Proof of Theorem 5.14

(IFC)There exists (H,S_sde) such that SDE (5.56) and (5.57) has a pathwise unique strong solution for each y_m∈H^m for each m∈Nand for P^upr_µ -a.s. X.

(TT)The tail σ-field T(S) is µ-trivial, that is,µ(A)∈ {0,1} for all A∈ T(S).

Feasible sufficient conditions for(IFC)were given in [53, Section 9.3]. The importance of (IFC) is that it yields the pathwise uniqueness of solutions of ISDE (5.18)–(5.19) to-gether with(µ-AC),(NBJ), and(TT)[53, Theorem 5.3 (2)]. All determinantal random point fields are tail trivial [50]. Hence (TT) is satisfied for sine2, Airy2, Bessel2, and Ginibre random point fields. We now quote a result from [53].

Lemma 5.16 ([53, Theorem 5.3 (2)]). Suppose that forµ-a.s.x, ISDE (5.18)–(5.19) has solutions with conditions(µ-AC)and(NBJ)and(IFC). Assume that(TT)holds. Then these solutions are pathwise unique for µ-a.s. x. That is, for µ-a.s. x, if there exist two such solutionsXandX^′ defined on the same probability space withX₀=X^′₀=l(x), then P(X_t=X^′_t for allt) = 1.

Corollary 5.17. Assume (A1)–(A6). Assume (IFC)and (TT). Then (5.54) holds.

We next consider the case that µis not tail trivial. We recall the decomposition of µ into µ^s given by (5.44)–(5.45). Then it is known that, if (A1) is satisfied, then µ^s is tail trivial for µ-a.s.s [53, Lemma 13.2].

Lemma 5.18 ([53, Theorem 5.4]). (1) Assume (A1)–(A6). Then for µ-a.s. s, ISDE (5.18)–(5.19) has solutions for µ^s-a.s.xsatisfying conditions(µ^s-AC)and (NBJ).

(2) Suppose that for µ-a.s. s, ISDE (5.18)–(5.19) has solutions for µ^s-a.s. x satisfying (µ^s-AC), (NBJ), and (IFC). Then these solutions are pathwise unique. That is, for µ-a.s.s, if there exist two such solutions X and X^′ defined on the same probability space withX0 =X^′₀ =l(x) forµ^s-a.s. initial starting pointsx, thenP(Xt=X^′_t for all t) = 1 for µ^s-a.s.x.

In [53], it was proved that solutions of ISDE in Lemma 5.18 (2) satisfy (IFC) if coefficients of ISDE comes from interaction potentials which are smooth outside the origin.

See Lemma 9.7 and Section 10 in [53] for details. We then deduce that, even ifµis not tail trivial, (5.54) holds for µ^s such thatµ^s satisfies the conditions mentioned in Lemma 5.18.

and s, we write X^Rs = (X^Rs,i)^∞_i=1 instead of X= (Xⁱ)^∞_i=1. Suppose that x(S_R)≥m. We set the m-labeled process X^Rs,[m]such that

X^Rs,[m]_t = (X_t^Rs,1, X_t^Rs,2, . . . , X_t^Rs,m,

∑∞ j=m+1

δ_XRs,j

t ), (5.59)

where we freeze particles outside S_R. Hence, X_t^Rs,i =X₀^Rs,i for all t ifi > x(S_R). From (5.59) we have consistency such that, if we denote by X_t^Rs,[m],i the i-th component of X^Rs,[m]_t from the beginning for 1≤i≤m to clarify the dependence on m, then

X_t^Rs,[m],i=X_t^Rs,[m+1],i =X_t^Rs,i (i= 1,2, . . . , m).

It is known [46] thatX^Rs,[m] is the diffusion process associated with the Dirichlet form E^Rs,[m](f, g) =

∫

S_R^m×S

{1 2

∑m i=1

∇if· ∇ig+DR[f, g]}(x,s)µ^Rs,[m](dxds)

on L²(S^m×S, µ^Rs,[m]), where the domainD^Rs,[m]is taken as the closure of {

f ∈C₀^∞(S^m)⊗ D_◦;E^Rs,[m](f, f)<∞}

∩L²(S^m×S, µ^Rs,[m]).

We set fi(x,s) =xi⊗1. We can thus write for 1≤i≤m

X_t^Rs,i−X₀^Rs,i =fi(X^Rs,[m]_t )−fi(X^Rs,[m]₀ ) =:A^[f_t^i],[m].

Because the coordinate function x_i = x_i ⊗1 belongs to D^Rs,[m], A^[fi],[m] is an additive functional of the m-labeled diffusionX^Rs,[m](see [16] for additive functional). We remark here that them-point correlation function ofµ^Rs vanishes outside SR.

Applying the Fukushima decomposition to f_i, the additive functional A^[f_t^i],[m] can be decomposed as a sum of a unique continuous local martingale additive functional M^Rs,i and an additive functional of zero energy N^Rs,i:

X_t^Rs,i−X₀^Rs,i=M_t^Rs,i+N_t^Rs,i. We refer to [16, Theorem 5.2.2] for the Fukushima decomposition.

We recall another decomposition ofA^[f_t^i],[m]called the Lyons–Zheng decomposition [16, Theorem 5.7.1]. LetrT :C([0, T];S)→C([0, T];S) be such thatrT(X)t=XT−t. Suppose that the distribution of X^Rs,[m]₀ is µ^Rs,[m], or more generally, absolutely continuous with respect toµ^Rs,[m]. Then from the Lyons–Zheng decomposition we obtain

X_t^Rs,i−X₀^Rs,i= 1

2M_t^Rs,i+1

2(M_T^Rs,i_−t(rT)−M_T^Rs,i(rT)) a.s. (5.60) From (5.28) andf_i(x,s) =x_i⊗1 we see thatM^Rs,i=Bⁱ for 1≤i≤x(S_R), and hence (5.60) becomes a simple form. That is, for 1≤i≤x(SR)

X_t^Rs,i−X₀^Rs,i= 1 2B_tⁱ+1

2(B_Tⁱ_−t(rT)−B_Tⁱ(rT)). (5.61)

Forx(S_R)< i <∞ we haveX_t^Rs,i=X₀^Rs,i=lⁱ(x) by definition. Thus (5.61) is enough for our purpose. The decomposition (5.61) will be the main tool in this section.

We set the maximal module variable X^Rs,m of the first m-particles by X^Rs,m= max

1≤i≤m sup

t∈[0,T]

|X_t^Rs,i|. (5.62)

Throughout this section we fix T ∈N. From (5.61) and (5.62) we obtain

Lemma 5.20. Assume that the distribution of X^Rs₀ is µ^Rs◦l⁻¹. Then there exists a positive constant c33 such that for 0≤t, u≤T

sup

R∈N

∑m i=1

E[|X_t^Rs,i−X_u^Rs,i|⁴]≤c₃₃m|t−u|². (5.63) Furthermore, for eachm∈N

alim→∞lim inf

R→∞ P(X^Rs,m≤a) = 1, (5.64)

and for each r∈N

ιlim→∞ inf

R∈NP(Ir,T(X^Rs)≤ι) = 1, (5.65) whereIr,T is defined by (5.22).

Proof. From (5.61), we obtain

2|X_t^Rs,i−X₀^Rs,i| ≤ |B_tⁱ|+|B_Tⁱ_−t(r_T)−B_Tⁱ(r_T)| a.s. (5.66) From (5.66) we easily obtain (5.63).

Recall that l(x) = (lⁱ(x))_i∈N∈S^Nis a label. From (5.46) we obtain for A∈ B(S^N)

Rlim→∞µ^Rs◦l⁻¹(A) =µ^s◦l⁻¹(A). (5.67) Equation (5.64) follows straightforwardly from (5.66) and (5.67).

We deduce from (5.66) P

( inf

t∈[0,T]|X_t^Rs,i| ≤r )≤P

(|X₀^Rs,i| −r≤ sup

t∈[0,T]

|X_t^Rs,i−X₀^Rs,i|)

(5.68)

≤P (

2{|X₀^Rs,i| −r} ≤ sup

t∈[0,T]

{|Bⁱ_t|+|B_Tⁱ_−t(r_T)−B_Tⁱ(r_T)|})

≤P(

|X₀^Rs,i| −r ≤ sup

t∈[0,T]|Bⁱ_t|) +P

(|X₀^Rs,i| −r ≤ sup

t∈[0,T]

|B_Tⁱ_−t(rT)−B_Tⁱ(rT)|)

=2P

(|lⁱ(x)| −r ≤ sup

t∈[0,T]

|B_tⁱ|)

≤4d

∫

S

Erf(|x| −r

√T )µ◦(lⁱ)⁻¹(dx).

Then we deduce from (5.22) and (5.68) that

R∈NsupP (

I_r,T(X^Rs)≥ι

)≤ ∑^∞

i>ι R∈NsupP

( inf

t∈[0,T]|X_t^Rs,i| ≤r )

(5.69)

≤4d

∑∞ i>ι

∫

S

Erf(|x| −r

√T )µ◦(lⁱ)⁻¹(dx).

From Lemma 5.9 we deduce

∑∞ i=1

∫

S

Erf(|x| −r

√T )µ◦(lⁱ)⁻¹(dx) =

∫

S

Erf(|x| −r

√T )ρ¹(x)dx <∞. (5.70)

From (5.69)–(5.70) we obtain (5.65).

From the conditions above we have the following lemma.

Lemma 5.21. Make the same assumption as Lemma 5.20. Then for eachi, a, R∈Nsuch thati≤m

P(

L^Rs,i_T = 0 ;X^Rs,m≤a)

= 1 fora < R. (5.71)

Proof. Recall that by (5.29) we have L^Rs,i_t =

∫ _t

0

1_∂SR(X_u^Rs,i)dL^Rs,i_u .

ThenL^R,i ={L^Rs,i_t }is non-negative and increases only when{X_t^Rs,i}touches the boundary

∂S_R={|x|=R}. HenceL^Rs,i_T = 0 for alla < Ron{X^Rs,m≤a}, which implies (5.71).

Let b_r,s,p be as in (A6)and put B^Rs,i_r,s,p(t) =

∫ _t

0

br,s,p(X_u^Rs,i,X^Rs,_u ^⋄i)du. (5.72) We set for m∈N

X^Rs,m= (X^Rs,i)^m_i=1,B^Rs,m_r,s,p = (B^Rs,i_r,s,p)^m_i=1, and L^Rs,m= (L^Rs,i)^m_i=1. LetX^Rs= (X^Rs,i)^∞_i=1 and consider random variables

V^Rs,mr,s,p = (X^Rs,m,B^Rs,m_r,s,p,L^Rs,m), (5.73) W^Rsr,s,p=(

(X^Rs,n,B^Rs,n_r,s,p,L^Rs,n)^∞_n=1,X^Rs)

. (5.74)

By construction, V^Rs,mr,s,p andW^Rs_r,s,pare functionals ofX^Rs. Hence we can regardV^Rs,mr,s,p

and W^Rs_r,s,p are defined on a common probability space. Let σ^Rs,m_a = inf{0≤t≤T; max

1≤i≤m

X_t^Rs,i≥a}.

Let Ξ^m =C([0, T];S^m)×C([0, T];R^dm)² and Ξ^m₀ =C([0, T];S^m)× BV × C+, where BV ={η= (ηⁱ)^m_i=1∈C([0, T];R^dm);η is bounded variation},

C+={ζ = (ζⁱ)^m_i=1∈C([0, T];R^dm);ζ is non-decreasing}.

We say a sequence of random variables is tight if for any subsequence we can choose a subsequence that is convergent in law. We also remark that tightness in C([0, T];S^N) for all T ∈Nis equivalent to tightness inC([0,∞);S^N) because we equip C([0,∞);S^N) with a compact uniform norm.

Lemma 5.22. Make the same assumption as Lemma 5.20. Then forµ-a.s.s, the following hold for allT ∈N.

(1){V^Rs,mr,s,p(· ∧σ^Rs,ma )}r,s,p,R∈N is tight inC([0, T]; Ξ^m) for each m, a∈N. (2){V^Rs,mr,s,p}r,s,p,R∈N is tight inC([0, T]; Ξ^m) for eachm∈N.

(3){

W^Rs_r,s,p}

r,s,p,R∈N is tight in∏_∞

n=1C([0, T]; Ξⁿ)×C([0, T];S^N).

Proof. We remark that tightness ofV^Rs,mr,s,p(· ∧σa^Rs,m) follows from that of each component X^Rs,m(·∧σ^Rs,ma ),B^Rs,mr,s,p(·∧σ^Rs,ma ), andL^Rs,m(·∧σa^Rs,m). Tightness of{X^Rs,m(·∧σa^Rs,m)}R∈N

follows from Lemma 5.20. Tightness of {L^Rs,m(· ∧σ^Rs,ma )}R∈N follows from Lemma 5.21.

Recall that br,s,p ∈Cb(S×S) by (A6). Then tightness of{B^Rs,mr,s,p(· ∧σ^Rs,ma )}r,s,p,R∈N

follows from (5.72) with a straightforward calculation. We thus obtain (1).

In general, a family of probability measuresm_a in a Polish space is compact under the topology of weak convergence if and only if for any ϵ > 0 there exists a compact set K such that infama(K)≥1−ϵ. Using this we conclude (2) from (1) combined with (5.64).

With the same reason as the proof of (1), we obtain (3) from (1) and (2).

Lemma 5.21 and Lemma 5.22 imply that for any subsequence of{

V^Rsr,s,p(·∧σ^Rs,ma )}

r,s,p,R∈N, {V^Rs_r,s,p}

r,s,p,R∈N, and{W^Rs_r,s,p}r,s,p,R∈Nthere exist convergent-in-law subsequences, denoted by the same symbols, such that the following convergence in law holds:

rlim→∞ lim

s→∞ lim

p→∞ lim

R→∞V^Rs,m_r,s,p(· ∧σ^Rs,m_a ) =(

X^s,m_a ,B^s,m_a ,0,X^s_a)

for each m∈N, (5.75)

rlim→∞ lim

s→∞ lim

p→∞ lim

R→∞V^Rs,mr,s,p =(

X^s,m,B^s,m,0,X^s)

for each m∈N, (5.76)

rlim→∞ lim

s→∞ lim

p→∞ lim

R→∞W^Rsr,s,p=(

(X^s,n,B^s,n,0)^∞_n=1,X^s)

. (5.77)

Here the subscript a in the right hand side of (5.75) denotes the dependence on a. We note that the convergence lim_R→∞L^Rs,m(· ∧σa^Rs,m) = 0 follows from Lemma 5.21. From Lemma 5.22 (3), we have consistency:

X^s,m= (X^s,1, . . . , X^s,m).

Here X^s,i in the right hand side is thei-th component ofX^s= (X^s,i)^∞_i=1. The same holds forB^s,n and we writeB^s,n= (B^s,i)ⁿ_i=1 This is the reason why we extend the state space in (3) of Lemma 5.22 from that in (1) and (2).

We next check consistency in a in the limits in (5.75) and (5.76). Without loss of generality, we can assume

P({X^s,m=a}) = 0. (5.78)

Indeed, if not, we can choose an increasing sequence{n(a)}a∈Nof positive numbers diverges to infinity such that P({X^Rs,m=n(a)}) = 0 instead of {a}a∈N. Let

σ_a^s,m= inf{0≤t≤T; max

1≤i≤m

X_t^s,i≥a}.

Then from (5.78) we deduce that the discontinuity points of the stopping time σ^s,ma is probability zero. Hence from convergence in (5.75) and (5.76) we have

(X^s,m_a ,B^s,m_a ,0,X^s_a)

(·) =(

X^s,m,B^s,m,0,X^s)

(· ∧σ_a^s,m). (5.79) We set X^Rs,_t ^⋄i =∑

j̸=iδ_XRs,j

t forX^Rs= (X_t^Rs,i)^∞_i=1. Using reversibility of diffusions, we obtain the following dynamic estimates from the static condition(A6).

Lemma 5.23. Make the same assumption as Lemma 5.20. Furthermore, we assume(A6).

Then for µ-a.s.s and for each i∈N

rlim→∞ lim

s→∞ lim

p→∞ sup

R≥r+s+1

E [ ∫ ^T

0

1Sr(X_t^Rs,i){br,s,p−b}(X_t^Rs,i,X^Rs,_t ^⋄i)dt ]

= 0, (5.80)

rlim→∞ lim

s→∞ lim

p→∞E [ ∫ ^T

0

1_Sr(X^s,i){b_r,s,p−b}(X_t^s,i,X^s,_t^⋄i)dt ]

= 0. (5.81)

Proof. LetX^Rsbe the unlabeled diffusion such thatX^Rs_t =∑_∞

i=1δ_XRs,i

t . Because the diffu-sion X^Rs is associated with the Dirichlet form (E_R^Rs,lwr,D_R^Rs,lwr) introduced in Section 5.2, X^Rs isµ^Rs-reversible. Then because of reversibility we have for all t

E[

1Sr(X_t^Rs,i){br,s,p−b}(X_t^Rs,i,X^Rs,_t ^⋄i)] (5.82)

≤E[∑^∞

i=1

1_Sr(X_t^Rs,i){b_r,s,p−b}(X_t^Rs,i,X^Rs,_t ^⋄i)]

=E[∑^∞

i=1

1Sr(X₀^Rs,i){br,s,p−b}(X₀^Rs,i,X^Rs,₀ ^⋄i)]

=

∫

S

∑

xi∈Sr

1_Sr(x_i){b_r,s,p−b}(x_i,

∑∞ j̸=i

δ_xj)µ^Rs(dx), where we set x=∑

iδ_xi ∈S. Then we obtain (5.80) from (5.47) and (5.82).

Recall that b∈L¹_loc(S×S, µ^[1]). Then br,s,p−b∈L¹_loc(S×S, µ^[1]). Hence b_r,s,p−b∈L¹_loc(S×S, µ^s,[1]) for µ-a.s.s.

From this and martingale convergence theorem, we obtain from (5.47) that

rlim→∞ lim

s→∞ lim

p→∞∥br,s,p−b∥L¹_loc(S×S, µ^s,[1])= 0 forµ-a.s.s.

Then we can prove (5.81) in the same way as (5.80).

Proof of Theorem 5.14. Forψ∈C₀^∞(S^m), letF : Ξ^m₀ →C([0, T];R) such that F(ξ, η, ζ)(t) =ψ(ξ(t))−ψ(ξ(0))−

∫ _t

0

∑m j=1

∇jψ(ξ(u))·dη^j(u) (5.83)

−

∫ _t

0

∑m j=1

∇jψ(ξ(u))·ζ^j(du)−

∫ _t

0

∑m j=1

1

2△jψ(ξ(u))du.

From Itˆo-Tanaka formula, (5.28)–(5.30), andd^µ= 2b, we deduce that for eachm∈N sup

R≥r+s+1

E [

sup

0≤t≤T

F(X^Rs,m,B^Rs,m_r,s,p,L^Rs,m)(t)−

∑m j=1

∫ _t

0

∇jψ(X^Rs,m_u )dB_u^j]

(5.84)

≤c34(s, m, r, s,p){∑^m

j=1

sup

x∈S^m

|∇jψ(x)|} , where we set

c₃₄(s, m, r, s,p) = sup

R≥r+s+1

∑m i=1

E [ ∫ ^T

0

1_Sr(X_t^Rs,i){b_r,s,p−b}(X_t^Rs,i,X^Rs,_t ^⋄i)dt ]

. (5.85) We deduce from (5.80) and (5.85) thatc34 satisfy forµ-a.s.s and for eachm∈N

rlim→∞ lim

s→∞ lim

p→∞c₃₄(s, m, r, s,p) = 0. (5.86) Takeψ=ψQ∈C₀^∞(S^m) such thatψQ(x1, . . . , xm) =xifor{|xi| ≤Q}. Leta, Q, R∈N be such that a < Q, R. Recall that L^Rs,m_t = 0 by Lemma 5.21. Then we deduce from (5.83) and Itˆo-Tanaka formula that

F(X^Rs,m,B^Rs,m_r,s,p,L^Rs,m)(t∧σ^Rs,m_a )−

∑m j=1

∫ t∧σ^Rs,ma

0

∇jψ_Q(X^Rs,m_u )dB_u^j (5.87)

=X^Rs,i(t∧σ_a^Rs,m)−X^Rs,i(0)−B^Rs,i_r,s,p(t∧σ^Rs,m_a )−Bⁱ

t∧σa^Rs,m,

where we writeYt=Y(t) for a stochastic processY ={Yt}. We also remark that{Bⁱ}^∞i=1

is (R^d)^N-valued Brownian motion taken to be independent ofR.

We writeX^s,m_a = (X_a^s,i)^∞_i=1 andX_a^s,i={X_a,t^s,i}. We set

r,s,p,Rlim = lim

r→∞ lim

s→∞ lim

p→∞ lim

R→∞. We have from (5.75), (5.87), (5.84), and (5.86) that

E[

0≤t≤Tsup

X_a,t^s,i−X_a,0^s,i −B^s,i_a,t−Bⁱ_t∧σ^s,m

a ]

= lim

r,s,p,RE[ sup

0≤t≤T

X^Rs,i(t∧σ_a^Rs,m)−X^Rs,i(0)−B^Rs,i_r,s,p(t∧σ_a^Rs,m)−Bⁱ

t∧σ^Rs,ma

] by (5.75)

= lim

r,s,p,RE[ sup

0≤t≤T

F(X^Rs,m,B^Rs,m_r,s,p,L^Rs,m)(t∧σ_a^Rs,m)−

∑m j=1

∫ _t∧σa^Rs,m

0

∇jψ_Q(X^Rs,m_u )dB_u^j] by (5.87)

= 0 by (5.84) and (5.86).

This implies

X_a,t^s,i−X_a,0^s,i −B^s,i_a,t−B_t∧σ^{i s,m}

a = 0 for all t. (5.88)

Then from (5.79) and (5.88) we have for alla∈N X_t^s,i_∧σs,m

a −X₀^s,i−B^s,i_t∧σs,m

a −B_tⁱ_∧σ^s,m

a = 0 for all t. (5.89)

FromP(lima→∞σ^s,ma =∞) = 1, (5.89) implies

X_t^s,i−X₀^s,i−B^s,i_t −Bⁱ_t= 0 for all t. (5.90) So it only remains to calculate the representation ofB^s,i.

We now recall B^Rs,mr,s,p = (B^Rs,ir,s,p)^m_i=1 and B^Rs,ir,s,p(t) = ∫_t

0 br,s,p(Xu^Rs,i,X^Rs,u ^⋄i)du by defini-tion. We then deduce from (5.73), (5.76), and (5.81) combined with b_r,s,p ∈ C_b(S×S) that

B^s,i_t = lim

r→∞ lim

s→∞ lim

p→∞ lim

R→∞B^Rs,i_r,s,p(t) by (5.73) and (5.76) (5.91)

= lim

r→∞ lim

s→∞ lim

p→∞ lim

R→∞

∫ _t

0

br,s,p(X_u^Rs,i,X^Rs,_u ^⋄i)du by definition

= lim

r→∞ lim

s→∞ lim

p→∞

∫ _t

0

b_r,s,p(X_u^s,i,X^s,_u^⋄i)du by b_r,s,p∈C_b(S×S)

= lim

r→∞

∫ _t

0

1_Sr(X_u^s,i)b(X_u^s,i,X^s,_u^⋄i)du by (5.81)

=

∫ _t

0

b(X_u^s,i,X^s,_u^⋄i)du in law.

Putting (5.90)–(5.91) together yields X_t^s,i−X₀^s,i−

∫ _t

0

b(X_u^s,i,X^s,_u^⋄i)du−B_tⁱ = 0.

We then complete the proof of Theorem 5.14.

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 97-104)