The branching chain with drift in space-time random environment (I): Model, Markov property, moments

Acta Mathematica Scientia 2010, 30B(5):1669–1678 http://actams.wipm.ac.cn

THE BRANCHING CHAIN WITH DRIFT IN SPACE-TIME RANDOM ENVIRONMENT (I): MODEL, MARKOV PROPERTY, MOMENTS∗


)

Hu Dihe (

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China E-mail: [email protected]

)

Hu Xiaoyu (

Graduate University, Chinese Academy of Sciences, Beijing 100049, China E-mail: [email protected]

Abstract There are three parts in this article. In Section 1, we establish the model of branching chain with drift in space-time random environment (BCDSTRE), i.e., the coupling of branching chain and random walk. In Section 2, we prove that any BCDSTRE must be a Markov chain in time random environment when we consider the distribution of the particles in space as a random element. In Section 3, we calculate the first-order moments and the second-order moments of BCDSTRE. Key words branching chain with drift in space-time random environment; random branching generating function; local random transition generating function; random drift law; random branching law 2000 MR Subject Classification

0

60J27; 60J35

Introduction

The investigation of branching processes and random walks has a long history by their strong physics background, they are two very important parts in the theory of stochastic processes. In the second half of the twentieth century, many authors studied the classical (determinate environment) branching processes. Especially, Sevastyanov [1], Athreya and Ney [2] and Harris [3] established a general theory of branching processes. Recently, Kalinkin [4] developed the theory of branching processes in a wider content by making use of the Kolmogrov equation. Athreya [5] and Cohn [6], [7] investigated the extinction probability and growth of branching processes in time random environment. Many authors studied the limit theory of random walks in random environments (time random environments, space random environments or space-time random environments) in recent years. (cf. [8–14]). Nawrotzki [15], Cog burn [16–18] and Orey [19] established the general theory of Markov chains in time random ∗ Received

March 5, 2008. Supported by the NSFC (10371092, 11771185, 10871200).

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environments. Yang and Hu [20] introduced the model of general Markov chains in space-time random environments and studied some basic properties. All of above works the time parameter sets of the processes are discrete. Recently, Hu [21], [22] studied the construction theory of Markov process with continuous time parameter set in random environment and the existence and uniqueness of q-processes in random environment, besides Hu [23] studied the extinction probability, polarization and proliferation rate of infinite-dimensional control Markov branching chains in time random environments. In this article, we introduce the model of branching chain with drift in space-time random environment and prove it must be a Markov chain in time random environment when we consider the distribution of the particles in space as a random element. Besides, we also introduce various generating functions and use them to calculate the moments of branching chain with drift in space-time random environment.

1

Notations, Definitions and Model

Let (Θ, B) be an abstract measurable space, Z d be the d-dimensional integer lattice, d ≥ 1, Z = Z 1 , Z+ = {0, 1, 2, · · ·}, N+ = {1, 2, · · ·}. For any function f from Z d to Z+ , let d Su (f ) = {x ∈ Z d : f (x) = 0} be the support set of f . Let Θ∗ = ΘZ×Z and E = {f |f : Z d →  Z+ with finite support Su (f )}, μf (A) = f (x), f ∈ E, A ⊂ Z d . Γ = {μf (.) : f ∈ x∈A∩Su (f )

E}, 0∗ be the zero function on E, 1C be the indicator function on set C. For any positive integer n and non-negative integers r1 , r2 , · · · , rn and A1 , A2 , · · · , An ⊂ Z d , we call {f ∈ E : μf (Ai ) = ri , 1 ≤ i ≤ n} be a μ-cylinder set, E be the σ-algebra generated by all μ-cylinder sets, the measurable space (E, E) will play the state space for some stochastic process which we want to study. Let ξ ∗ = {ξn,x , n ∈ Z, x ∈ Z d } be a random field on complete probability space (Ω, F , P) with state space (Θ, B), ξn = {ξn,x , x ∈ Z d }, ξ[x] = {ξn,x , n ∈ Z}ξnk = {ξi , n − 1 < i < k + 1} (−∞ ≤ n ≤ k ≤ ∞). Consider the branching chains for particles with drift, a position may be occupied by several particles. Let η(n, u, i) be the position at time n + 1 for i-th particle staying on position u at time n, and W (n, u, i; x) = 1η(n,u,i)=x , Y (n, u, i) be the number of particles at time n + 1 branched by the i-th particle at position u at time n. Xn,u be the number of particles at position u at time n, where n ∈ Z+ , i ∈ N+ , x, u ∈ Z d . For singleton set {j}, δi,j ≡ 1{j} (i). Definition 1.1 For any n ∈ Z+ , x ∈ Z d , let Xn+1,x =

n,u  X

W (n, u, i; x)Y (n, u, i),

(1.1)

u∈Z d i=1

Y = {Y (n, u, i), n ∈ Z+ , u ∈ Z d , i ∈ N+ }, Γ = {η(n, u, i), n ∈ Z+ , u ∈ Z d , i ∈ N+ }. If {X0,x , Y (n, u, i), η(m, v, j) : m, n ∈ Z+ , x, u, v ∈ Z d , i, j ∈ N+ } are independent for P, Y is a collection of i.i.d. random variables with distribution  P (Y (n, u, i) = k) = b(k) ≥ 0, b(k) = 1; (1.2) k≥0

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Γ is a collection of random variables such that P (η(n, u, i) = x) = q(x − u) ≥ 0,



q(v) = 1,

(1.3)

v∈Z d

then we call X = {Xn,x , n ∈ Z+ , x ∈ Z d } a classical (or determinate environment) branching with drift. We want now to introduce the concept of branching chain with drift in space-time random environment. Definition 1.2 Let ξ ∗ = {ξn,x , n ∈ Z, x ∈ Z d } be a random field defined as before, X ∗ = {Xn,u , n ∈ Z+ , u ∈ Z d } defined as in equation (1.1), Y = {Y (n, u, i) : n ∈ Z+ , u ∈ Z d , i ∈ N+ } and Γ = {η(n, u, i) : n ∈ Z+ , u ∈ Z d , i ∈ N+ } be two collections of random variables on (Ω, F , P) taking values in Z+ and Z d respectively, W = {W (n, u, i; x)=1 ˆ η(n,u,i)=x , n ∈ Z+ , i ∈ N+ , x, u ∈ Z d }. If (1) {X0,x , Y (n, u, i), η(n, u, i) : m, n ∈ Z+ , u, v, x ∈ Z d , i, j ∈ N+ } are conditionally ∗ independent of given ξ ∗ , i.e., they are independent for probability measure P ξ (.)=P ˆ (.|ξ ∗ ); (2) P (Y (n, u, i) = k|ξ ∗ ) = b(ξn,u ; k), where b(θ; k) is B-measurable as a function of θ for any fixed k ∈ Z+ and

(1.4) 

b(θ; k) = 1 for

k∈Z+

any fixed θ ∈ Θ; (3) P (η(n, u, i) = x|ξ ∗ ) = q(ξn,u ; x − u), where q(θ; v) is B-measurable as a function of θ for any fixed v ∈ Z d and

(1.5) 

q(θ; v) = 1 for

v∈Z d

any fixed θ ∈ Θ. Then, we call (X ∗ , ξ ∗ ) a branching chain with drift in space-time random environment (BCDSTRE) and ξ ∗ the space-time random environment, X ∗ the original chain. b(θ) = (b(θ; k), k ∈ Z+ ) is called the random branching law (distribution), and q(θ) = (q(θ; v), v ∈ Z d ) the random drift (or random walk) law (distribution). Definition 1.3 Let ξ ∗ , X ∗ , Y, Γ, W, b(θ) and q(θ) be defined as in Definition 1.2. We call ϕ(θ; s)= ˆ

∞ 

b(θ; k)sk (|s| ≤ 1, θ ∈ Θ),

(1.6)

k=0

the random branching generating function (RBGT). Let ψn+1,x (s; f, θ∗ )= ˆ

∞  k=0

P (Xn+1,x = k|ξ ∗ = θ∗ , Xn = f )sk ,

(1.7)

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where n ∈ Z+ , x ∈ Z d , |s| ≤ 1, f ∈ E, θ∗ ∈ Θ∗ . We call ψn+1,x (s; f, θ∗ ) the local random transition generating function (LRTGF). Proposition 1.1 We always have  ψn+1,x (s; f, θ∗ ) = [1 − q(θn,v ; x − v) + q(θn,v ; x − v)ϕ(θn,v ; s)]f (v) , (1.8) v∈Su (f )

where n ∈ Z+ , x ∈ Z d , f ∈ E, θ∗ = (θn,v , n ∈ Z, v ∈ Z d ) ∈ Θ∗ . Proof By the independence (1) in Definition 1.2 and the definition of Xn+1,x , we have ψn+1,x (s; f, θ∗ ) =

∞ 

P (Xn+1,x = k|Xn = f, ξ ∗ = θ∗ )sk

k=0

=

(v) ∞    f  P W (n, v, i; x)Y (n, v, i) = k|ξ ∗ = θ∗ sk v∈Z d i=1

k=0

=

∞ 

P

k=0

=



 

f (v)



 W (n, v, i; x)Y (n, v, i) = k|ξ ∗ = θ∗ sk

v∈Su (f ) i=1 f (v) ∞ 

P (W (n, v, i; x)Y (n, v, i) = k|ξ ∗ = θ∗ )sk .

(1.9)

v∈Su (f ) i=1 k=0

But ∞ 

=

k=0 ∞ 

P (W (n, v, i; x)Y (n, v, i) = k|ξ ∗ = θ∗ )sk P (W (n, v, i; x) = 1, Y (n, v, i) = k|ξ ∗ = θ∗ )sk

k=1

+P ({W (n, v, i; x) = 0} ∪ {Y (n, v, i) = 0}|ξ ∗ = θ∗ ) ∞  q(θn,v ; x − u)b(θn,v ; k)sk + 1 = k=1

−[P (W (n, v, i; x) = 0|ξ ∗ = θ∗ ) · P (Y (n, v, i) = 0|ξ ∗ = θ∗ )] ∞  q(θn,v ; x − u)b(θn,v ; k)sk + 1 − [q(θn,v ; x − u)(1 − b(θn,v ; 0))] = k=1

= q(θn,v ; x − u)ϕ(θn,v ; s) + 1 − q(θn,v ; x − v).

(1.10)

Proposition 1.1 is proved by (1.9) and (1.10). Remark 1.1 If W (n, u, i; x) ≡ δu,x = 1{u} (x), i.e., there is no drift, then the chain {Xn,0 , n ∈ Z+ } defined as in Definition 1.1 is the simplest branching chain, the G-W chain. Similarly, the chain {Xn,0 , n ∈ Z+ } defined as in Definition 1.2 is the branching chain in space-time random environment ξ ∗ . Remark 1.2 In Definition 1.1, if Y (n, u, i) ≡ 1, X0,0∗ ≡ 1, X0,x ≡ 0 (x = 0∗ ), i.e., there is no branching and there is only one particle on the original point0∗ at the initial time 0, and  then is always 0 or 1. Let An,x = {Xn,x = 1}, ηn = x1An,x be the position of the particle x∈Z d

at time n. It is easy to see that  An,x An,y = ∅, (∀x = y),

 x∈Z d

An,x ≡ Ω.

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Since {η(n, u, i), n ∈ Z+ , u ∈ Z d , i ∈ N+ } are independent, P (ηn+1 = x|ηn = u, ηn−1 = un−1 , · · · , η0 = u0 ) = P (ηn+1 = x|ηn = u) = P (η(n, u, 1) = x) = q(x − u)

(∀n ∈ Z+ , x, u ∈ Z d ).

It means that {ηn , n ∈ Z+ } is a time homogeneous and space homogeneous Markov chain, i.e., {ηn } is a classical random walk. Similarly, if Y (n, u, i), Xn,u satisfy the conditions as above, then, by the independence of {η(n, u, i), n ∈ Z+ , u ∈ Z d , i ∈ N+ } w.r.t. the probability ∗ measure P ξ (.) = P (.|ξ ∗ ), we have ∗

= P ξ (ηn+1 = x|ηn = u, ηn−1 = un−1 , · · · , η0 = u0 ) ∗

∗

= P ξ (ηn+1 = x|ηn = u) = P ξ (η(n, u, 1) = x) = q(ξn,u ; x − u), i.e., ηn , n ∈ Z+ , is a random walk in space-time random environment ξ ∗ with random drift law q(θ; v).

2

Markov Property

Now, we consider Xn being the distribution of particles at time n, i.e., Xn is a random element on(Ω, F, P) with state apace (E, E), and we want top rove {Xn , n ∈ N+ } has Markov property in certain sense. Let (X , A) and (Λ, D) be two abstract measurable spaces, V ∗ = {Vn , n ∈ Z+ } and ∗ ζ = {ζn , n ∈ Z} be two collections of random elements on Ω taking values in X and Λ

k = {Vm , n − 1 < m < k + 1}, 0 ≤ n ≤ k ≤ ∞, ζ k = {ζm , n − 1 < m < respectively. Set V n n k + 1}, − ∞ ≤ n ≤ k ≤ ∞. Definition 2.1 Let p(λ; f, A) : Λ × X × A → [, ∞]. If (1) p(.; ., A) is measurable w.r.t. σ-algebra D × A for any fixed A ∈ A; (2) p(λ; f, .) is a probability measure on A for any fixed λ ∈ Λ and f ∈ X , then we call p(λ; f, A) a random Markov kernel (RMK).

k and ζ k be defined as above, p(λ; f, A) Definition 2.2 Let (Ω, F , P), (Λ, D), (X , A), V n n be aRMK. If the following conditions are satisfied

n , ζ ∗ ) = p(ζn ; Vn , A); (M0 ) : P (Vn+1 ∈ A|V 0 0 (M1 ) : P (V0 ∈ A|ζ ∗ ) = P (V0 ∈ A|ζ −∞ ),

where n ∈ Z+ , A ∈ A, B ∈ B, then we call (V ∗ , ζ ∗ ) a Markov chain in time random environment (MCTRE) with random Markov kernel (RMK) p(λ; f, A). In the rest of this article we always take Vn = Xn = {Xn,x , x ∈ Z d }, (X , A) = (E, E),   ζn = ξn = {ξn,x , x ∈ Z d }, (Λ, D) = (×Z , B Z ). b(θ), q(θ), ϕ(θ; s) and ψn+1,x (s; f, θ∗ ) are defined as in Section 1. Y (n, u, i), η(n, u, i) and W (n, u, i; x) are defined as in Section 1 and satisfying the conditions as described in Section 1.

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Note

Vol.30 Ser.B

Since Z d is a denumerable set and

E = {f |f : Z d → Z+ , Su (f ) < ∞} =

∞ 

{f |f : Z d → Z+ , Su (f ) = n},

n=0

hence {f |f : Z d → Z+ , Su (f ) = n}  = {f |f : Z d → Z+ , Su (f ) = {x1 , · · · , xn }} {x1 ,···,xn }⊂Z d ,xi =xj

is a denumerable set of functions for every n, and then E is as well. Hence the random Markov kernel d p(λ; f, A) : ΘZ × E × E → [0, 1] d

is determined by the random Markov matrixp(λ) = (p(λ; f, g), f, g ∈ E), where λ ∈ ΘZ , p(λ; f, g) = p(λ; f, {g}). Definition 2.3 Let μ1 , μ2 , · · · , μn be n finite measures on linear measurable space (L, L), n μ= μi be the product measure of μ1 , μ2 , · · · , μn , then we define the convolution measure of i=1

μ1 , μ2 , · · · , μn by (μ1 ∗ μ2 ∗ · · · ∗ μn )(A) = μ({(f1 , · · · , fn ) ∈ Ln :

n 

fi ∈ A}), A ∈ L.

i=1

If μi = ν (i ≤ i ≤ n), we write ν n∗ = μ1 ∗ μ2 ∗ · · · ∗ μn . Theorem 2.1 Let (X ∗ , ξ ∗ ) be a BCDSTRE with random branching law b(θ) and random drift law q(θ) defined as in Definition 1.2, then ({Xn , n ∈ Z+ }, {ξn , n ∈ Z}) must be a MCTRE with random Markov matrix (RMM) p(λ) = (p(λ; f, g), f, g ∈ E) satisfying P (Xn+1 = g|ξ ∗ = θ∗ , Xn = f ) = p(θn ; f, g) =

 

({xv,i ∈A: xv,i =x})∗

(b(θn,v ))



(g(x))

f (v)



 q(θn,v ; xv,i − v) ,

(2.1)

v∈Su (f ) i=1

x∈Z d

xv,i ∈Z d



1≤i≤f (v) v∈Su (f )

where n ∈ Z+ , f, g ∈ E, θ∗ = (θn,v , n ∈ Z, v ∈ Z d ), θn = (θn,v , v ∈ Z d ), A = xv,i : 1 ≤ i ≤ f (v), v ∈ Su (f ).      means ··· , has the similar meaning. xk ∈B xk−1 ∈B

xi ∈B 1≤i≤k

x1 ∈B

xv,i ∈Z d 1≤i≤f (v) v∈Su (f )

Proof If we can prove equation (2.1) is true, then conditions (M0 ) and (M1 ) are obviously true, and then Theorem 2.1 is true. In fact, ∗

P (Xn+1 = g| Xn = f, ξ ∗ = θ∗ ) = P θ (Xn+1 = g| Xn = f ) =

 ∈Z d

xv,i 1≤i≤f (v) v∈Su (f )

∗

P θ (Xn+1 = g,



f (v)



v∈Su (f ) i=1

{W (n, v, i; xv,i ) = 1}| Xn = f )

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=

Pθ

∗

   x∈Z d

xv,i ∈Z d 1≤i≤f (v) v∈Su (f )

 

=

xv,i ∈Z d 1≤i≤f (v) v∈Su (f )

Pθ

xv,i ∈Z d 1≤i≤f (v) v∈Su (f )

 

f (v)





{η(n, v, i) = xv,i }



v∈Su (f ) i=1

xv,i ∈A xv,i =x



Y (n, v, i) = g(x)

f (v)





 ∗ P θ (η(n, v, i) = xv,i )

v∈Su (f ) i=1

xv,i ∈A xv,i =x

x∈Z d

 

=

∗

Y (n, v, i) = g(x) ,

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(b(θn,v ))({xv,i ∈A:

xv,i =x})∗



(g(x))



f (v)



 q(θn,v ; xv,i − v) .

v∈Su (f ) i=1

x∈Z d

Theorem 2.1 is proved.

3

Moments

Let (X ∗ , ξ ∗ ) be the BCDSTRE with random branching law b(θ) and random drift law q(θ) ∞ ∞   as in Definition 1.2, ϕ(θ; s) = b(θ; k)sk . Let B(θ) = kb(θ; k) = ϕ (θ; 1), B2 (θ) = ϕ (θ; 1), k=0

k=0

p(λ; f, g) be the RMM defined as in (2.1). Definition 3.1 We call ˆ m(θ∗ ; n, f, n + 1, x)=

∞ 

P (Xn+1,x = k|Xn = f, ξ ∗ = θ∗ )k

k=0

the local conditional expectation at x given Xn = f and ξ ∗ = θ∗ (θ∗ ∈ Θ∗ , n ∈ Z+ , f ∈ E, x ∈ Z d ). Proposition 3.1 We always have  m(θ∗ ; n, f, n + 1, x) = f (v)q(θn,0 ; x − v)B(θn,v ), (3.1) v∈Su (f )

where θ∗ = (θn,v , n ∈ Z, v ∈ Z d ) ∈ Θ∗ , n ∈ Z, f ∈ E, x ∈ Z d , B(θ) = Proof

∞ 

kb(θ; k) = ϕ, (θ; 1).

k=0

Let β(s; θ, v, x) = 1 − q(θ; x − v) + q(θ; x − v)ϕ(θ; s),

then ψn+1,x (s; f, θ∗ ) =



β(s; θn,v , v, x)f (v)

v∈Su (f )

But β(1; θ, v, x) = 1, β (1; θ, v, x) = q(θ; x − v)ϕ (θ; 1), hence m(θ∗ ; n, f, n + 1, x) 

d ψn+1,x (s; f, θ∗ ) = ds s=1  = f (v)β (1; θn,v , v, x)β(1; θn,v , v, x)f (v)−1 v∈Su (f )

=



v∈Su (f )

 ω∈Su (f )−{v}

f (v)B(θn,v )q(θn,v ; x − v).

β(1; θn,ω , ω, x)f (ω)

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Definition 3.2 For any θ∗ ∈ Θ∗ , fm , fm+n ∈ E, m, n ∈ Z+ , define 

∗

p(θ ; m, fm , m + n, fm+n ) =



m+n−1 

fm+n−1 ∈E

i=m

···

fm+1 ∈E

p(θi ; fi , fi+1 ).

(3.2)

It is easy to prove by the Markov property p(θ∗ ; m, f, m + n, g) = P (Xm+n = g| Xm = f, ξ ∗ = θ∗ )

(3.3)

is the n-step transition probability for the non-time homogeneous Markov chain Xn under the ∗ probability measure P θ . We write simply p(n) (θ∗ ; f, g) = p(θ∗ ; 0, f, n, g). (3.4) Definition 3.3 We call mμ (θ∗ ; s, f, t, A)= ˆ



p(θ∗ ; s, f, t, g)μg (A)

(3.5)

g∈E

the conditional expectation of the total number of particles in region A at time t for given ξ ∗ = θ∗ and Xs = f , where θ∗ ∈ Θ∗ , 0 ≤ s ≤ t, s, t ∈ Z+ , f ∈ E, A ∈ E. We write simply mμ (θ∗ ; s, f, t, {x}) = mμ (θ∗ ; s, f, t, x), mμ (θ∗ ; s, 1{x} , t, A) = mμ (θ∗ ; s, x, t, A). Proposition 3.2 We always have (1) mμ (θ∗ ; s, f, s + t, .) is a measure on E; (2) mμ (θ∗ ; s, f, s + 1, x) = m(θ∗ ; s, f, s + 1, x);  (3) mμ (θ∗ ; s, f, s + t + u, A) = p(θ∗ ; s, f, s + t, g)mμ (θ∗ ; s + t, g, s + t + u, A), g∈E

where s, t, u ∈ Z+ , f ∈ E, x ∈ Z d , A ∈ E. Proof (1) is obvious by the definition of mμ . (2) Let Ak (x) = {g ∈ E|g(x) = k}, x ∈ Z d , k ∈ Z+ , then mμ (θ∗ ; s, f, s + 1, x) = =

 g∈E ∞ 

p(θ∗ ; s, f, s + t, g)g(x) 

∗

P θ (Xs+1 = g| Xs = f )g(x)

k=0 g∈Ak (x)

= =

∞  k=0 ∞ 

∗

P θ (Xs+1 ∈ Ak (x)| Xs = f )k ∗

P θ (Xs+1,x = k| Xs = f )k

k=0

= m(θ∗ ; s, f, s + 1, x). (3) By the definition of p(θ∗ ; s, f, t, g), it satisfies the random Kolmogorov-Chapmann equation:  p(θ∗ ; s, f, t, h)p(θ∗ ; t, h, u, g) = p(θ∗ ; s, f, u, g), h∈E

(θ∗ ∈ Θ∗ , s ≤ t ≤ u, s, t, u ∈ Z+ , f, g ∈ E).

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Hence (3) is obviously true. Note Using propositions 3.1 and 3.2, we can use the random branching law b(θ) and the random drift law q(θ) to calculate various moments. Similarly, we can define the second-order moments as follows. Definition 3.4 We call m2 (θ∗ ; n, f, n + 1, x) =

∞ 

∗

P θ (Xn+1,x = k| Xn = f )k 2

k=0

the local conditional second-order moment at x given Xn = f and ξ ∗ = θ∗ ; and call  2 m2,μ (θ∗ ; s, f, t, A)= ˆ p(θ∗ ; s, f, t, g)μg (A) g∈E

the conditional second order moment of the total number of particles in region A at time t for given ξ ∗ = θ∗ and Xx = f . Proposition 3.3 We always have (1) m2 (θ∗ ; n, f, n + 1, x)



f (v)(f (v) − 1)(q(θn,v ; x − v)B(θn,v ))2

= m(θ∗ ; n, f, n + 1, x) +

v∈Su (f )

+f (v)q(θn,v ; x − v)B2 (θn,v )   + f (v)f (ω)q(θn,v ; x − v)q(θn,v ; x − ω)B(θn,v )B(θn,ω )

(3.6)

ω∈Su (f )−{v} ∗

(2) m2,μ (θ ; s, f, s + t, .) is measurable on E; (3) m2,μ (θ∗ ; n, f, s + t + u, x) = m2 (θ∗ ; n, f, n + 1, x);  (4) m2,μ (θ∗ ; s, f, s + t + u, A) = p(θ∗ ; s, f, s + t, g)m2,μ (θ∗ ; s + t, g, s + t + u, A), g∈E

where n, s, t, u ∈ Z+ , θ∗ ∈ Θ∗ , f ∈ E, A ∈ E, x ∈ E. Proof (1) Let β(s; θ, v, x) = 1 − q(θ; x − v) + q(θ; x − v)ϕ(θ; s), be defined as before, we have proved ψn+1,x (s; f, θ∗ ) = β(s; θn,v , v, x)f (v) , v∈Su (f )

d ψn+1,x (s; f, θ∗ ) ds f (v)β(s; θn,v , v, x)f (v)−1 β (s; θn,v , v, x) = v∈Su (f )



β(s; θn,ω , ω, x)f (ω) .

ω∈Su (f )−{v}

Hence, by β(1; θ, v, x) ≡ 1 and β (1; θ, v, x) = q(θ; x − v)B(θ), 

d2 ∗ ψ (s; f, θ ) n+1,x ds2 s=1  = f (v)(f (v) − 1)β (1; θn,v , v, x)2 +



v∈Su (f )

v∈Su (f )

v∈Su (f )

v∈Su (f )

f (v)β (1; θn,v , v, x)

   d + β(s; θn,ω , ω, x)f (ω) [f (v)β (1; θn,v , v, x)] ds s=1 v∈Su (f ) ω∈Su (f )−{v}   = f (v)(f (v) − 1)β (1; θn,v , v, x)2 + f (v)β (1; θn,v , v, x)

1678

ACTA MATHEMATICA SCIENTIA

+



[f (v)β (1; θn,v , v, x)]

v∈Su (f )

=



Vol.30 Ser.B

 f (ω)B(θn,ω )q(θn,ω ; x − ω)

 ω∈Su (f )−{v}

[f (v)(f (v) − 1)(q(θn,v ; x − v)B(θn,v ))2 + f (v)q(θn,v ; x − v)B2 (θn,v )

v∈Su (f )

+(f (v)q(θn,v ; x − v)B(θn,v )



f (ω)q(θn,ω ; x − ω)B(θn,ω ))].

ω∈Su (f )−{v}

(1) is proved. The proofs of (2)–(4) are similar to that in Proposition 3.2. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

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