}*r*be the

*m-particles density of*

*µ*

*on*

^{N}*S*

*with respect to the Lebesgue measure. We set*

_{r}*E**r,k** ^{N}* (f) =

∑*k*
*m=1*

∫

*S*_{r}^{m}

D[f]σ^{N,m}_{r}*dx**m**.*

Hereafter, *E*(f) and D[f] denote *E*(f, f) and D[f, f], respectively. We remark that, if
*f* *∈ B*^{b}_{r}*∩ D*_{◦}* ^{µ}*, then by (6.9) and (6.11) we have

*E**r,k** ^{N}*(f) =

∑*k*
*m=1*

∫

S

D^{m}*r* [f](s)dµ* ^{N}* =

∑*k*
*m=1*

*E**r** ^{N,m}*(f).

From*µ** ^{N}*(s(S) =

*N*) = 1 we have

*D*

*=*

_{◦}*D*

^{µ}

_{◦}*, where*

^{N}*D*

^{µ}

_{◦}*is defined by (6.8) with*

^{N}*µ*

*. 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 of*

^{N}*N*-particles dynamics, we assume:

**(M1)**For any*N* *∈*N, (E^{N}*,D** _{◦}*) is closable on

*L*

^{2}(µ

*). Furthermore, the closure (E*

^{N}

^{N}*,D*

*) of (*

^{N}*E*

^{N}*,D*

*◦*) is a regular Dirichlet form on

*L*

^{2}(µ

*).*

^{N}Let X* ^{N}* and X be the diffusions associated with the Dirichlet space (

*E*

^{N}*,D*

^{N}*, L*

^{2}(µ

*)) and (*

^{N}*E,D, L*

^{2}(µ)), respectively. We assume the initial distributions satisfy:

**(M2)** The distributions of X^{N}_{0} and X0 have densities *ξ*^{N}*∈* *L*^{2}(µ* ^{N}*) and

*ξ*

*∈*

*L*

^{2}(µ) with respect to

*µ*

*and*

^{N}*µ, respectively, and satisfy*

*N*lim*→∞**ξ** ^{N}* =

*ξ*strongly in the sense of Definition 6.15.

We assume density functions *σ**r** ^{N,m}* and

*σ*

^{m}*of*

_{r}*µ*

*and*

^{N}*µ*defined in (6.10) satisfy:

**(M3)**For each *r, m∈*N

*N*lim*→∞*

*σ**r*^{N,m}

*σ*_{r}^{m}*−*1

*S*^{m}* _{r}* = 0. (6.12)

Here *∥ · ∥**S*^{m}* _{r}* denotes the

*L*

*(S*

^{∞}

_{r}

^{m}*, dx)-norm.*

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

*N*lim*→∞*X* ^{N}* =Xin distribution in

*C([0,∞*);S). (6.13) The density

*σ*

^{m}*in (6.12) may vanish in general. Then we introduce the condition*

_{r}Cap
( ∪^{∞}

*m,r=1*

*{*s*∈*S^{m}* _{r}* ;

*σ*

_{r}*(s) = 0*

^{m}*}*)

= 0. (6.14)

Here, *σ*^{m}* _{r}* =

*σ*

_{r}*(s1*

^{m}*, . . . , s*

*m*) is regarded as a function on S

^{m}*=*

_{r}*{*s

*∈*S;s(S

*r*) =

*m}*such that

*σ*

_{r}*(s) =*

^{m}*σ*

^{m}*(s*

_{r}_{1}

*, . . . , s*

*) fors(*

_{m}*· ∩S*

*) =∑*

_{r}

_{m}*i=1**δ*_{s}* _{i}*, and Cap is the capacity associated
with (E,

*D) on*

*L*

^{2}(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 each*r, m∈*N,

*N*lim*→∞*

*σ*_{r}^{N,m}*−σ*_{r}^{m}

*S*^{m}* _{r}* = 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 *ρ** ^{n}* :

*S*

^{n}*→*[0,

*∞*) is called the

*n-point*correlation function of a random point field

*µ*on

*S*with respect to the Lebesgue measure if

*ρ*

*satisfies*

^{n}∫

*A*^{k}_{1}^{1}*×···×**A*^{km}*m*

*ρ** ^{n}*(x1

*, . . . , x*

*n*)dx1

*· · ·dx*

*n*=

∫

S

∏*m*
*i=1*

s(A*i*)!

(s(A*i*)*−k**i*)!*dµ*

for any sequence of disjoint bounded measurable sets*A*_{1}*, . . . , A*_{m}*∈ B*(S) and a sequence of
natural numbers*k*1*, . . . , k**m* satisfying *k*1+*· · ·*+*k**m* =*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 constants*c*_{39} and *c*_{40}
satisfying *c*_{39}*>*0 and *c*_{40}*<*1 such that

sup

**x***n**∈**S*_{r}^{n}

*ρ** ^{n}*(x

*)*

_{n}*≤c*

^{n}_{39}

*n*

^{c}^{40}

^{n}*,*

then

*σ*_{r}* ^{m}*(x

*) =*

_{m}∑*∞*
*j=0*

(*−*1)^{j}*j!*

∫

*S*_{r}^{m}

*ρ** ^{m+j}*(x

_{m}*,*

**y**

*)m(dy*

_{j}*).*

_{j}Let *ρ** ^{N,n}* be the

*n-correlation function of*

*µ*

*. We shall obtain (6.15) from uniform convergence of*

^{N}*ρ*

*to*

^{N,m}*ρ*

*on*

^{m}*S*

_{r}*.*

^{m}**(M3”)**Correlation functions*ρ** ^{N,n}* and

*ρ*

*satisfy*

^{n}*N*lim*→∞**ρ*^{N,m}*−ρ*^{m}

*S*_{r}* ^{m}* = 0 for each

*r, m∈*N

*,*(6.16) sup

*N**∈N* sup

**x***n**∈**S*_{r}^{n}

*ρ** ^{N,n}*(x

*n*)

*≤c*

^{n}_{39}

*n*

^{c}^{40}

^{n}*.*(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 *σ*_{r}* ^{m}* 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 :*S*^{N}*→* S be the unlabeling map given byu(s) =∑

*i**δ*_{s}* _{i}*, where s = (s

*)*

_{i}

_{i}*. We assume the following:*

_{∈N}**(A4)** Each particle is non-explosion and non-collision.

Because of**(A4), we can construct the labeled dynamicsX**= (X* ^{i}*)

_{i}

_{∈N}*∈C([0,∞*);

*S*

^{N}) such that X

*=∑*

_{t}*i**∈N**δ*_{X}*i*

*t* with initial label l(X_{0}) = **X**_{0}. 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 dynamics**X*** ^{N}* and

**X**satisfy for each

*m∈*N,

*N*lim*→∞**µ*^{N}*◦*(l^{N,1}*, . . . ,*l* ^{N,m}*)

^{−}^{1}=

*µ◦*(l

^{1}

*, . . . ,*l

*)*

^{m}

^{−}^{1}(6.18) weakly. Then for each

*m∈*N,

*N*lim*→∞*(X^{N,1}*, . . . , X** ^{N,m}*) = (X

^{1}

*, . . . , X*

*) (6.19) in distribution in*

^{m}*C([0,∞*);

*S*

*).*

^{m}*Proof.* From**(A4)**we can construct the labeled dynamics**X*** ^{N}* and

**X**such that

**X**

^{N}_{0}=l

*(s) and*

^{N}**X**

_{0}=l(s). Note that the initial distribution is in

*L*

*(µ*

^{p}*) for some 1*

^{N}*< p. Then using*Lyons-Zheng decomposition, we see the tightness of

*{*(X

*)*

^{N,i}

^{m}

_{i=1}*}*

*N*

*∈N*in

*C([0,∞*);

*S*

*) for each*

^{m}*m.*

The convergence of the finite-dimensional distributions of **X*** ^{N}* follows from the weak
convergence of the unlabeled processes X

*and the convergence of the labeled initial distributions (6.18).*

^{N}Collecting these we obtain Theorem 6.10.

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

We write *f* *∈L*^{p}_{loc}(µ^{[1]}) if *f* *∈L** ^{p}*(S

*r*

*×*S, µ

^{[1]}) for all

*r*

*∈*N. Let

*C*

_{0}

*(S)*

^{∞}*⊗ D*

*◦*be the algebraic tensor product of

*C*

_{0}

*(S) and*

^{∞}*D*

*, that is,*

_{◦}*C*_{0}* ^{∞}*(S)

*⊗ D*

*◦*=

*{*

∑*N*
*i=1*

*f**i*(x)g*i*(y) ;*f**i* *∈C*_{0}* ^{∞}*(S), g

*i*

*∈ D*

*◦*

*, N*

*∈*N}

*.*

**Definition 6.11** ([47]). An R* ^{d}*-valued function d

^{µ}*∈*

*L*

^{1}

_{loc}(µ

^{[1]})

*is called*

^{d}*the logarithmic*

*derivative*of

*µ*if, for all

*f*

*∈C*

_{0}

*(S)*

^{∞}*⊗ {D*

_{◦}*∩L*

*(µ)*

^{∞}*}*,

∫

*S**×*S

d* ^{µ}*(x,y)f(x,y)µ

^{[1]}(dxdy) =

*−*

∫

*S**×*S

*∇**x**f*(x,y)µ^{[1]}(dxdy).

**Lemma 6.12** ([47]). Assume **(A1)–(A4). Assume the logarithmic derivatives** d* ^{µ}* of

*µ*exists. Then, the following ISDE has a solution.

*dX*_{t}* ^{i}* =

*dB*

_{t}*+ 1*

^{i}2d* ^{µ}*(X

_{t}

^{i}*, X*

_{t}

^{i,}*)dt (i*

^{♢}*∈*N). (6.20) Here

*X*

_{t}

^{i,}*denotes ∑*

^{♢}*j̸=i**δ*_{x}^{j}

*t*

Assume the logarithmic derivatived^{µ}* ^{N}* of

*µ*

*exists. Then the finite particle dynamics*

^{N}**X**

*= (X*

^{N}

^{N,1}*, . . . , X*

*)*

^{N,N}*∈C([0,∞*) ;

*S*

*) are solutions of SDEs such that*

^{N}*dX*_{t}* ^{N,i}*=

*dB*

^{N,i}*+1*

_{t}2d^{µ}* ^{N}*(X

_{t}

^{N,i}*, X*

_{t}

^{N,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].