Preliminaries

is established by assuming the limits of the lower and upper Dirichlet forms in (5.1) are equal in addition to a certain strong convergence of the random point fields. Hence our main theorem (Theorem 3.2) plays a crucial role in the dynamical universality of random matrices in the sense given above.

The organization of the paper is as follows: In Section 5.2, we prepare the two schemes of the Dirichlet forms describing interacting Brownian motions, and quote related results.

In Section 5.3, we state the main theorems (Theorem 5.14 and Theorem 5.15). In Sec-tion 5.4, we prove Theorem 5.14. In SecSec-tion 5.5, we prove Theorem 5.15. In SecSec-tion 5.6, we comment on a generalization to the uniformly elliptic case. In Section 5.7, we con-struct cut-off coefficients b_r,s,p appearing in (A6). In Section 5.8, we present examples.

In Appendix (Section 5.9) we present a setD_• used in Section 5.2 and prove Lemma 5.7.

In Section 5.10, we give concluding remarks with some open questions.

are independent of s. In this case we often writef_r^m instead off_r,s^m. We set Br(S) ={f :S→R; f is σ[πr]-measurable}, B_∞(S) =

∪∞ r=1

Br(S), D◦ ={f ∈ B∞(S) ; f_r,s^m is smooth onS_r^m for all m, r∈N,s∈S}.

Note that D◦∩L²(S, µ) is dense in L²(S, µ) and D◦ ⊂C(S), where C(S) is the set of all continuous functions on S. We remark that, if f ∈ D◦ and f is σ[π_r]-measurable, then f_r,s^m(x₁, . . . , x_m) is constant in x_m on the boundary ∂S_r for each (x₁, . . . , x_m−1) ∈S_r^m−1, and its derivatives vanishes on∂S_r^m.

For s=∑

iδ_si we set ∇si = (_∂s^∂

i1, . . . ,_∂s^∂

id). Forf, g∈ D◦ let D^mr [f, g](s) =

{1 2

∑

i;si∈Sr∇sif_r,s^m(x^m_r (s))· ∇sig_r,s^m(x^m_r (s)) fors∈S^m_r ,

0 fors∈/S^m_r . (5.7)

Moreover, we set

Dr =

∑∞ m=1

D^mr . (5.8)

Note that D^m_r [f, g] is independent of the choice of the S_r^m-coordinate x^m_r (s) and is well-defined. We now define bilinear forms onD◦:

E_r^µ,m(f, g) =

∫

S

D^m_r [f, g](s)µ(ds) and E_r^µ=

∫

S

Dr[f, g](s)µ(ds). (5.9) Then clearlyEr^µ=∑_∞

m=1Er^µ,m.

Let (E^µ,D◦^µ) be a bilinear form on L²(S, µ) with domain D◦^µ defined by E^µ(f, f) = lim

r→∞E_r^µ(f, f), (5.10)

D^µ_◦ ={f ∈ D_◦∩L²(S, µ) ;E^µ(f, f)<∞}.

We note that Er^µ(f, f) is nondecreasing in r, and hence the limit in (5.10) exists. We assume

(E_r^µ,m,D_◦^µ) is closable onL²(S, µ) for eachm, r∈N. (5.11) We present later a sufficient condition regarding (5.11); see (A1)in Section 5.2.

Lemma 5.1. ([44, Lemma 2.2, Theorem 2]) Assume (5.11). Then the following hold.

(1) (E^µ,D_◦^µ∩ Br(S)) and (Er^µ,D^µ_◦) are closable on L²(S, µ) for each r.

(2) (E^µ,D◦^µ) is closable onL²(S, µ).

For symmetric bilinear forms (E,D) and (E^′,D^′) we write (E,D) ≤(E^′,D^′) ifD ⊃ D^′ and E(f, f) ≤ E^′(f, f) for all f ∈ D^′. We say a sequence of symmetric bilinear forms {(En,Dn)}n∈Nis increasing if (En,Dn)≤(En+1,Dn+1) for alln. Replacing≤by≥, we call {(En,Dn)}n∈Ndecreasing. Let (E^upr,D^uprr ) and (Er^lwr,Dr^lwr) denote the closures of (E^µ,D◦^µ∩ Br(S)) and (Er^µ,D^µ_◦) onL²(S, µ), respectively. Then we quote:

Lemma 5.2 ([44, Lemma 2.1]). Assume (5.11). Then (1){(E_r^lwr,D^lwr_r )}r∈N is increasing.

(2){(E^upr,D^uprr )}r∈Nis decreasing.

Let (E^lwr,D^lwr) be the increasing limit of {(Er^lwr,Dr^lwr)}r∈N, that is, E^lwr(f, f) = lim

r→∞Er^lwr(f, f), and D^lwr={f ∈ ∩

r∈N

Dr^lwr; lim

r→∞Er^lwr(f, f)<∞}. Then (E^lwr,D^lwr) is closed onL²(S, µ) by construction. Recall that (E^µ,D^µ◦) is closable on L²(S, µ) by Lemma 5.1. We then denote by (E^upr,D^upr) the closure of (E^µ,D^µ_◦) onL²(S, µ).

Let G^lwr_r,α, G^lwr_α , G^upr_r,α, and G^upr_α be resolvent of (E_r^lwr,D^lwr_r ), (E^lwr,D^lwr), (E^upr,D^uprr ), and (E^upr,D^upr) onL²(S, µ), respectively.

Lemma 5.3. ([44, Lemma 2.1, Theorem 2]) Assume (5.11). Then

(1){G^lwr_r,α}r∈N converges toG^lwr_α strongly in L²(S, µ) as r→ ∞ for each α >0.

(2){G^uprr,α}r∈N converges toG^uprα strongly inL²(S, µ) as r→ ∞ for each α >0.

By construction we have for each r

(E_r^lwr,D^lwr_r )≤(E^upr,D^upr_r ). (5.12) Hence takingr→ ∞ we see that

(E^lwr,D^lwr)≤(E^upr,D^upr). (5.13) We call (E^lwr,D^lwr) the lower Dirichlet form and (E^upr,D^upr) the upper Dirichlet form. We also call {(Er^lwr,D^lwrr )}r∈N a lower scheme and {(E^upr,D^uprr )}r∈N an upper scheme. The relations (5.12) and (5.13) justify the names of these schemes.

5.2.2 Quasi-Gibbs measures, unlabeled diffusions, and labeled dynamics Let Λ_r be the Poisson random point field whose intensity is the Lebesgue measure on S_r and set Λ^m_r = Λ_r(· ∩S^m_r ). Let Φ :S → R∪ {∞}and Ψ : S² →R∪ {∞} be measurable functions such that Ψ(x, y) = Ψ(y, x). Following [48, 49] we quote:

Definition 5.4. A random point field µ is called a (Φ,Ψ)-quasi Gibbs measure if its regular conditional probabilities

µ^m_r,s=µ(πr(x)∈ · |π^c_r(x) =π_r^c(s),x(Sr) =m) satisfy, for allr, m∈Nand µ-a.s.s,

c⁻₃₀¹e^−Hmr(x)Λ^m_r (dx)≤µ^m_r,s(dx)≤c30e^−Hmr(x)Λ^m_r (dx). (5.14) Here c₃₀=c₃₀(r, m,s) is a positive constant depending on r, m,s. For two measures µ, ν on a σ-field F, we write µ ≤ ν if µ(A) ≤ ν(A) for all A ∈ F. Moreover, H^mr is the Hamiltonian on S_r defined by

H^m_r (x) = ∑

xi∈Sr 1≤i≤m

Φ(x_i) + ∑

xj ,xk∈Sr 1≤j<k≤m

Ψ(x_j, x_k) forx=∑

i

δ_xi.

Remark 5.5. (1) From (5.14), we see that for all r, m ∈ N and µ-a.s. s, µ^m_r,s(dx) have (unlabeled) Radon–Nikodym densities m^m_r,s(x) with respect to Λ^m_r . Clearly, the canonical Gibbs measures µ with potentials (Φ,Ψ) are quasi-Gibbs measures, and their densities m^m_r,s(x) with respect to Λ^m_r are given by the Dobrushin–Lanford–Ruelle (DLR) equation.

That is, for µ-a.s.s=∑

jδ_sj m^m_r,s(x) = 1

Zr,s^m

exp{−H^mr (x)− ∑

xi∈Sr , sj∈Scr 1≤i≤m

Ψ(x_i, s_j)}.

HereZr,s^m is the normalizing constant. For random point fields appearing in random matrix theory, interaction potentials are logarithmic functions, where the DLR equations do not make sense as stated because the term ∑

xi∈Sr, sj∈S^crΨ(xi, sj) diverges. The notion of a quasi-Gibbs measure still makes sense for logarithmic potentials.

(2) We refer to [48, 49] for sufficient conditions of quasi-Gibbs measures. These conditions give us the quasi-Gibbs property of random point fields appearing in random matrix theory, such as sine_β, Airy_β (β= 1,2,4), and Bessel2,α (1≤α), and Ginibre random point fields [48, 49, 54, 17].

We make the following assumption.

(A1)µis a (Φ,Ψ)-quasi Gibbs measure. Furthermore, there exists upper semi-continuous functions ( ˆΦ,Ψ) and positive constantsˆ c31 and c32 satisfying

c⁻₃₁¹Φ(x)ˆ ≤Φ(x)≤c₃₁Φ(x),ˆ c⁻₃₂¹Ψ(x, y)ˆ ≤Ψ(x, y)≤c₃₂Ψ(x, y),ˆ where Ψ and ˆΨ satisfy Ψ(x, y) = Ψ(y, x) and ˆΨ(x, y) = ˆΨ(y, x).

If these interaction potentials are translation invariant, we often write Ψ(x, y) = Ψ(x− y) and ˆΨ(x, y) = ˆΨ(x−y). The importance of(A1)lies in the fact that it gives a sufficient condition of the basic assumption (5.11). We quote:

Lemma 5.6 ([48, 45-46pp]). Assume (A1). Then (Er^µ,m,D_◦^µ) is closable on L²(S, µ) for each m, r∈N. In particular, (Er^µ,D◦^µ) is closable onL²(S, µ).

We now recall two basic notions on random point fields: correlation functions and density functions.

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

∫

A^k₁¹×···×A^kmm

ρⁿ(x₁, . . . , x_n)dx₁· · ·dx_n=

∫

S

∏m i=1

s(A_i)!

(s(Ai)−ki)!dµ

for any sequence of disjoint bounded measurable setsA1, . . . , Am∈ B(S) and a sequence of positive integersk₁, . . . , k_msatisfyingk₁+· · ·+k_m =n. Whens(A_i)−k_i <0, according to our interpretation, s(Ai)!/(s(Ai)−ki)! = 0 by convention. We assume that µhasn-point correlation functionρⁿ for each n∈N.

A symmetric function σ^k_r : S_r^k → [0,∞) is called the k-point density function of a random point field µ on S_r with respect to the Lebesgue measure if for all non-negative, bounded σ[πr]-measurable function f with S_r^k-representation f_r^k

1 k!

∫

S^k_r

f_r^kσ_r^kdx^k=

∫

S^k_r

f dµ <∞.

Let S^m_r ={s∈S;s(S_r) =m} as before. We make the following assumption.

(A2) ∑_∞

m=1mµ(S^m_r )<∞ for all r∈N.

A family of probability measures {Px}x∈S on C([0,∞);S) is called a diffusion if the canonical process X = {Xt} under Px is a continuous process having a strong Markov property starting at x. Furthermore, {P_x}x∈S is called conservative if it has an invariant probability measure.

Assume(A1). Then we deduce from Lemma 5.1 and Lemma 5.6 that the non-negative form (E^µ,D^µ_◦) is closable onL²(S, µ). Therefore, let (E^upr,D^upr) be its closure onL²(S, µ).

The next result is a refinement of [44, 119p. Corollary 1] and can be proved in a similar fashion. We postpone the proof to Appendix (see Section 5.9).

Lemma 5.7. Assume(A1)and(A2). Then there exists aµ-reversible diffusion{P^uprx }x∈S

associated with the Dirichlet form (E^upr,D^upr) on L²(S, µ).

We note that (A2) is used to guarantee the existence of the diffusion. The µ-reversibility of the diffusion follows from 1∈ D^upr and symmetry of (E^upr,D^upr).

By construction, such a family of diffusion measuresP^upr ={P^upr_x }with quasi-continuity in x is unique for quasi-everywhere starting point x. Equivalently, there exists a set S0

such that the complement of S₀ has capacity zero and the family of diffusion measures P^upr ={P^upr_x }associated with the Dirichlet space above with quasi-continuity inxis unique for all x∈S0 and P^uprx (Xt∈S0 for allt) = 1 for all x∈S0.

We next lift the unlabeled dynamics X to a labeled dynamics X = (Xⁱ)_i∈N. Under P^upr = {P^upr_x }, we can write X_t = ∑_∞

i=1δ_Xi

t, where each Xⁱ = {X_tⁱ} is a continuous process with time parameter of the form [0, b) or (a, b). We callXⁱ tagged particles and X = (Xⁱ)_i∈N labeled dynamics. Note that for a given unlabeled process X, there exist plural labeled dynamics in general. We next give a condition such that X = (Xⁱ)_i∈N is determined uniquely. For this purpose, we impose the following condition:

(A3) Under P^upr = {P^uprx }, each tagged particle {Xⁱ}i∈N does not collide with another.

Furthermore,{Xⁱ}i∈Ndoes not hit the boundary ∂S ofS.

This condition is equivalent to both the capacity of multiple points and that of config-urations with particles on the boundary∂S being zero:

Cap^µ({s∈S;s({x})≥2 for some x∈S}) = 0, Cap^µ({s∈S;s(∂S)≥1}) = 0.

Here Cap^µ denotes the one-capacity with respect to the Dirichlet space (E^µ,D^µ) on L²(S, µ) (see [16] for the definition of capacity).

Let Erf(t) =∫_∞

t (1/√

2π)e^−|x|2/2dxbe the error function. We further assume:

(A4) There exists aT >0 such that, for eachR >0,

rlim→∞Erf

( r

√(r+R)T ) ∫

|x|≤r+R

ρ¹(x)dx= 0.

From(A4), we deduce the non-explosion of each tagged particle [46, Theorem 2.5]. We hence see from (A3)and (A4) that under P^upr ={P^uprx }each tagged particle of{Xⁱ}i∈N

neither collide each other nor hit the boundary∂S nor explode.

We call u the unlabeling map if u((x_i)) =∑

iδ_xi. We call la label if l:S →S^N is a measurable map defined forµ-a.s. x, and u◦l(x) =x. For simplicity, we take las

|lⁱ(x)|<|lⁱ⁺¹(x)| for all i∈N

throughout the paper. Because µhas an m-point correlation function for each m,l(x) is well defined for µ-a.s.x.

Lemma 5.8 ([46, 48]). Assume that (A1)–(A4). Let l be a label. Then under P^upr = {P^uprx } there exists a unique, labeled dynamics X = (Xⁱ)i∈N ∈ C([0,∞);S^N) such that X₀ =l(X₀) and thatX_t=∑

i∈Nδ_Xⁱ

t for all t.

Once the initial labellis assigned, the particles are marked forever because they neither collide nor explode. We hence determine the labeled dynamics X from the unlabeled dynamicsXand the labelluniquely. We have thus had a natural correspondence between X and (X,l) under the conditions (A3) and (A4). We remark here that Xt ̸= l(Xt) for t >0 in general.

The next lemma will be used in the proof of Lemma 5.12 and Lemma 5.20.

Lemma 5.9. Assume (A4). Then for each r, T ∈N, the following holds.

∫

S

Erf

(|x| −r

√T )

ρ¹(x)dx <∞. (5.15)

Proof. LetF(u) =∫

Suρ¹(x)dx. Then from (A4)we deduce F(u) =o

( 1 Erf

(u√−R uT

))

asu→ ∞.

Hence we obtain

∫

S

Erf

(|x| −r

√T )

ρ¹(x)dx=

∫ _∞

0

Erf (u−r

√T )

F^′(u)du

= [

Erf (u−r

√T )

F(u) ]_∞

0 −

∫ _∞

0

∂

∂uErf (u−r

√T )

F(u)du

<∞.

This implies (5.15).

5.2.3 ISDE-representation: Logarithmic derivative

We next present the ISDE describing the labeled dynamics given by Lemma 5.8. The key notion for this is the logarithmic derivative of µto be introduced below.

We first recall two new measures arising from random point fieldµ. The first concerns the conditioning ofµ, the second its disintegration.

Forx= (x₁, . . . , x_k)∈S^k a random point fieldµ_x is called the reduced Palm measure of µconditioned atx∈S^k ifµx is the regular conditional probability defined as

µ_x=µ(· −

∑k i=1

δ_xi|s({x_i})≥1 for i= 1, . . . , k).

A Radon measureµ^[k]on S^k×Sis called the k-Campbell measure of µifµ^[k]is given by µ^[k](dxds) =ρ^k(x)µx(ds)dx.

We set L¹_loc(S×S, µ^[1]) =∩_∞

r=1L¹(S×S, µ^[1]r ), whereµ^[1]r (·) =µ^[1](· ∩Sr×S). We set C₀^∞(S)⊗ D◦ ={

∑m i=1

fi(x)gi(y) ;fi ∈C₀^∞(S), gi∈ D◦, m∈N}. We now recall the notion of the logarithmic derivative of µ[47].

Definition 5.10. An R^d-valued function d^µ∈L¹_loc(S×S, µ^[1])^d is called the logarithmic derivative ofµ if, for allφ∈C₀^∞(S)⊗ D_◦,

∫

S×S

d^µ(x,y)φ(x,y)µ^[1](dxdy) =−

∫

S×S

∇xφ(x,y)µ^[1](dxdy). (5.16) We make the following assumption:

(A5) µhas a logarithmic derivatived^µ.

The next lemma reveals the importance of logarithmic derivative.

Lemma 5.11 ([47]). Assume (A1)–(A5). LetX andlbe as in Lemma 5.8. Assume b= 1

2d^µ. (5.17)

Then there existsS₀ ⊂Ssuch that µ(S₀) = 1 and that the labeled dynamicsX= (Xⁱ)_i∈N underP^uprx solves the ISDE for each x∈S0

dX_tⁱ =dB_tⁱ+b(X_tⁱ,X^⋄_tⁱ)dt, i∈N, (5.18)

X₀ =l(x), (5.19)

where B = (Bⁱ)_i∈N is the (R^d)^N-valued standard Brownian motion, and X^⋄i_t = ∑

j̸=iδ_Xj t

forX= (Xⁱ)_i∈N.

Let X = (Xⁱ)_i∈N be a solution of ISDE (5.18) and denote by X_t = ∑_∞

i=1δ_Xi t the associated unlabeled process. Let µ_t be the distribution of X_t. We make the following assumptions on X.

(µ-AC)Theµ-absolutely continuous condition is satisfied. That is, ifX₀=µin law, then

µt≺µ for all t≥0, (5.20)

whereµt≺µ meansµtis absolutely continuous with respect to µ.

(NBJ) The no-big-jump condition is satisfied. That is, if the distribution ofX₀ equals to µ, then for eachr, T ∈N

P(I_r,T(X)<∞) = 1, (5.21)

whereI_r,T is the maximal label with which the particle intersects S_r defined by

I_r,T(X) = max{i∈N∪ {∞};|X_tⁱ| ≤r for some 0≤t≤T}. (5.22) Lemma 5.12 ([53]). Assume (A1)–(A5). Then under P^upr = {P^uprx }x∈S the labeled dynamicsX satisfies the conditions (µ-AC), and (NBJ).

Proof. Because the unlabeled dynamicsXis µ-reversible, µt=µ for allt. Hence (µ-AC) is obvious. The second claim follows from the Lyons–Zheng decomposition and Lemma 5.9 (see [53, Lemma 9.4] for detail).

5.2.4 Finite systems in SR of interacting Brownian motions with reflecting boundary condition

We give the SDE representation of the unlabeled process X associated with the Dirichlet form (E_R^lwr,D^lwr_R ) on L²(S, µ). We denote by P^lwr_R = {P^lwr_R,x} the family of the diffusion measures given by (E_R^lwr,D_R^lwr) on L²(S, µ). The Dirichlet form (E_R^lwr,D^lwr_R ) is dominated by (E^upr,D^upr), that is,

(ER^lwr,D^lwrR )≤(E^upr,D^upr). (5.23) Then from (5.23) we see that the capacity of (E_R^lwr,D^lwr_R ) is dominated by that of (E^upr,D^upr).

Hence non-collision of tagged particles underP^lwr_R follows from that of the limit diffusionX given by (E^upr,D^upr), which is assumed by(A3). With the same reason, tagged particles under P^lwr_R do not hit the set (∂S)∩S_R. Non-explosion of tagged particles under P^lwr_R is obvious because they are reflecting diffusion on SRand frozen outside SR.

We now denote by X= (Xⁱ)^∞_i=1 the labeled process associated withX and the labell.

Then X={X_t} is given by X_t =∑_∞

i=1δ_Xi

t from X = (Xⁱ)^∞_i=1. By definitionX₀ =l(X₀).

The processX underP^lwr_R describes the system of interacting Brownian motions in which 1. each particle in SR moves inSR and when it hits the boundary ∂SR, it reflects and

enter the domain S_R immediately,

2. the particles out ofS_R stay the initial positions forever.

We denote by µ^Rs the regular conditional distribution defined by

µ^Rs(·) =µ(· |σ[π_R^c])(s) forµ-a.s. s. (5.24) Let S^Rs ={y∈S;π_R^c(y) =π^c_R(s)}. Then µ^Rs is a probability measure supported on S^Rs. Letµ^Rs,[1] be the 1-Campbell measure ofµ^Rs. Then we have

µ^Rs,[1](dxdy) =ρ^Rs,1(x)µ^Rs_x (dy)dx forx∈SR, (5.25) where ρ^Rs,1 is the one-point correlation function of µ^Rs and µ^Rs_x is the reduced Palm measure ofµ^Rs conditioned atx. By the Green formula, we see for allφ∈C₀^∞(S)⊗ D◦,

−

∫

S_R×S

∇xφ(x,y)µ^Rs,[1](dxdy) =

∫

S_R×S

d^µ(x,y)φ(x,y)µ^Rs,[1](dxdy) (5.26) +

∫

∂SR×S

φ(x,y)n^R(x)SR(dx)µ^Rs_x (dy),

whereSR is the Lebesgue surface measure on the boundary∂SRand n^R(x) is the inward normal unit vectors at x ∈ ∂S_R. Hence for µ-a.s. s and for µ^Rs_x -a.s. y, the logarithmic derivatived^Rs of µ^Rs coincides with the sum of

d^µ(x, π_R(y) +π_R^c(s)) forx∈S_R

and a singular part associated with the the boundary∂SR. We then obtain informally d^Rs(x,y) = 1SR(x)d^µ(x, πR(y) +π_R^c(s)) +n^R(x)1∂S_R(x)δx. (5.27) Here we naturally extend the domain of d^Rs(x,y) to S×S by taking d^Rs(x,y) = 0 for x̸∈SR. This is reasonable because particles outsideSR are fixed.

By definition x(S_R) coincides with the number of particles in S_R for a given configu-rationx. From the Green formula (5.26) we see that X= (Xⁱ)^∞_i=1 is the system of infinite number of particles such that only particles in SR move and satisfies the following SDE:

Forµ-a.s. s=∑

iδ_si and for µ^Rs-a.s.x=∑

iδ_xi dX_tⁱ =dBⁱ_t+1

2d^µ(X_tⁱ,X^⋄_tⁱ)dt+1

2n^R(X_tⁱ)dL^R,i_t , 1≤i≤x(SR), (5.28) dL^R,i_t =1_∂SR(X_tⁱ)dL^R,i_t , 1≤i≤x(SR), (5.29)

X_tⁱ =X₀ⁱ, i >x(SR), (5.30)

X₀ =l(x), (5.31)

where l(s) = (si)^∞_i=1,l(x) = (xi)^∞_i=1, X^⋄_tⁱ = ∑

j̸=iδ_X^j

t, and L^R,i ={L^R,i_t } are non-negative increasing processes; see for instance [8]. The particles outside S_R are frozen by (5.30).

Hence L^R,i_t = 0 for i >x(S_R). We remark that s_i=x_i for alli >x(S_R) for µ^s-a.s.x.

Let (E_R^lwr,D_R^lwr) be the Dirichlet form introduced in Lemma 5.2. Then we can easily deduce from(A1)and (A2)that (E_R^lwr,D_R^lwr) is a quasi-regular Dirichlet form onL²(S, µ) and that there exists the associated diffusionX. The capacity for (E_R^lwr,D_R^lwr) is dominated by that for (E^upr,D^upr). Hence from(A3)we deduce thatXhas also non-collision property.

Clearly, each tagged particle of X does not explode because of the definition of E_R^lwr. We

have thus obtained the labeled process X from the unlabeled process X and the label l.

Moreover, using the Fukushima decomposition and taking (5.27) into account, we see that X is a solution of SDE (5.28)–(5.31).

Let µ^Rs be as (5.24). Then

µ(·) =

∫

S

µ^Rs(·)µ(ds) (5.32)

and µ^Rs(S^Rs) = 1, whereS^Rs={y∈S;π^c_R(y) =π^c_R(s)} as before. We set ER^Rs(f, g) :=

∫

S

DR[f, g]dµ^Rs=

∫

S

DR[f_Rs, g_Rs]dµ^Rs, (5.33) D_◦^Rs={f ∈ D◦∩L²(S, µ^Rs) ;ER^Rs(f, f)<∞},

where we sethRs(·) =h(πR(·) +π^c_R(s)) for a functionhonS. The second equality in (5.33) is clear because forµ^Rs-a.s.x

DR[f, g](x) =DR[f, g]Rs(x) =DR[fRs, gRs](x).

From (A1) we easily deduce that (E_R^Rs,D^Rs_◦ ) is closable on L²(S, µ^Rs). We then denote by (E_R^Rs,lwr,D_R^Rs,lwr) the closure of (E_R^Rs,D_◦^Rs) on L²(S, µ^Rs). Furthermore, we see that (E_R^Rs,lwr,D_R^Rs,lwr) is a quasi-regular Dirichlet form on L²(S^Rs, µ^Rs). Hence there exists a diffusion X associated with (E_R^Rs,lwr,D^Rs,lwr_R ) on L²(S^Rs, µ^Rs). Using the Fukushima de-composition, we deduce that the associated labeled diffusion is a solution of the SDE (5.28)–(5.31) forµ^Rs-a.s.xforµ-a.s.s.

Lemma 5.13. Let X^lwr be the solution of the SDE (5.28)–(5.31) given by the Dirichlet form (E_R^lwr,D^lwr_R ) on L²(S, µ). LetX^Rs,lwr be the solution of the SDE (5.28)–(5.31) given by the Dirichlet form (E_R^Rs,lwr,D^Rs,lwr_R ) on L²(S^Rs, µ^Rs). Recall the disintegration µ =

∫

Sµ^Rsµ(ds) given by (5.32). Then, forµ-a.s.s,X^lwr =X^Rs,lwr in distribution forµ^Rs-a.s.x.

Proof. LetT_R,t^lwrandT_R,t^Rs,lwrbe the semi-groups associated with the Dirichlet forms (E_R^lwr,D^lwr_R ) and (E_R^Rs,lwr,D_R^Rs,lwr) onL²(S, µ) andL²(S, µ^Rs), respectively. Then to prove Lemma 5.13 it is sufficient to prove the coincidence of these two semi-groups.

There exists a countable subset D_• of L²(S, µ),D_R^lwr, and D_R^Rs,lwr forµ-a.s.s such that D• is dense in L²(S, µ),D^lwr_R , and D^Rs,lwr_R forµ-a.s.s with respect toL²(S, µ)-norm,E_R,1^lwr -norm, and E_R,1^Rs,lwr-norm forµ-a.s.s, respectively. (see Section 5.9). HereE_R,1^lwr-norm off is given byE_R^lwr(f, f)^1/2+∥f∥L²(S,µ). We setE_R,1^Rs,lwr-norm similarly.

From (5.9), (5.32) and (5.33) we have for f, g∈ D_• ER^lwr(f, g) =

∫

S

E_R^Rs,lwr(fRs, gRs)µ(ds). (5.34) Then we see forf, g∈ D•

∫

S

f(s)g(s)µ(ds)−

∫

S

T_R,t^lwrf(s)g(s)µ(ds) =

∫ _t

0

ER^lwr(T_R,u^lwrf, g)du, (5.35)

∫

S

fRs(x)gRs(x)µ^Rs(dx)−

∫

S

T_R,t^Rs,lwr(fRs)(x)gRs(x)µ^Rs(dx) =

∫ _t

0

E_R^Rs,lwr(T_R,u^Rs,lwr(fRs), gRs)du (5.36)

for µ-a.s. s. We recall that D_• is countable. Hence, for µ-a.s. s, (5.36) holds for all f, g ∈ D_•. Because the particles outside S_R are frozen, we easily see that for f, g ∈ D_• and for µ-a.s.s

T_R,t^lwr(f_Rs) (x) = (T_R,t^lwrf)_Rs(x) forµ^Rs-a.s.x. (5.37) In the following we write T_R,t^lwrf_Rs=T_R,t^lwr(f_Rs) and T_R,t^Rs,lwrf_Rs=T_R,t^Rs,lwr(f_Rs).

From (5.32) and the definitionh_Rs(·) =h(π_R(·) +π_R^c(s)) we have

∫

S

f(s)g(s)µ(ds) =

∫

S

{ ∫

S

fRs(x)gRs(x)µ^Rs(dx) }

µ(ds). (5.38)

Replacing f withT_R,t^lwrf in (5.38) and using (5.37) we have

∫

S

T_R,t^lwrf(s)g(s)µ(ds) =

∫

S

∫

S

(T_R,t^lwrf)_Rs(x)g_Rs(x)µ^Rs(dx)µ(ds) (5.39)

=

∫

S

∫

S

T_R,t^lwrf_Rs(x)g_Rs(x)µ^Rs(dx)µ(ds).

Note thatE_R^lwr(T_R,u^lwrf, T_R,u^lwrf)≤ E_R^lwr(f, f). With the same reason as (5.34) and from (5.37) we have for all f, g∈ D•

E_R^lwr(T_R,u^lwrf, g) =

∫

S

E_R^Rs,lwr((T_R,u^lwrf)Rs, gRs)µ(ds) (5.40)

=

∫

S

E_R^Rs,lwr(T_R,u^lwrfRs, gRs)µ(ds).

Putting (5.38)–(5.40) into (5.35) and using the Fubini theorem we obtain

∫

S

{ ∫

S

fRs(x)gRs(x)µ^Rs(dx) }

µ(ds)−

∫

S

{ ∫

S

T_R,t^lwrfRs(x)gRs(x)µ^Rs(dx) }

µ(ds) (5.41)

=

∫

S

{ ∫ ^t

0

E_R^Rs,lwr(T_R,u^lwrfRs, gRs)du }

µ(ds).

Hence we from (5.41) for µ-a.s.s

∫

S

f_Rs(x)g_Rs(x)µ^Rs(dx)−

∫

S

T_R,t^lwrf_Rs(x)g_Rs(x)µ^Rs(dx) =

∫ _t

0

E_R^Rs,lwr(T_R,u^lwrf_Rs, g_Rs)du.

(5.42) Compare (5.42) with (5.36). Then we see thatf 7→T_R,t^lwrfRsis the semi-group associated with the Dirichlet form (E_R^Rs,lwr,D_R^Rs,lwr) onL²(S, µ^Rs). We note here f =fRs forµ^Rs-a.s.

Therefore, for µ-a.s.s,

T_R,t^lwrf(x) =T_R,t^lwrf_Rs(x) =T_R,t^Rs,lwrf(x) forµ^Rs-a.s.x. (5.43) Here we usedµ^Rs(S^Rs) = 1, where S^Rs ={y∈S;π_R^c(y) =π_R^c(s)} as before.

The solutions X^lwr and X^Rs,lwr are associated with the semi-groups T_R,t^lwr and T_R,t^Rs,lwr, respectively. Hence from (5.43) we deduce that these are equivalent in distribution. We thus see that the solutions of SDE (5.28)–(5.31) given by these Dirichlet forms are the same.

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 83-94)