where 1 γ α. It is easy to see that
dE s,t 1−E 0,t
is a probability measure on [0, t], by Jessen’s inequality, we have
Z
t
t − s
1 γ
∨ 1dE s, t
= Z
t
t − s ∨ 1 dE s, t
1 − E 0, t
1 γ
1 − E 0, t ≤
Z
t
t − s ∨ 1dE s, t
1 γ
≤ Z
t
E s, tds
1 γ
+ tE 0, t ≤ Cǫ, γ. On the other hand, by Doob’s martingale inequality and
α-stable property 2.2, for all N ∈ N, we have
E sup
1≤t≤2
N
Zt t
1 γ
≤ E
N
X
i=1
sup
2
i−1
≤t≤2
i
Zt t
1 γ
≤
N
X
i=1
E sup
2
i−1
≤t≤2
i
|Zt| 2
i−1γ
≤ C
N
X
i=1
2
i α
2
i−1γ
≤ Cα, γ. From the above three inequalities, we immediately have
E Z
t
Zt − Zs dE s, t ≤ Cα, γ, ǫ.
Collecting all the above estimates, we conclude the proof of 2.9.
3 Existence of Infinite Dimensional Interacting
α-stable Systems
In order to prove the existence theorem of the equation 1.1, we shall first study its Galerkin approximation, and uniformly bound some approximate quantities. To pass to the Galerkin approx-
imation limit, we need to apply a well known estimate in interacting particle systems – finite speed of propagation of information property.
3.1 Galerkin Approximation
Denote Γ
N
:= [−N , N ]
d
, which is a cube in Z
d
centered at origin. We approximate the infinite dimensional system by
d X
N i
t = [J
i
X
N i
t + I
N i
X
N
t]d t + d Z
i
t, X
N i
0 = x
i
, 3.1
for all i ∈ Γ
N
, where x
N
= x
i i∈Γ
N
and I
N i
x
N
= I
i
x
N
, 0. It is easy to see that 3.1 can be written in the following vector form
d X
N
t = [J
N
X
N
t + I
N
X
N
t]d t + d Z
N
t, X
N
0 = x
N
3.2 2001
The infinitesimal generator of 3.2 [4], [33] is L
N
= X
i∈Γ
N
∂
α i
+ X
i∈Γ
N
J
i
x
N i
+ I
N i
x
N
∂
i
, it is easy to see that
[∂
k
, L
N
] =
∂
k
J
k
x
N k
∂
k
+ X
i∈Γ
N
∂
k
I
N i
x
N
∂
i
. 3.3
We shall study the mild solution of Eq. 3.2 in the sense that for each i ∈ Γ
N
, X
i
t = E
i
0, tx
i
+ Z
t
E
i
s, tI
N i
X
N
sds + Z
t
E
i
s, td Z
i
s, 3.4
where E
i
s, t = exp{ R
t s
J
i
X
N i
r X
N i
r
d r} with
J
i
:= J
′
i
0. The following proposition is important for proving the main theorems. 3 is the key estimates for
obtaining the limiting semigroup of 1.1, while 2 plays the crucial role in proving the ergodicity.
Proposition 3.1. Let I
i
, J
i
satisfy Assumption 2.2, together with 2.3 and 2.4, then 1. Eq. 3.2 has a unique mild solution X
N
t in the sense of 3.4. 2. For all x ∈ B
R, ρ
, if c η with c, η defined in 3 of Assumption 2.2, we have
E
x
[|X
N i
t|] ≤ Cρ, R, d, η, c1 + |i|
ρ
. 3. For all x ∈ B
R, ρ
, we have E
x
[|X
N i
t|] ≤ Cρ, R, d1 + |i|
ρ
1 + te
1+ηt
. 4. For any f ∈ C
2 b
R
Γ
N
, R, define P
N t
f x = E
x
[ f X
N
t], we have P
N t
f x ∈ C
2 b
R
Γ
N
, R. Proof. To show 1, we first formally write down the mild solution as in 1, then apply the classical
Picard iteration [9], Section 5.3. We can also prove 1 by some other method as in the appendix of [34].
For the notational simplicity, we shall drop the index N of the quantities if no confusions arise. By 1, we have
X
i
t = E
i
0, tx
i
+ Z
t
E
i
s, tI
i
X
N
sds + Z
t
E
i
s, td Z
i
s. 3.5
By 1 of Assumption 2.2 w.l.o.g. we assume I
i
0 = 0 for all i, |X
i
t| ≤ X
j∈Γ
N
δ
ji
|x
j
| + Z
t
E
j
s, td Z
j
s +
Z
t
e
−ct−s
X
j∈Γ
N
a
ji
|X
j
s|ds. 3.6
2002
We shall iterate the the above inequality in two ways, i.e. the following Way 1 and Way 2, which are the methods to show 2 and 3 respectively. The first way is under the condition c
η, which is crucial for obtaining a upper bound of E|X
i
t| uniformly for t ∈ [0, ∞, while the second one is without any restriction, i.e. c ≥ 0, but one has to pay a price of an exponential growth in t.
Way 1: The case of c η. By the definition of c, η in 3 of Assumption 2.2, 3.6 and Proposition
2.7, E
|X
i
t| ≤ X
j∈Z
d
δ
ji
|x
j
| + Cc + Z
t
e
−ct−s
X
j∈Z
d
a
ji
E |X
j
s|ds. 3.7
Iterating 3.7 once, one has E
|X
i
t| ≤ X
j∈Z
d
δ
ji
|x
j
| + Cc + X
j∈Z
d
a
ji
c |x
j
| + Cc +
Z
t
e
−ct−s
Z
s
e
−cs−r
X
j∈Z
d
a
2 ji
E |X
j
r|d r ds, 3.8
where Cc 0 is some constant only depending on c and α but we omit α since it does not play
any crucial role here. Iterating 3.7 infinitely many times, we have E
|X
i
t| ≤
M
X
n=0
1 c
n
X
j∈Z
d
a
n ji
|x
j
| + Cc + R
M
≤
∞
X
n=0
1 c
n
X
j∈Z
d
a
n ji
|x
j
| + Cc
1 − ηc
3.9
where R
M
is an M -tuple integral see the double integral in 3.8 and lim
M →∞
R
M
= 0. To estimate the double summation in the last line, we split the sum ’
P
j∈Z
d
· · · ’ into two pieces, and control them by 2.6 and
1 c
n
respectively. More precisely, let Λi, n ⊂ Z
d
be a cube centered at i such that d isti, Λ
c
i, n = n
2
up to some O1 correction, one has
∞
X
n=1
1 c
n
X
j∈Z
d
a
n ji
|x
j
| =
∞
X
n=1
1 c
n
X
j∈Λi,n
+ X
j∈Λ
c
i,n
a
n ji
|x
j
|. 3.10
2003
Since x ∈ B
R, ρ
, we have by 2.6 with c = 0 therein
∞
X
n=0
1 c
n
X
j∈Λ
c
i,n
a
n ji
|x
j
| ≤ R
∞
X
n=0
1 c
n
X
j∈Λ
c
i,n
a
n ji
| j|
ρ
+ 1 ≤ CR, ρ
∞
X
n=0
1 c
n
X
j∈Λ
c
i,n
a
n ji
| j − i|
ρ
+ |i|
ρ
+ 1 ≤ CR, ρ
∞
X
n=0
η
n
c
n
X
j∈Λ
c
i,n
X
k≥| j−i|
2k
nd
e
−
1 2
k
e
−
1 2
k
| j − i|
ρ
+ |i|
ρ
+ 1 ≤ CR, ρ
∞
X
n=1
η
n
c
n
X
k≥n
2
2k
nd
e
−
1 2
k
X
j∈Λ
c
i,n
e
−
1 2
| j−i|
| j − i|
ρ
+ |i|
ρ
+ 1 ≤ Cρ, R, d1 + |i|
ρ
3.11
where the last inequality is by the fact P
k≥n
2
2k
nd
e
−
1 2
k
≤ P
k≥1
e
−
1 2
k+nd log2k
∞ and the fact P
j∈Λ
c
i,n
e
−
1 2
| j−i|
| j − i|
ρ
≤ P
j∈Z
d
e
−
1 2
| j−i|
| j − i|
ρ
∞. For the other piece, one has
∞
X
n=0
1 c
n
X
j∈Λi,n
a
n ji
|x
j
| ≤ CR, ρ
∞
X
n=0
1 c
n
X
j∈Λi,n
a
n ji
| j − i|
ρ
+ |i|
ρ
+ 1 ≤ CR, ρ
∞
X
n=0
η
n
c
n
|Λi, n|
n
2 ρ
+ |i|
ρ
+ 1
≤ Cρ, R
∞
X
n=0
η
n
c
n
n
2d
n
2 ρ
+ |i|
ρ
+ 1
≤ CR, ρ, η, c1 + |i|
ρ
. 3.12
Collecting 3.9, 3.11 and 3.12, we immediately obtain 2. Way 2: The general case of c ≥ 0. By the integration by parts, Doob’s martingale inequality and the
2004
easy relation dE
j
s, t = E
j
s, t[−L
j
X s]ds where L
j
x =
J
j
x x
, we have E
Z
t
E
j
s, td Z
j
s ≤ E|Z
j
t| + E Z
t
E
j
s, tL
j
X sZ
j
sds ≤ C t
1 α
+ E
sup
0≤s≤t
|Z
j
s| Z
t
E
j
s, t−L
j
X sds
≤ C t
1 α
+ E sup
0≤s≤t
|Z
j
s| ≤ C t
1 α
. 3.13
By 3.6 and 3.13, one has E
|X
i
t| ≤ X
j∈Z
d
δ
ji
|x
j
| + C t
1 α
+ Z
t
X
j∈Z
d
δ + a
ji
E |X
j
s|ds 3.14
Iterating the above inequality infinitely many times, E
|X
i
t| ≤
∞
X
n=0
t
n
n X
j∈Z
d
[δ + a
n
]
ji
|x
j
| + C e
1+ηt
t
1 α
, 3.15
By estimating the double summation in the last line by the same method as in Way 1, we finally obtain 3.
4 immediately follows from Proposition 5.6.10 and Corollary 5.6.11 in [9].
3.2 Finite speed of propagation of information property
