A Recursive Inequality of Empirical Measures Associated with EDM

‒ 253 ‒

A Recursive Inequality of Empirical Measures Associated with EDM

DÔKU, Isamu

Faculty of Education, Saitama University

Summary

In this paper we consider a random model related to stochastic interacting systems, named an environment-dependent spatial model (EDM). As a matter of fact, this stochastic model is deeply connected with some Markov processes investigated by Liggett [18]. We shall show that rescaled processes of the empirical measures derived from EDMs satisfy some applicationally important recursive inequality.

Key Words: environment-dependent model, random model, stochastic interacting systems, empiri- cal measures, rescaled process, recursive inequality.

1. Introduction

In this section we shall introduce an environment-dependent random model (EDM)[12]. Let Z

^d

be a d-dimensional lattice space, and we suppose that each site on Z

^d

is occupied by all means by either one of the two species. Just after a random time period, a particle dies out and is replaced by a new one, but the random time and the type chosen of the species are assumed to be deter- mined by the environment conditions around the particle. The random function ξ

t

≡ ξ

t

(x): Z

^d

→ {0, 1} denotes the state at time t, and each number of {0, 1} denotes the label of the type chosen of the two species. When we set  y 

_∞

:= max

i

y

i

for y = (y

₁

, . . . , y

_d

), then the R-neighborhood of x is defined by

A Recursive Inequality of Empirical Measures Associated with EDM

D ˆ OKU, Isamu

Faculty of Education, Saitama University

Summary

In this paper we consider a random model related to stochastic interacting systems, named an environment-dependent spatial model (EDM). As a matter of fact, this stochastic model is deeply connected with some Markov processes investigated by Liggett [18]. We shall show that rescaled processes of the empirical measures derived from EDMs satisfy some applicationally important recursive inequality.

Key Words: environment-dependent model, random model, stochastic interacting systems, empirical measures, rescaled process, recursive inequality.

1. Introduction

In this section we shall introduce an environment-dependent random model (EDM)[12].

Let Z

^d

be a d-dimensional lattice space, and we suppose that each site on Z

^d

is occupied by all means by either one of the two species. Just after a random time period, a particle dies out and is replaced by a new one, but the random time and the type chosen of the species are assumed to be determined by the environment conditions around the particle. The random function ξ

t

≡ ξ

t

(x) : Z

^d

→ { 0, 1 } denotes the state at time t, and each number of { 0, 1 } denotes the label of the type chosen of the two species. When we set ∥ y ∥

_∞

:= max

i

y

i

for y

= (y

1

, . . . , y

d

), then the R-neighborhood of x is defined by

N

x

:= x + { y : 0 < ∥ y ∥

∞

R } , (1) where R is a positive constant given. For i = 0, 1, let f

i

(x, ξ) be a frequency of appearance of type i in N

x

for ξ. More precisely, it can be expressed as

f

i

(x) ≡ f

i

(x, ξ) := # { y : ξ

t

(y) = i ; y ∈ N

^x

}

# N

x

. (2)

For non-negative parameters α

_ij

≥ 0, the dynamics of ξ

_t

is defined as follows. The state ξ makes transition 0 → 1 at rate

λf

₁

(f

₀

+ α

₀₁

f

₁

) λf

1

+ f

0

, (3)

and it makes transition 1 → 0 at rate

f

0

(f

1

+ α

10

f

0

) λf

1

+ f

0

. (4)

1

(1) where R is a positive constant given. For i = 0, 1, let f

i

(x, ξ) be a frequency of appearance of type

i in N

x

for ξ. More precisely, it can be expressed as

A Recursive Inequality of Empirical Measures Associated with EDM

D ˆ OKU, Isamu

Faculty of Education, Saitama University

Summary

In this paper we consider a random model related to stochastic interacting systems, named an environment-dependent spatial model (EDM). As a matter of fact, this stochastic model is deeply connected with some Markov processes investigated by Liggett [18]. We shall show that rescaled processes of the empirical measures derived from EDMs satisfy some applicationally important recursive inequality.

Key Words: environment-dependent model, random model, stochastic interacting systems, empirical measures, rescaled process, recursive inequality.

1. Introduction

In this section we shall introduce an environment-dependent random model (EDM)[12].

Let Z

^d

be a d-dimensional lattice space, and we suppose that each site on Z

^d

is occupied by all means by either one of the two species. Just after a random time period, a particle dies out and is replaced by a new one, but the random time and the type chosen of the species are assumed to be determined by the environment conditions around the particle. The random function ξ

t

≡ ξ

t

(x) : Z

^d

→ { 0, 1 } denotes the state at time t, and each number of { 0, 1 } denotes the label of the type chosen of the two species. When we set ∥ y ∥

∞

:= max

i

y

i

for y

= (y

1

, . . . , y

d

), then the R-neighborhood of x is defined by

N

^x

:= x + { y : 0 < ∥ y ∥

∞

R } , (1) where R is a positive constant given. For i = 0, 1, let f

i

(x, ξ) be a frequency of appearance of type i in N

^x

for ξ. More precisely, it can be expressed as

f

i

(x) ≡ f

i

(x, ξ) := # { y : ξ

_t

(y) = i ; y ∈ N

x

}

# N

x

. (2)

For non-negative parameters α

ij

≥ 0, the dynamics of ξ

t

is defined as follows. The state ξ makes transition 0 → 1 at rate

λf

1

(f

0

+ α

01

f

1

) λf

1

+ f

0

, (3)

and it makes transition 1 → 0 at rate

f

0

(f

1

+ α

10

f

0

)

λf

₁

+ f

₀

. (4)

1

(2) For non-negative parameters α

ij

≥ 0, the dynamics of ξ

t

is defined as follows. The state ξ makes transition 0 → 1 at rate

A Recursive Inequality of Empirical Measures Associated with EDM

D ˆ OKU, Isamu

Faculty of Education, Saitama University

Summary

In this paper we consider a random model related to stochastic interacting systems, named an environment-dependent spatial model (EDM). As a matter of fact, this stochastic model is deeply connected with some Markov processes investigated by Liggett [18]. We shall show that rescaled processes of the empirical measures derived from EDMs satisfy some applicationally important recursive inequality.

Key Words: environment-dependent model, random model, stochastic interacting systems, empirical measures, rescaled process, recursive inequality.

1. Introduction

In this section we shall introduce an environment-dependent random model (EDM)[12].

Let Z

^d

be a d-dimensional lattice space, and we suppose that each site on Z

^d

is occupied by all means by either one of the two species. Just after a random time period, a particle dies out and is replaced by a new one, but the random time and the type chosen of the species are assumed to be determined by the environment conditions around the particle. The random function ξ

t

≡ ξ

t

(x) : Z

^d

→ { 0, 1 } denotes the state at time t, and each number of { 0, 1 } denotes the label of the type chosen of the two species. When we set ∥ y ∥

∞

:= max

i

y

i

for y

= (y

1

, . . . , y

d

), then the R-neighborhood of x is defined by

N

x

:= x + { y : 0 < ∥ y ∥

∞

R } , (1) where R is a positive constant given. For i = 0, 1, let f

_i

(x, ξ) be a frequency of appearance of type i in N

x

for ξ. More precisely, it can be expressed as

f

i

(x) ≡ f

i

(x, ξ) := # { y : ξ

t

(y) = i ; y ∈ N

x

}

# N

^x

. (2)

For non-negative parameters α

ij

≥ 0, the dynamics of ξ

t

is defined as follows. The state ξ makes transition 0 → 1 at rate

λf

1

(f

0

+ α

01

f

1

) λf

1

+ f

0

, (3)

and it makes transition 1 → 0 at rate

f

0

(f

1

+ α

10

f

0

) λf

1

+ f

0

. (4)

1

(3) and it makes transition 1 → 0 at rate

A Recursive Inequality of Empirical Measures Associated with EDM

D ˆ OKU, Isamu

Faculty of Education, Saitama University

Summary

In this paper we consider a random model related to stochastic interacting systems, named an environment-dependent spatial model (EDM). As a matter of fact, this stochastic model is deeply connected with some Markov processes investigated by Liggett [18]. We shall show that rescaled processes of the empirical measures derived from EDMs satisfy some applicationally important recursive inequality.

Key Words: environment-dependent model, random model, stochastic interacting systems, empirical measures, rescaled process, recursive inequality.

1. Introduction

In this section we shall introduce an environment-dependent random model (EDM)[12].

Let Z

^d

be a d-dimensional lattice space, and we suppose that each site on Z

^d

is occupied by all means by either one of the two species. Just after a random time period, a particle dies out and is replaced by a new one, but the random time and the type chosen of the species are assumed to be determined by the environment conditions around the particle. The random function ξ

t

≡ ξ

t

(x) : Z

^d

→ { 0, 1 } denotes the state at time t, and each number of { 0, 1 } denotes the label of the type chosen of the two species. When we set ∥ y ∥

∞

:= max

i

y

i

for y

= (y

1

, . . . , y

d

), then the R-neighborhood of x is defined by

N

^x

:= x + { y : 0 < ∥ y ∥

∞

R } , (1) where R is a positive constant given. For i = 0, 1, let f

i

(x, ξ) be a frequency of appearance of type i in N

^x

for ξ. More precisely, it can be expressed as

f

_i

(x) ≡ f

_i

(x, ξ) := # { y : ξ

t

(y) = i ; y ∈ N

^x

}

# N

x

. (2)

For non-negative parameters α

ij

≥ 0, the dynamics of ξ

t

is defined as follows. The state ξ makes transition 0 → 1 at rate

λf

1

(f

0

+ α

01

f

1

) λf

1

+ f

0

, (3)

and it makes transition 1 → 0 at rate

f

0

(f

1

+ α

10

f

0

)

λf

₁

+ f

₀

. (4)

1

(4)

J. Saitama Univ. Fac. Educ., 65（2）：253－259（2016）

‒ 254 ‒

The interpretation of the above rate is as follows. The particle of type i dies at rate f

i

+ α

ij

f

j

, and is replaced instantaneously by either one of the two species chosen at random, according to the proliferation rate of type 0 and the interaction (= the competitive result) with the particle of type 1.

We assume that competitive two species possess the same intensity of intraspecific interaction. The exchange of particles after death is described in the form being proportional to the weighted densi- ty between the two species, expressed by a parameter λ. Assume usually that λ ≥ 1. The case of λ

= 1 means that the contribution to a local appearance rate between the two competitive species is equivalent.

2. Scaling, rescaled process and empirical measure

For simplicity we shall treat a simple case λ = 1 only in what follows. For N = 1, 2, . . . , let M

N

∈ N , and we put 

N

:= M

N

 N, and — S

^N

:= Z

^d

/

N

. And also W

N

= (W

N¹

, . . . , W

N^d

) ∈ ( Z

^d

/M

N

) \ { 0 } is defined as a random vector satisfying

The interpretation of the above rate is as follows. The particle of type i dies at rate f

i

+α

ij

f

j

, and is replaced instantaneously by either one of the two species chosen at random, according to the proliferation rate of type 0 and the interaction (= the competitive result) with the particle of type 1. We assume that competitive two species possess the same intensity of intraspecific interaction. The exchange of particles after death is described in the form being proportional to the weighted density between the two species, expressed by a parameter λ. Assume usually that λ ≥ 1. The case of λ = 1 means that the contribution to a local appearance rate between the two competitive species is equivalent.

2. Scaling, rescaled process and empirical measure

For simplicity we shall treat a simple case λ = 1 only in what follows. For N = 1, 2, . . . , let M

N

∈ N , and we put ℓ

N

:= M

N

√ N , and S

^N

:= Z

^d

/ℓ

N

. And also W

N

= (W

_N¹

, . . . , W

_N^d

)

∈ ( Z

^d

/M

N

) \ { 0 } is defined as a random vector satisfying

(i) L (W

N

) = L ( − W

N

); (ii) E(W

_Nⁱ

W

_N^j

) → δ

ij

σ

²

( ≥ 0) (as N → ∞ );

(iii) {| W

N

|

²

} (N ∈ N ) is uniformly integrable.

Here L (Y ) indicates the law of a random variable Y . For the kernel p

N

(x) := P (W

N

/ √ N = x), x ∈ S

^N

and ξ ∈ { 0, 1 }

^SN

, we define the scaled frequency f

_i^N

as

f

_i^N

(x, ξ) = ∑

y∈SN

p

N

(y − x)1

_{ξ(y)=i}

, (i = 0, 1). (5) Actually, ξ

_t^N

is given by ξ

_t^N

= ξ

N t

(x √

N). As a matter of fact, the rescaled process ξ

_t^N

: S

^N

∋ x �→ ξ

_t^N

(x) ∈ { 0, 1 } is determined by the following state transition law, nemaly, it makes transition 0 → 1 at rate N f

₁^N

(f

₀^N

+ α

^N₀

f

₁^N

), or else it makes transition 1 → 0 at rate N f

₀^N

(f

₁^N

+ α

^N₁

f

₀^N

). We also denote the rescaled process ξ

_t^N

by the symbol Res(p

N

, α

^N_i

).

On this account, we may define the associated measure-valued process (or its corresponding empirical measure) as

X

_t^N

:= 1 N

∑

x∈SN

ξ

_t^N

(x)δ

x

. (6)

For the initial value X

₀^N

, we assume that sup

N

⟨ X

₀^N

, 1 ⟩ < ∞ , X

₀^N

→ X

0

in M

F

( R

^d

) (N → ∞ ), (7) where M

F

( R

^d

) is the totality of all the finite measures on R

^d

, equipped with the topology of weak convergence. For a finite measure µ ∈ M

F

(E) with a topological space E, we use the notation ⟨ µ, φ ⟩ = ∫

E

φ(x)µ(dx) for integral of a measurable function φ over E with respect to a measure µ on E. Note that the convergence in (7) is that in the sense of weak convergence for measures [17].

2

Here L (Y) indicates the law of a random variable Y. For the kernel p

N

(x) := P (W

N

/  —

N =x), x ∈ S

^N

and ξ ∈ {0, 1}

^SN

, we define the scaled frequency f

^Ni

as

The interpretation of the above rate is as follows. The particle of type i dies at rate f

i

+α

ij

f

j

, and is replaced instantaneously by either one of the two species chosen at random, according to the proliferation rate of type 0 and the interaction (= the competitive result) with the particle of type 1. We assume that competitive two species possess the same intensity of intraspecific interaction. The exchange of particles after death is described in the form being proportional to the weighted density between the two species, expressed by a parameter λ. Assume usually that λ ≥ 1. The case of λ = 1 means that the contribution to a local appearance rate between the two competitive species is equivalent.

2. Scaling, rescaled process and empirical measure

For simplicity we shall treat a simple case λ = 1 only in what follows. For N = 1, 2, . . . , let M

_N

∈ N , and we put ℓ

_N

:= M

_N

√

N , and S

^N

:= Z

^d

/ℓ

_N

. And also W

_N

= (W

_N¹

, . . . , W

_N^d

)

∈ ( Z

^d

/M

N

) \ { 0 } is defined as a random vector satisfying

(i) L (W

N

) = L ( − W

N

); (ii) E(W

_Nⁱ

W

_N^j

) → δ

ij

σ

²

( ≥ 0) (as N → ∞ );

(iii) {| W

N

|

²

} (N ∈ N ) is uniformly integrable.

Here L (Y ) indicates the law of a random variable Y . For the kernel p

N

(x) := P (W

N

/ √ N = x), x ∈ S

^N

and ξ ∈ { 0, 1 }

^SN

, we define the scaled frequency f

_i^N

as

f

_i^N

(x, ξ) = ∑

y∈SN

p

N

(y − x)1

_{ξ(y)=i}

, (i = 0, 1). (5) Actually, ξ

_t^N

is given by ξ

_t^N

= ξ

N t

(x √

N). As a matter of fact, the rescaled process ξ

_t^N

: S

^N

∋ x �→ ξ

_t^N

(x) ∈ { 0, 1 } is determined by the following state transition law, nemaly, it makes transition 0 → 1 at rate N f

₁^N

(f

₀^N

+ α

^N₀

f

₁^N

), or else it makes transition 1 → 0 at rate N f

₀^N

(f

₁^N

+ α

^N₁

f

₀^N

). We also denote the rescaled process ξ

_t^N

by the symbol Res(p

N

, α

^N_i

).

On this account, we may define the associated measure-valued process (or its corresponding empirical measure) as

X

_t^N

:= 1 N

∑

x∈SN

ξ

_t^N

(x)δ

x

. (6)

For the initial value X

₀^N

, we assume that sup

N

⟨ X

₀^N

, 1 ⟩ < ∞ , X

₀^N

→ X

0

in M

F

( R

^d

) (N → ∞ ), (7) where M

F

( R

^d

) is the totality of all the finite measures on R

^d

, equipped with the topology of weak convergence. For a finite measure µ ∈ M

_F

(E) with a topological space E, we use the notation ⟨ µ, φ ⟩ = ∫

E

φ(x)µ(dx) for integral of a measurable function φ over E with respect to a measure µ on E. Note that the convergence in (7) is that in the sense of weak convergence for measures [17].

2

(5) Actually, ξ

^Nt

is given by ξ

^Nt

= ξ

Nt

(x  —

N ). As a matter of fact, the rescaled process ξ

^Nt

: S

^N

 x → ξ

^Nt

(x) ∈ { 0, 1 } is determined by the following state transition law, nemaly, it makes transition 0 → 1 at rate Nf

^N₁

(f

^N₀

+ α

^N₀

f

^N₁

), or else it makes transition 1 → 0 at rate Nf

^N₀

(f

^N₁

+ α

^N₁

f

^N₀

). We also de- note the rescaled process ξ

^Nt

by the symbol Res(p

N

, α

^Ni

). On this account, we may define the asso- ciated measure-valued process (or its corresponding empirical measure) as

The interpretation of the above rate is as follows. The particle of type i dies at rate f

_i

+ α

_ij

f

_j

, and is replaced instantaneously by either one of the two species chosen at random, according to the proliferation rate of type 0 and the interaction (= the competitive result) with the particle of type 1. We assume that competitive two species possess the same intensity of intraspecific interaction. The exchange of particles after death is described in the form being proportional to the weighted density between the two species, expressed by a parameter λ. Assume usually that λ ≥ 1. The case of λ = 1 means that the contribution to a local appearance rate between the two competitive species is equivalent.

2. Scaling, rescaled process and empirical measure

For simplicity we shall treat a simple case λ = 1 only in what follows. For N = 1, 2, . . . , let M

N

∈ N , and we put ℓ

N

:= M

N

√ N, and S

^N

:= Z

^d

/ℓ

N

. And also W

N

= (W

_N¹

, . . . , W

_N^d

)

∈ ( Z

^d

/M

N

) \ { 0 } is defined as a random vector satisfying

(i) L (W

N

) = L ( − W

N

); (ii) E(W

_Nⁱ

W

_N^j

) → δ

ij

σ

²

( ≥ 0) (as N → ∞ );

(iii) {| W

N

|

²

} (N ∈ N ) is uniformly integrable.

Here L (Y ) indicates the law of a random variable Y . For the kernel p

N

(x) := P(W

N

/ √ N = x), x ∈ S

^N

and ξ ∈ { 0, 1 }

^SN

, we define the scaled frequency f

_i^N

as

f

_i^N

(x, ξ) = ∑

y∈SN

p

_N

(y − x)1

_{ξ(y)=i}

, (i = 0, 1). (5) Actually, ξ

^N_t

is given by ξ

_t^N

= ξ

_{N t}

(x √

N ). As a matter of fact, the rescaled process ξ

_t^N

: S

^N

∋ x �→ ξ

_t^N

(x) ∈ { 0, 1 } is determined by the following state transition law, nemaly, it makes transition 0 → 1 at rate N f

₁^N

(f

₀^N

+ α

^N₀

f

₁^N

), or else it makes transition 1 → 0 at rate N f

₀^N

(f

₁^N

+ α

^N₁

f

₀^N

). We also denote the rescaled process ξ

^N_t

by the symbol Res(p

N

, α

^N_i

).

On this account, we may define the associated measure-valued process (or its corresponding empirical measure) as

X

_t^N

:= 1 N

∑

x∈SN

ξ

_t^N

(x)δ

x

. (6)

For the initial value X

₀^N

, we assume that sup

N

⟨ X

₀^N

, 1 ⟩ < ∞ , X

₀^N

→ X

0

in M

F

( R

^d

) (N → ∞ ), (7) where M

F

( R

^d

) is the totality of all the finite measures on R

^d

, equipped with the topology of weak convergence. For a finite measure µ ∈ M

F

(E) with a topological space E, we use the notation ⟨ µ, φ ⟩ = ∫

E

φ(x)µ(dx) for integral of a measurable function φ over E with respect to a measure µ on E. Note that the convergence in (7) is that in the sense of weak convergence for measures [17].

2

(6) For the initial value X

^N₀

, we assume that

The interpretation of the above rate is as follows. The particle of type i dies at rate f

_i

+ α

_ij

f

_j

, and is replaced instantaneously by either one of the two species chosen at random, according to the proliferation rate of type 0 and the interaction (= the competitive result) with the particle of type 1. We assume that competitive two species possess the same intensity of intraspecific interaction. The exchange of particles after death is described in the form being proportional to the weighted density between the two species, expressed by a parameter λ. Assume usually that λ ≥ 1. The case of λ = 1 means that the contribution to a local appearance rate between the two competitive species is equivalent.

2. Scaling, rescaled process and empirical measure

For simplicity we shall treat a simple case λ = 1 only in what follows. For N = 1, 2, . . . , let M

N

∈ N , and we put ℓ

N

:= M

N

√ N , and S

^N

:= Z

^d

/ℓ

N

. And also W

N

= (W

_N¹

, . . . , W

_N^d

)

∈ ( Z

^d

/M

N

) \ { 0 } is defined as a random vector satisfying

(i) L (W

N

) = L ( − W

N

); (ii) E(W

_Nⁱ

W

_N^j

) → δ

ij

σ

²

( ≥ 0) (as N → ∞ );

(iii) {| W

N

|

²

} (N ∈ N ) is uniformly integrable.

Here L (Y ) indicates the law of a random variable Y . For the kernel p

N

(x) := P (W

N

/ √ N = x), x ∈ S

^N

and ξ ∈ { 0, 1 }

^SN

, we define the scaled frequency f

_i^N

as

f

_i^N

(x, ξ) = ∑

y∈SN

p

_N

(y − x)1

_{ξ(y)=i}

, (i = 0, 1). (5) Actually, ξ

_t^N

is given by ξ

_t^N

= ξ

_{N t}

(x √

N ). As a matter of fact, the rescaled process ξ

^N_t

: S

^N

∋ x �→ ξ

^N_t

(x) ∈ { 0, 1 } is determined by the following state transition law, nemaly, it makes transition 0 → 1 at rate N f

₁^N

(f

₀^N

+ α

^N₀

f

₁^N

), or else it makes transition 1 → 0 at rate N f

₀^N

(f

₁^N

+ α

^N₁

f

₀^N

). We also denote the rescaled process ξ

_t^N

by the symbol Res(p

N

, α

^N_i

).

On this account, we may define the associated measure-valued process (or its corresponding empirical measure) as

X

_t^N

:= 1 N

∑

x∈SN

ξ

_t^N

(x)δ

x

. (6)

For the initial value X

₀^N

, we assume that sup

N

⟨ X

₀^N

, 1 ⟩ < ∞ , X

₀^N

→ X

0

in M

F

( R

^d

) (N → ∞ ), (7) where M

F

( R

^d

) is the totality of all the finite measures on R

^d

, equipped with the topology of weak convergence. For a finite measure µ ∈ M

F

(E) with a topological space E, we use the notation ⟨ µ, φ ⟩ = ∫

E

φ(x)µ(dx) for integral of a measurable function φ over E with respect to a measure µ on E. Note that the convergence in (7) is that in the sense of weak convergence for measures [17].

2

(7) where M

F

( R

^d

) is the totality of all the finite measures on R

^d

, equipped with the topology of weak convergence. For a finite measure µ ∈ M

F

(E) with a topological space E, we use the notation  µ,

  = 

^E

φ (x)µ(dx) for integral of a measurable function  over E with respect to a measure µ on E. Note that the convergence in (7) is that in the sense of weak convergence for measures [17].

3. Main theorem : recursive inequality

In this section we shall introduce the principal result on an estimate of the maximum of the

moment of total mass process for the empirical measure. To prove it we need some precise esti-

‒ 255 ‒

mate of the quantity in question, and in fact, that can be realized by a certain recursive type in- equality for the empirical measures.

Theorem 1. (Main Result) Let F (N) be a function of N that satisfies 1  F (N)  N and lim

_N→∞

F (N)/N=0. If the condition N

^5/7

/F (N) → 0 holds as N → ∞, then for any p > 1 and T

> 0, there exists a finite constant c (p , T) > 0 such that 3. Main theorem : recursive inequality

In this section we shall introduce the principal result on an estimate of the maximum of the moment of total mass process for the empirical measure. To prove it we need some precise estimate the quantity in question, and in fact, that can be realized by a certain recursive type inequality for the empirical measures.

Theorem 1. (Main Result) Let F (N ) be a function of N that satisfies 1 F (N ) N and lim

_N→∞

F(N )/N = 0. If the condition N

^5/7

/F (N ) → 0 holds as N → ∞ , then for any p > 1 and T > 0, there exists a finite constant c(p, T ) > 0 such that

E[sup

tT

⟨ X

_t^N

, 1 ⟩

^p

] c(p, T ) ( N

F (N )

)

p−1/2

( ⟨ X

₀^N

, 1 ⟩

^p

+ 1). (8)

4. Sketch of proof of main result

Step 1. In this section we shall introduce a sketch of proof of our main result Theorem 1. First of all, we begin with showing a useful equality.

Lemma 2. The following equality holds for every t > 0:

E[ ⟨ X

_t^N

, 1 ⟩ ] = ⟨ X

₀^N

, 1 ⟩ . (9) Proof. First we consider a bounded function ψ : S

^N

→ R . For a continuous time random walk B

_t^x,N

with rate N and step distribution p

N

starting at x,

ϕ

_s

(x) ≡ ϕ(s, x) = P

_t^N_−s

ψ(x) := E[ψ(B

^x,N_t−s

)] (10) defines a semigroup. Indeed, this newly defined function ϕ satisfies a differential equation

∂

_s

ϕ(s) + A

N

ϕ(s) = 0 by virtue of the backward equation argument for continuous time Markov chains. Recall that A

N

is its generator, and is given by

A

N

ϕ(x) := N ∑

y

p

N

(y − x)(ϕ(y) − ϕ(x)). (11) According to the theory of semimartingales [21], we may apply Itˆo’s formula in stochastic calculus [14] to a relationship of rescaled EDMs to obtain

⟨ X

_t^N

, ϕ

t

⟩ = ⟨ X

₀^N

, ϕ

0

⟩ +

∫

t 0

X

_s^N

(∂

s

ϕ(s) + A

^N

ϕ(s))ds + M

_t^N

(ϕ), (12) for 0 t T , where M

_t^N

(ϕ) is a martingale term and X

t

(ϕ) denotes an integral of the test function ϕ relative to the measure-valued process dX

_t

. Taking the expectation operation E[ · ] at the both sides of (12), we can get the equality E[X

_t^N

(ϕ

t

)] = E[X

₀^N

(ϕ

0

)] = E[X

₀^N

(P

_t^N

ψ)],

3

(8)

4. Sketch of proof of main result

Step 1. In this section we shall introduce a sketch of proof of our main result Theorem 1.

First of all, we begin with showing a useful equality.

Lemma 2. The following equality holds for every t > 0:

3. Main theorem : recursive inequality

In this section we shall introduce the principal result on an estimate of the maximum of the moment of total mass process for the empirical measure. To prove it we need some precise estimate the quantity in question, and in fact, that can be realized by a certain recursive type inequality for the empirical measures.

Theorem 1. (Main Result) Let F (N ) be a function of N that satisfies 1 F (N ) N and lim

N→∞

F(N )/N = 0. If the condition N

^5/7

/F (N ) → 0 holds as N → ∞ , then for any p > 1 and T > 0, there exists a finite constant c(p, T ) > 0 such that

E[sup

tT

⟨ X

_t^N

, 1 ⟩

^p

] c(p, T ) ( N

F (N )

)

p−1/2

( ⟨ X

₀^N

, 1 ⟩

^p

+ 1). (8)

4. Sketch of proof of main result

Step 1. In this section we shall introduce a sketch of proof of our main result Theorem 1. First of all, we begin with showing a useful equality.

Lemma 2. The following equality holds for every t > 0:

E[ ⟨ X

_t^N

, 1 ⟩ ] = ⟨ X

₀^N

, 1 ⟩ . (9) Proof. First we consider a bounded function ψ : S

^N

→ R . For a continuous time random walk B

_t^x,N

with rate N and step distribution p

N

starting at x,

ϕ

s

(x) ≡ ϕ(s, x) = P

_t^N_−s

ψ(x) := E[ψ(B

^x,N_t−s

)] (10) defines a semigroup. Indeed, this newly defined function ϕ satisfies a differential equation

∂

s

ϕ(s) + A

N

ϕ(s) = 0 by virtue of the backward equation argument for continuous time Markov chains. Recall that A

^N

is its generator, and is given by

A

^N

ϕ(x) := N ∑

y

p

N

(y − x)(ϕ(y) − ϕ(x)). (11) According to the theory of semimartingales [21], we may apply Itˆo’s formula in stochastic calculus [14] to a relationship of rescaled EDMs to obtain

⟨ X

_t^N

, ϕ

_t

⟩ = ⟨ X

₀^N

, ϕ

₀

⟩ +

∫

t 0

X

_s^N

(∂

_s

ϕ(s) + A

N

ϕ(s))ds + M

_t^N

(ϕ), (12) for 0 t T , where M

_t^N

(ϕ) is a martingale term and X

t

(ϕ) denotes an integral of the test function ϕ relative to the measure-valued process dX

t

. Taking the expectation operation E[ · ] at the both sides of (12), we can get the equality E[X

_t^N

(ϕ

t

)] = E[X

₀^N

(ϕ

0

)] = E[X

₀^N

(P

_t^N

ψ)],

3

(9) Proof. First we consider a bounded function ψ : S

^N

→ R . For a continuous time random walk B

^x,Nt

with rate N and step distribution p

N

starting at x,

3. Main theorem : recursive inequality

In this section we shall introduce the principal result on an estimate of the maximum of the moment of total mass process for the empirical measure. To prove it we need some precise estimate the quantity in question, and in fact, that can be realized by a certain recursive type inequality for the empirical measures.

Theorem 1. (Main Result) Let F (N ) be a function of N that satisfies 1 F (N) N and lim

N→∞

F (N )/N = 0. If the condition N

^5/7

/F (N ) → 0 holds as N → ∞ , then for any p > 1 and T > 0, there exists a finite constant c(p, T ) > 0 such that

E[sup

tT

⟨ X

_t^N

, 1 ⟩

^p

] c(p, T ) ( N

F (N )

)

p−1/2

( ⟨ X

₀^N

, 1 ⟩

^p

+ 1). (8)

4. Sketch of proof of main result

Step 1. In this section we shall introduce a sketch of proof of our main result Theorem 1. First of all, we begin with showing a useful equality.

Lemma 2. The following equality holds for every t > 0:

E[ ⟨ X

_t^N

, 1 ⟩ ] = ⟨ X

₀^N

, 1 ⟩ . (9) Proof. First we consider a bounded function ψ : S

^N

→ R . For a continuous time random walk B

_t^x,N

with rate N and step distribution p

N

starting at x,

ϕ

s

(x) ≡ ϕ(s, x) = P

_t^N_−s

ψ(x) := E[ψ(B

_t^x,N_−s

)] (10) defines a semigroup. Indeed, this newly defined function ϕ satisfies a differential equation

∂

s

ϕ(s) + A

^N

ϕ(s) = 0 by virtue of the backward equation argument for continuous time Markov chains. Recall that A

N

is its generator, and is given by

A

N

ϕ(x) := N ∑

y

p

N

(y − x)(ϕ(y) − ϕ(x)). (11) According to the theory of semimartingales [21], we may apply Itˆo’s formula in stochastic calculus [14] to a relationship of rescaled EDMs to obtain

⟨ X

_t^N

, ϕ

t

⟩ = ⟨ X

₀^N

, ϕ

0

⟩ +

∫

t 0

X

_s^N

(∂

s

ϕ(s) + A

^N

ϕ(s))ds + M

_t^N

(ϕ), (12) for 0 t T , where M

_t^N

(ϕ) is a martingale term and X

t

(ϕ) denotes an integral of the test function ϕ relative to the measure-valued process dX

t

. Taking the expectation operation E[ · ] at the both sides of (12), we can get the equality E[X

_t^N

(ϕ

_t

)] = E[X

₀^N

(ϕ

₀

)] = E[X

₀^N

(P

_t^N

ψ)],

3

(10) defines a semigroup. Indeed, this newly defined function  satisfies a di ff erential equation ∂

s

(s) + A

^N

 (s) = 0 by virtue of the backward equation argument for continuous time Markov chains.

Recall that A

^N

is its generator, and is given by

3. Main theorem : recursive inequality

In this section we shall introduce the principal result on an estimate of the maximum of the moment of total mass process for the empirical measure. To prove it we need some precise estimate the quantity in question, and in fact, that can be realized by a certain recursive type inequality for the empirical measures.

Theorem 1. (Main Result) Let F (N ) be a function of N that satisfies 1 F(N ) N and lim

N→∞

F (N )/N = 0. If the condition N

^5/7

/F (N ) → 0 holds as N → ∞ , then for any p > 1 and T > 0, there exists a finite constant c(p, T ) > 0 such that

E[sup

tT

⟨ X

_t^N

, 1 ⟩

^p

] c(p, T ) ( N

F(N )

)

p−1/2

( ⟨ X

₀^N

, 1 ⟩

^p

+ 1). (8)

4. Sketch of proof of main result

Step 1. In this section we shall introduce a sketch of proof of our main result Theorem 1. First of all, we begin with showing a useful equality.

Lemma 2. The following equality holds for every t > 0:

E[ ⟨ X

_t^N

, 1 ⟩ ] = ⟨ X

₀^N

, 1 ⟩ . (9) Proof. First we consider a bounded function ψ : S

^N

→ R . For a continuous time random walk B

_t^x,N

with rate N and step distribution p

N

starting at x,

ϕ

s

(x) ≡ ϕ(s, x) = P

_t^N_−s

ψ(x) := E[ψ(B

_t^x,N_−s

)] (10) defines a semigroup. Indeed, this newly defined function ϕ satisfies a differential equation

∂

s

ϕ(s) + A

^N

ϕ(s) = 0 by virtue of the backward equation argument for continuous time Markov chains. Recall that A

^N

is its generator, and is given by

A

^N

ϕ(x) := N ∑

y

p

N

(y − x)(ϕ(y) − ϕ(x)). (11) According to the theory of semimartingales [21], we may apply Itˆo’s formula in stochastic calculus [14] to a relationship of rescaled EDMs to obtain

⟨ X

_t^N

, ϕ

t

⟩ = ⟨ X

₀^N

, ϕ

0

⟩ +

∫

t 0

X

_s^N

(∂

s

ϕ(s) + A

N

ϕ(s))ds + M

_t^N

(ϕ), (12) for 0 t T , where M

_t^N

(ϕ) is a martingale term and X

t

(ϕ) denotes an integral of the test function ϕ relative to the measure-valued process dX

t

. Taking the expectation operation E[ · ] at the both sides of (12), we can get the equality E[X

_t^N

(ϕ

t

)] = E[X

₀^N

(ϕ

0

)] = E[X

₀^N

(P

_t^N

ψ)],

3

(11) According to the theory of semimartingales [21], we may apply Itô’s formula in stochastic calculus [14] to a relationship of rescaled EDMs to obtain

3. Main theorem : recursive inequality

In this section we shall introduce the principal result on an estimate of the maximum of the moment of total mass process for the empirical measure. To prove it we need some precise estimate the quantity in question, and in fact, that can be realized by a certain recursive type inequality for the empirical measures.

Theorem 1. (Main Result) Let F (N ) be a function of N that satisfies 1 F (N ) N and lim

N→∞

F(N )/N = 0. If the condition N

^5/7

/F (N ) → 0 holds as N → ∞ , then for any p > 1 and T > 0, there exists a finite constant c(p, T ) > 0 such that

E[sup

tT

⟨ X

_t^N

, 1 ⟩

^p

] c(p, T ) ( N

F (N )

)

p−1/2

( ⟨ X

₀^N

, 1 ⟩

^p

+ 1). (8)

4. Sketch of proof of main result

Step 1. In this section we shall introduce a sketch of proof of our main result Theorem 1. First of all, we begin with showing a useful equality.

Lemma 2. The following equality holds for every t > 0:

E[ ⟨ X

_t^N

, 1 ⟩ ] = ⟨ X

₀^N

, 1 ⟩ . (9) Proof. First we consider a bounded function ψ : S

^N

→ R . For a continuous time random walk B

_t^x,N

with rate N and step distribution p

_N

starting at x,

ϕ

s

(x) ≡ ϕ(s, x) = P

_t^N_−s

ψ(x) := E[ψ(B

^x,N_t−s

)] (10) defines a semigroup. Indeed, this newly defined function ϕ satisfies a differential equation

∂

s

ϕ(s) + A

N

ϕ(s) = 0 by virtue of the backward equation argument for continuous time Markov chains. Recall that A

^N

is its generator, and is given by

A

^N

ϕ(x) := N ∑

y

p

N

(y − x)(ϕ(y) − ϕ(x)). (11) According to the theory of semimartingales [21], we may apply Itˆo’s formula in stochastic calculus [14] to a relationship of rescaled EDMs to obtain

⟨ X

_t^N

, ϕ

t

⟩ = ⟨ X

₀^N

, ϕ

0

⟩ +

∫

t 0

X

_s^N

(∂

s

ϕ(s) + A

N

ϕ(s))ds + M

_t^N

(ϕ), (12) for 0 t T , where M

_t^N

(ϕ) is a martingale term and X

t

(ϕ) denotes an integral of the test function ϕ relative to the measure-valued process dX

t

. Taking the expectation operation E[ · ] at the both sides of (12), we can get the equality E[X

_t^N

(ϕ

t

)] = E[X

₀^N

(ϕ

0

)] = E[X

₀^N

(P

_t^N

ψ)],

3

(12) for 0  t  T, where M

^Nt

() is a martingale term and X

t

() denotes an integral of the test func- tion  relative to the measure-valued process dX

t

. Taking the expectation operation E[ · ] at the both sides of (12), we can get the equality E[X

^Nt

(

t

)] = E[X

^N₀

(

₀

)] = E[X

^N₀

(P

^Nt

ψ)], and besides we readily obtain the desired expression (9) with E[X

^Nt

(ψ)] = E[X

^N₀

(P

^Nt

ψ)] by changing the function

(t) to a general one ψ, where we have substituted ·(x) ≡ 1 instead of ψ and we also have made use of

and besides we readily obtain the desired expression (9) with E[X

_t^N

(ψ)] = E[X

₀^N

(P

_t^N

ψ)] by changing the function ϕ(t) to a general one ψ, where we have substituted ϕ

_·

(x) ≡ 1 instead of ψ and we also have made use of

ϕ(s, x) = ϕ

s

(x) = P

_t^N_−s

1(x) = E[1(B

_t^x,N_−s

)] = E[1(x)] = 1. (13)

This finishes the proof of lemma.

Step 2. Recall standard results for stochastic integrals with respect to Poisson processes N

s

with the intensity E[N

_s

] = λ

_s

. Since ˆ N

_s

= N

_s

− λ

_s

is a martingale, the stochastic integral M

s

= ∫

t

0

Ψ(s, ω)d N ˆ

s

becomes a martingale. Furthermore, it follows that E | M

t

|

²

= E

� �

� �

∫

t 0

Ψ(s, ω)d N ˆ

s

� �

� �

2

= E

∫

t 0

Ψ

²

(s, ω)dλ

s

. (14)

Let { Λ

^N_t

(x, y) : x, y ∈ S

^N

} be a family of independent Poisson processes with rate N · p

N

(y − x) defined on a complete probability space. Note that its compensated process

Λ ˆ

^N_t

(x, y) = Λ

^N_t

(x, y) − N · p

N

(y − x)t (15) are ( F

t

)-martingale. On this account, for every test function ψ ≡ ψ(s, x) ∈ M

b

([0, T ] × S

^N

) with T < ∞ ,

M

_t^N

(ψ) := 1 F (N )

∑

s

∑

y

∫

t 0

ψ

s

(x)(ξ

s−

(y) − ξ

s−

(x))d Λ ˆ

s

(x, y) (16) is a cadlag L

²

( F

^t

)-martingale, and its predictable square function is given by

⟨ M

^N

(ψ) ⟩

^t

= N F(N )

²

∫

t 0

∑

x

∑

y

ψ

s

(x)

²

(ξ

s

(y) − ξ

s

(x))

²

p

N

(y − x)ds, t ∈ [0, T ] (17) where M

b

(D) is the totality of all bounded measurable functions defined on a proper space D, and the summation ∑

x

is taken over the whole space S

^N

. In particular, the equality

⟨ M

^N

(1) ⟩

t

= 2

∫

t

0

⟨ X

_s^N

, N

F (N ) V

N

(s, x) ⟩ ds (18)

holds, where V

N

(t, x) = ∑

y

p

N

(y − x)1 { ξ

t

(y) = 0 } . An application of the results (16), (17) and (18) with the expression (12) leads to X

_t^N

(1) = X

₀^N

(1) + M

_t^N

(1).

Lemma 3. The random quantity ⟨ X

_t^N

, 1 ⟩ is an L

²

-martingale such that

⟨ X

^N

(1) ⟩

t

= 2N F(N )

∫

t 0

1 F(N )

∑

x

ξ

s

(x)V

N

(s, x)ds 2N F (N )

∫

t

0

⟨ X

_s^N

, 1 ⟩ ds (19) holds.

(13)

This finishes the proof of lemma. □

Step 2. Recall standard results for stochastic integrals with respect to Poisson processes N

s

with the