Strong law of large number of a class of super-diffusions
Rong-Li Liu,
Yan-Xia Ren∗ and Renming Song
Abstract
In this paper we prove that, under certain conditions, a strong law of large numbers holds for a class of super-diffusions X corresponding to the evolution equation
∂t ut = Lut + βut − ψ(ut ) on a bounded domain D in Rd , where L is the generator
the underlying diffusion and the branching
mechanism ψ(x, λ) = 12 α(x)λ2 +
R ∞ of−λr
R∞
− 1 + λr)n(x, dr) satisfies supx∈D 0 (r ∧ r2 )n(x, dr) < ∞.
0 (e
Keywords Super-diffusion, martingale, point process, principal eigenvalue, strong
law of large numbers
2010 MR Subject Classification
1
60J68, 60G55, 60G57
Introduction
1.1
Motivation
Recently many people (see [3, 4, 6, 7, 8, 9, 20] and the references therein) have studied limit theorems
for branching Markov processes or super-processes using the principal eigenvalue and ground state
of the linear part of the characteristic equations. All the papers above, except [8], assumed that
the branching mechanisms satisfy a second moment condition. In [8], a (1 + θ)-moment condition,
θ > 0, on the branching mechanism is assumed instead.
In [1], Asmussen and Hering established a Kesten-Stigum L log L type theorem for a class
branching diffusion processes under a condition which is later called a positive regular property
in [2]. In [16, 17] we established Kesten-Stigum L log L type theorems for super-diffusions and
branching Hunt processes respectively.
This paper is a natural continuation of [16, 17]. The main purpose of this paper is to establish
a strong law of large numbers for a class of super-diffusions. The main tool of this paper is the
stochastic integral representation of super-diffusions.
Throughout this paper, we will use the following notations. For any positive integer k, Cbk (Rd )
denotes the family of bounded functions on Rd whose partial derivatives of order up to k are
bounded and continuous, C0k (Rd ) denotes the family of functions of compact support on Rd whose
∗
The research of this author is supported by NSFC (Grant No. 10871103 and 10971003) and Specialized Research
Fund for the Doctoral Program of Higher Education.
1
partial derivatives of order up to k are continuous. For any open set D ⊂ Rd , the meanings of
Cbk (D) and C0k (D) are similar. We denote by MF (D) the space of finite measures on D equipped
with the topology of weak convergence. We will use MF (D)0 to denote the subspace of nontrivial
measures in MF (D). The integral of a function ϕ with respect to a measure µ will often be denoted
as hϕ, µi.
For convenience we use the following convention throughout this paper: For any probability
measure P , we also use P to denote the expectation with respect to P .
1.2
Model
Suppose that aij ∈ Cb1 (Rd ), i, j = 1, · · · , d, and that the matrix (aij ) is symmetric and satisfies
X
0 < a|υ|2 ≤
aij υi υj , for all x ∈ Rd and υ ∈ Rd
i,j
for some positive constant a. We assume that bi , i = 1, · · · , d, are bounded Borel functions on Rd .
We will use (ξ, Πx , x ∈ Rd ) to denote a diffusion process on Rd corresponding to the operator
1
L = ∇ · a∇ + b · ∇.
2
In this paper we will always assume that D is a bounded domain in Rd . We will use (ξ D , Πx , x ∈
D) to denote the process obtained by killing ξ upon exiting from D, that is,
ξt if t < τ,
D
ξt =
∂, if t ≥ τ,
where τ = inf{t > 0; ξt ∈
/ D} is the first exit time of D and ∂ is a cemetery point. Any function f
on D is automatically extended to D ∪ {∂} by setting f (∂) = 0.
We will always assume that β is a bounded Borel function on Rd . We will use {PtD }t≥0 to
denote the following Feynman-Kac semigroup
Z t
D
D
D
Pt f (x) = Πx exp
β(ξs )ds f (ξt ) , x ∈ D.
0
It is well known that the semigroup {PtD }t≥0 is strongly continuous in L2 (D) and, for any t > 0,
PtD has a bounded, continuous and strictly positive density pD (t, x, y).
Let {PbtD }t≥0 be the dual semigroup of {PtD }t≥0 defined by
Z
PbtD f (x) =
pD (t, y, x)f (y)dy, x ∈ D.
D
It is well known that {PbtD }t≥0 is also strongly continuous in L2 (D).
b be the generators of the semigroups {P D }t≥0 and {PbD }t≥0 in L2 (D) respectively.
Let A and A
t
t
b resp.) denote the spectrum of A (A,
b resp.). It follows from Jentzsch’s theorem
Let σ(A) (σ(A)
([19, Theorem V.6.6, p. 337]) and the strong continuity of {PtD }t≥0 and {PbtD }t≥0 that the common
2
b is an eigenvalue of multiplicity 1 for both A and A,
b and
value λ1 := sup Re(σ(A)) = sup Re(σ(A))
that an eigenfunction φ of A associated with λ1 can be chosen to be strictly positive a.e. on D
b associated with λ1 can be chosen to be strictly positive a.e. on D. By
and an eigenfunction φe of A
[13, Proposition 2.3] we know that φ and φe are bounded and continuous on D, and they are in fact
R
e
strictly positive everywhere on D. We choose φ and φe so that D φ(x)φ(x)dx
= 1.
Throughout this paper we assume the following
Assumption 1 The semigroups {PtD }t≥0 and {PbtD }t≥0 are intrinsically ultracontractive, that is,
for any t > 0, there exists a constant ct > 0 such that
e
pD (t, x, y) ≤ ct φ(x)φ(y),
for all (x, y) ∈ D × D.
Assumption 1 is a very weak regularity assumption on D. It follows from [13, 14] that Assumption 1 is satisfied when D is a bounded Lipschitz domain. For other, more general, examples
of domain D for which Assumption 1 is satisfied, we refer our readers to [14] and the references
therein.
Define
e−λ1 t D
p (t, x, y) φ(y).
(1.1)
pφ (t, x, y) =
φ(x)
Then it follows from [13, Theorem 2.7] that if Assumption 1 holds, then for any σ > 0 there are
positive constants C(σ) and ν such that
e−λ1 t pD (t, x, y)φ(y)
φ
e
e
e
− φ(y)φ(y) ≤ C(σ)e−νt φ(y)φ(y),
x, y ∈ D, t > σ.
p (t, x, y) − φ(y)φ(y) = φ(x)
(1.2)
φ
e
By the definition of φ and φ, it is easy to check that, for any t > 0, p (t, ·, ·) is a probability density
and that φφe is its unique invariant probability density. (1.2) shows that pφ (t, ·, x) converges to
e
φ(x)φ(x)
uniformly with exponential rate. Denote by Ptφ the semigroup with density pφ (t, ·, ·) and
Πφh the probability generated by (Ptφ )t≥0 with initial distribution h(x)dx on D. Then (ξ D , Πφ e) is
φφ
e
a diffusion with initial distribution φ(x)φ(x)dx.
The super-diffusion (X, Pµ ), µ ∈ MF (D)0 , we are going to study is a (ξ D , ψ(λ) − βλ)-superprocess, which is a measure-valued Markov process with underlying spatial motion ξ D , branching
rate dt and branching mechanism ψ(λ) − βλ, where
Z ∞
1
2
ψ(x, λ) = α(x)λ +
e−rλ − 1 + λr n(x, dr),
λ > 0,
2
0
for some nonnegative bounded measurable function α on D and for some σ-finite kernel n from
(D, B(D)) to (R+ , B(R+ )), that is, n(x, dr) is a σ-finite measure on R+ for each fixed x, and n(·, B)
is a measurable function for each Borel set B ⊂ R+ . The measure µ here is the initial value of X.
In this paper we will always assume that
Z ∞
sup
(r ∧ r2 )n(x, dr) < ∞.
(1.3)
x∈D
0
3
Note that this assumption implies, for any fixed λ > 0, ψ(·, λ) is bounded on D. Define a new
kernel nφ (x, dr) from (D, B(D)) to (R+ , B(R+ )) such that for any nonnegative measurable function
f on R+ ,
Z
Z
∞
∞
f (r)nφ (x, dr) =
0
f (rφ(x))n(x, dr),
x ∈ D.
Then, by (1.3) and the boundedness of φ, nφ satisfies
Z ∞
(r ∧ r2 )nφ (x, dr) < ∞.
sup
x∈D
1.3
(1.4)
0
(1.5)
0
Stochastic Integral Representation and Main Result
Let (Ω, F, Pµ , µ ∈ MF (D)0 ) be the underlying probability space equipped with the filtration (Ft ),
which is generated by X and is completed as usual with the F∞ −measurable and Pµ −negligible
sets for every µ ∈ MF (D)0 . It is known (cf. [5, Section 6.1]) that the super-diffusion X is a solution
to the following martingale problem: for any ϕ ∈ C02 (D) and h ∈ Cb2 (R),
Z t
Z
1 t 00
h(hϕ, Xt i) − h(hϕ, µi) −
h0 (hϕ, Xs i)hAϕ, Xs ids −
h (hϕ, Xs i)hαϕ2 , Xs ids
2
0
0
Z tZ Z
−
h(hϕ, Xs i + rϕ(x)) − h(hϕ, Xs i) − h0 (hϕ, Xs i)rϕ(x) n(x, dr)Xs (dx)ds
0
D
(1.6)
(0,∞)
is a martingale. Let J denote the set of all jump times of X and δ denote the Dirac measure. It is
easy to see from the martingale problem (1.6) that the only possible jumps of X are point measures
rδx (·) with r being a positive real number and x ∈ D. (1.6) implies that the compensator of the
random measure
X
N :=
δ(s,4Xs )
s∈J
b on R+ × MF (D)0 such that for any nonnegative predictable function F on
is a random measure N
R+ × Ω × MF (D)0 ,
Z Z
Z ∞ Z
Z ∞
b (ds, dν) =
F (s, ω, ν)N
ds
Xs (dx)
F (s, ω, rδx )n(x, dr),
(1.7)
0
D
0
where n(x, dr) is the kernel of the branching mechanism ψ. Therefore we have
#
"
Z ∞ Z
Z ∞
X
Pµ
ds
Xs (dx)
F (s, ω, rδx )n(x, dr).
F (s, ω, 4Xs ) = Pµ
0
s∈J
D
0
See [5, p. 111]. Let F be a predictable function on R+ × Ω × MF (D)0 satisfying
Pµ
X
2
F (s, ∆Xs )
1/2 < ∞,
s∈[0,t],s∈J
4
for all µ ∈ MF (D)0 .
(1.8)
b
Then the stochastic integral of F with respect to the compensated random measure N − N
Z tZ
b )(ds, dν),
F (s, ν)(N − N
0
MF (D)0
can be defined (cf. [15] and the reference therein) as the unique purely discontinuous martingale
(vanishing at time 0) whose jumps are indistinguishable from 1J (s)F (s, ∆Xs ).
Suppose that ϕ is a measurable function on R+ × D. Define
Z
ϕ(s, x)ν(dx), ν ∈ MF (D)
(1.9)
Fϕ (s, ν) :=
D
whenever the integral above makes sense. We write
Z tZ
Z tZ
ϕ(s, x)S J (ds, dx) :=
StJ (ϕ) =
0
0
D
MF (D)0
b )(ds, dν),
Fϕ (s, ν)(N − N
(1.10)
whenever the right hand of (1.10) makes sense. If ϕ is bounded on R+ × D, then StJ (ϕ) is well
defined. Indeed, we only need to check that

1/2 
X


Pµ 
Fϕ (s, ∆Xs )2   < ∞,
for all µ ∈ MF (D)0 .
(1.11)
s∈[0,t],s∈J
Note that, for any µ ∈ MF (D)0 ,
1/2 P
2
Pµ
s∈[0,t],s∈J Fϕ (s, ∆Xs )
2 1/2
R
P
= Pµ
ϕ(s, x)(∆Xs )(dx)
s∈[0,t],s∈J

X

≤ kϕk∞ Pµ 
h1, ∆Xs i2 I{h1,∆Xs i≤1} +
s∈[0,t],s∈J


≤ kϕk∞ Pµ 
X
X
1/2 

h1, ∆Xs i2 I{h1,∆Xs i>1}  
s∈[0,t],s∈J
1/2 

h1, ∆Xs i2 I{h1,∆Xs i≤1}  
s∈[0,t],s∈J


+kϕk∞ Pµ 
X
1/2 

h1, ∆Xs i2 I{h1,∆Xs i>1}   .
s∈[0,t],s∈J
Here and throughout this paper, for any set A, IA stands for the indicator function of A. Using
the first two displays on [15, p. 203], we get (1.11). Thus for any bounded function ϕ on R+ × D.
(StJ (ϕ))t≥0 is a martingale.
For any ϕ ∈ C02 (D) and µ ∈ MF (D),
Z t
J
C
hϕ, Xt i = hϕ, µi + St (ϕ) + St (ϕ) +
hAϕ, Xs ids,
(1.12)
0
5
where StC (ϕ) is a continuous local martingale with quadratic variation
Z t
C
hαϕ2 , Xs ids.
hS (ϕ)it =
(1.13)
0
In fact, according to [10, 11], the above is still valid when A is replaced by L + β, where L is the
weak generator of ξ D in the sense of [10, Section 4]. Using this, [11, Corollary 2.18] and applying
a limit argument, one can show that for any bounded function g on D,
Z tZ
Z tZ
D
D
J
D
hg, Xt i = hPt g, µi +
Pt−s g(x)S (ds, dx) +
Pt−s
g(x)S C (ds, dx).
(1.14)
0
D
0
D
In particular, taking g = φ in (1.14), where φ is the positive eigenfunction of A defined in Section
1.1, we get that
Z
Z t
Z
Z t
−λ1 s
J
−λ1 s
−λ1 t
φ(x)S C (ds, dx).
(1.15)
e
φ(x)S (ds, dx) +
e
e
hφ, Xt i = hφ, µi +
0
D
0
D
e−λ1 t hφ, Xt i.
Set Mt (φ) :=
Then Mt (φ), t ≥ 0, is a nonnegative martingale. Denote by M∞ (φ) the
almost sure limit of Mt (φ) as t → ∞. In [16], we studied the relationship between the degeneracy
property of M∞ (φ) and the function l:
Z ∞
l(y) :=
r ln rnφ (y, dr).
(1.16)
1
Theorem 1.1 [16, Theorem 1.1] Suppose that Assumption 1 holds, λ1 > 0 and that X is a
(ξ D , ψ(λ) − βλ)−super-diffusion. Then the following assertions hold:
R
e
< ∞, then M∞ (φ) is non-degenerate under Pµ for any µ ∈ MF (D)0 , and
(1) If D l(y)φ(y)dy
M∞ (φ) is also the L1 (Pµ ) limit of Mt (φ).
R
e
= ∞, then M∞ (φ) = 0, Pµ −a.s. for any µ ∈ MF (D)0 .
(2) If D l(y)φ(y)dy
Remark 1.2 In [16, Theorem 1.1], we only stated that in case (1) under the extra assumption
α ≡ 0, M∞ (φ) is non-degenerate under Pµ for any µ ∈ MF (D)0 . But actually in this case we have
Pµ M∞ (φ) = Pµ M0 (φ) (see [16, Lemma 3.4]), and therefore Mt (φ) converges to M∞ (φ) in L1 (Pµ ).
For general α ≥ 0, by the L2 maximum inequality, and using the fact that α and φ are bounded
in D, we have
"
Z t
2 #
Z
−λ1 s
C
Pµ sup
e
φ(x)S (ds, dx)
t≥0
0
D
2
t
≤4 sup Pµ
e−λ1 s
φ(x)S C (ds, dx)
t≥0
0
D
Z ∞
Z
=4Pµ
e−2λ1 s ds
α(x)φ2 (x)Xs (dx)
D
Z ∞0
Z
Z
−λ1 s
=4
e
ds
φ(y)µ(dy)
pφ (s, y, x)α(x)φ(x)dx
Z
0
Z
D
D
<∞.
6
(1.17)
R
R
t
Thus the martingale 0 e−λ1 s D φ(x)S C (ds, dx)
converges almost surely and in L1 (Pµ ). Det≥0
R∞
R
note the limit by 0 e−λ1 s D φ(x)S C (ds, dx). Furthermore, we obtain that when λ1 > 0 and
R
R t −λ s R
J
e
1
D l(x)φ(x)dx < ∞, the martingale R0 e
DRφ(x)S (ds, dx) converges almost surely and in
∞
L1 (Pµ ) as well. Denote the limit by 0 e−λ1 s D φ(x)S J (ds, dx). Thus it follows from (1.15)
that Mt (φ) converges to a non-degenerate M∞ (φ) Pµ -almost surely and in L1 (Pµ ) for every µ ∈
MF (D)0 .
The main goal of this paper is to establish the following almost sure convergence result.
Theorem 1.3 Suppose that Assumption 1 holds, λ1 > 0 and that X is a (ξ D , ψ(λ) − βλ)−superdiffusion. Then there exists Ω0 ⊂ Ω of probability one (that is, Pµ (Ω0 ) = 1 for every µ ∈ MF (D)0 )
such that, for every ω ∈ Ω0 and for every nontrivial nonnegative bounded Borel function f on D
with compact support whose set of discontinuous points has zero Lebesgue measure, we have
hf, Xt i(ω)
M∞ (φ)(ω)
=
.
t→∞ Pµ hf, Xt i
hφ, µi
lim
(1.18)
As a consequence of this theorem we immediately get the following
Corollary 1.4 Suppose that Assumption 1 holds, λ1 > 0 and that X is a (ξ D , ψ(λ) − βλ)−superdiffusion. Then there exists Ω0 ⊂ Ω of probability one (that is, Pµ (Ω0 ) = 1 for every µ ∈ MF (D)0 )
such that, for every ω ∈ Ω0 and every relatively compact Borel subset B in D of positive Lebesgue
measure whose boundary is of Lebesgue measure zero, we have
Xt (B)(ω)
M∞ (φ)(ω)
=
.
t→∞ Pµ [Xt (B)]
hφ, µi
lim
Remark 1.5 (i) Although we assumed in this paper that the underlying motion ξ D is a diffusion
process in a bounded domain D, the arguments of this paper can be easily extended to the case when
the underlying motion is a Hunt process on a locally compact separable metric space E satisfying
[17, Assumption 1.1] for some measure m with full support and with m(E) < ∞, and the analogue
of Assumption 1 above.
(ii) In [3, 6, 7, 9, 20], the branching mechanism is assumed to be binary, while in the present
paper we deal with general branching mechanism. [8] considers a general branching mechanism
under a (1 + θ)-moment condition, θ > 0, while in the present paper, we only assume a L log L
condition. In [3] the underlying motion is assumed to be a symmetric Hunt process, while in the
present paper, our underlying process needs not be symmetric.
2
Proof of Theorem 1.3
A main step in proving Theorem 1.3 is the following result.
7
Theorem 2.1 Suppose that Assumption 1 holds, λ1 > 0 and that X is a (ξ D , ψ(λ) − βλ)−superdiffusion. Then for any f ∈ Bb (D) and µ ∈ MF (D)0 ,
Z
−λ1 t
e
φ(y)φ(y)f
(y)dy,
Pµ −a.s.
(2.1)
lim e
hφf, Xt i = M∞ (φ)
t→∞
D
We will prove this result first. According to Theorem 1.1 (2), when
have
M∞ (φ) = lim e−λ1 t hφ, Xt i = 0,
Pµ −a.s.
R
D
e
φ(x)l(x)dx
= ∞, we
t→∞
For any f ∈
Bb+ (D),
lim sup e−λ1 t hφf, Xt i ≤ kf k∞ lim sup e−λ1 t hφ, Xt i = 0
t→∞
t→∞
and (2.1) follows immediately from the nonnegativity of f .
R
e
It remains to prove the case when D φ(x)l(x)dx
< ∞. In the remainder of this section, we
assume that the assumptions of Theorem 2.1 hold and that f ∈ Bb+ (D) is fixed. Define
S(ds, dx) = S J (ds, dx) + S C (ds, dx).
Using (1.14), we get
e−λ1 t hφf, Xt i = e−λ1 t hPtD (φf ), µi + e−λ1 t
Z tZ
0
D
(Pt−s
(φf ))(x)S(ds, dx).
(2.2)
D
It follows from (1.2) that
lim e−λ1 t hPtD (φf ), µi = hφ, µi
t→∞
Z
e
φ(x)φ(x)f
(x)dx.
D
Therefore, to prove Theorem 2.1, it suffices to show that
Z tZ
Z
Z
D
e
(Pt−s
(φf ))(x)S(ds, dx) =
φφ(x)f
(x)dx
lim e−λ1 t
t→∞
0
D
D
∞
e−λ1 s
0
Z
φ(x)S(ds, dx),
Pµ −a.s.
D
(2.3)
To prove the above result, we need some lemmas first.
Lemma 2.2 For any µ ∈ MF (D)0 , we have
X
I{∆Xs (φ)>eλ1 s } < ∞,
Pµ −a.s.
(2.4)
s>0
R∞
Proof: First note that (1.5) implies that supx∈D 1 rnφ (x, dr) < ∞. It is well known that for
any g ∈ B + (D),
Pµ hg, Xt i = hPtD g, µi.
(2.5)
8
By the definition of nφ given by (1.4), we have
#
"
X
I{∆Xs (φ)>eλ1 s }
Pµ
Z
s>0
∞
Z Z
∞
I{rφ(x)>eλ1 s } Xs (dx)n(x, dr)
ds
= Pµ
Z ∞
Z ∞0 Z D 0 Z
D
p (s, y, x)dx
nφ (x, dr).
µ(dy)
ds
=
eλ1 s
D
D
0
It follows from (1.2) that there is a constant C > 0 such that
e
pD (t, y, x) ≤ Ceλ1 t φ(y)φ(x),
∀ t > 1, x, y ∈ D.
(2.6)
Therefore we have,
"
#
X
Pµ
I{∆Xs (φ)>eλ1 s }
s>0
Z
1
Z
Z
∞
Z
Z
ds
D
φ
∞
Z
Z
D
Z
∞
µ(dy)
p (s, y, x)dx
n (x, dr) +
ds
µ(dy)
p (s, y, x)dx
nφ (x, dr)
λ1 s
λ1 s
0
D D
e
1
D
D
e
Z ∞
Z
Z
Z ∞
Z
∞ φ
λ1 s
kβk∞
e
+C
e
ds
φ(y)µ(dy)
φ(x)dx
nφ (x, dr)
n
(x,
dr)
≤ e
µ(D) λ
s
D
D
e 1
Z1
Z ∞∞
Z0 ∞
e
=: C1 + Chφ, µi
φ(x)dx
eλ1 s ds
nφ (x, dr).
=
D
eλ1 s
0
Using Fubini’s theorem we get,
Z ∞
Z
eλ1 s ds
∞
nφ (x, dr) ≤
eλ1 s
0
1
λ1
∞
Z
rnφ (x, dr).
1
Hence we have
Pµ
hX
I{∆Xs (φ)>eλ1 s }
i
s>0
Chφ, µi
≤
λ1
Z
Z
∞
rnφ (x, dr) < ∞.
e
φ(x)dx
D
1
2
Consequently, (2.4) holds.
Define
(1)
X
Nφ :=
(2)
δ(s, 4Xs )
X
and Nφ :=
0<4Xs (φ)<eλ1 s
δ(s, 4Xs ) ,
4Xs (φ)≥eλ1 s
(1)
(2)
b (1) and N
b (1) respectively. Then for any
and denote the compensators of Nφ and Nφ by N
φ
φ
nonnegative predictable function F on R+ × Ω × MF (D),
Z
0
∞Z
(1)
b (ds, dν) =
F (s, ν)N
φ
Z
∞
Z
Z
ds
0
Xs (dx)
D
9
0
eλ1 s
F (s, rφ(x)−1 δx )nφ (x, dr),
(2.7)
and
Z
∞Z
0
(1)
Let Jφ
b (2) (ds, dν) =
F (s, ν)N
φ
∞
Z
Z
Z
Xs (dx)
ds
D
0
(1)
(2)
∞Z
(2.8)
F (s, ω, 4Xs ),
(2.9)
F (s, ω, 4Xs ),
(2.10)
(1)
s∈Jφ
∞Z
(2)
X
F (s, ν)Nφ (ds, dν) =
0

F (s, rφ(x)−1 δx )nφ (x, dr).
the set of jump times of Nφ . Then
X
F (s, ν)Nφ (ds, dν) =
0
Z
(1)
eλ1 s
(2)
denote the set of jump times of Nφ , and Jφ
Z
∞
(2)
s∈Jφ

∞
Z
 X

Pµ 
F (s, ω, 4Xs ) = Pµ
Z
ds
Xs (dx)
0
(1)
s∈Jφ
eλ1 s
Z
D
F (s, ω, rφ(x)−1 δx )nφ (x, dr),
(2.11)
F (s, ω, rφ(x)−1 δx )nφ (x, dr).
(2.12)
0
and


Z

 X
F (s, ω, 4Xs ) = Pµ
Pµ 
∞
Z
0
(2)
s∈Jφ
Z
ds
Xs (dx)
D
∞
eλ1 s
(1)
We can construct two martingale measures S J,(1) (ds, dx) and S J,(2) (ds, dx) respectively from Nφ (ds, dν)
(2)
and Nφ (ds, dν), similar to the way we constructed S J (ds, dx) from N (ds, dν). Then for any
bounded measurable function g on R+ × D,
Z tZ
Z tZ
J,(1)
(1)
(1)
b (1) )(ds, dν),
St (g) =
g(s, x)S (ds, dx) =
Fg (s, ν)(Nφ − N
(2.13)
φ
0
D
MF (D)
0
and
J,(2)
St (g)
Z tZ
=
g(s, x)S
0
(2)
Z tZ
(ds, dx) =
D
MF (D)
0
(2)
b (2) )(ds, dν),
Fg (s, ν)(Nφ − N
φ
R
where Fg (s, ν) = g(s, x)ν(dx).
For any m, n ∈ N, σ > 0 and f ∈ Bb+ (D), define
−λ1 (n+m)σ
(n+m)σ
Z
Z
H(n+m)σ (f ) := e
0
and
−λ1 (n+m)σ
Z
D
(n+m)σ
Z
L(n+m)σ (f ) := e
0
D
10
D
P(n+m)σ−s
(φf )(x)S J,(1) (ds, dx)
D
P(n+m)σ−s
(φf )(x)S J,(2) (ds, dx).
(2.14)
Lemma 2.3 If
R
D
e
l(x)φ(x)dx
< ∞, then for any m ∈ N, σ > 0, µ ∈ MF (D)0 and f ∈ Bb+ (D),
∞
X
2
Pµ H(n+m)σ (f ) − Pµ (H(n+m)σ (f )Fnσ ) < ∞
(2.15)
n=1
and
lim H(n+m)σ (f ) − Pµ [H(n+m)σ (f )Fnσ ] = 0,
n→∞
Pµ −a.s.
(2.16)
Proof: Since PtD (φf ) is bounded in [0, T ] × D for any T > 0, the process
Z tZ
D
−λ1 (n+m)σ
P(n+m)σ−s
(φf )(x)S J,(1) (ds, dx),
t ∈ [0, (n + m)σ]
Ht (f ) := e
0
D
is a martingale with respect to (Ft )t≤(n+m)σ . Thus
Z nσ Z
D
−λ1 (n+m)σ
(φf )(x)S J,(1) (ds, dx),
P(n+m)σ−s
Pµ (H(n+m)σ (f ) Fnσ ) = e
0
D
and hence
H(n+m)σ (f ) − Pµ (H(n+m)σ (f )Fnσ ) = e−λ1 (n+m)σ
Z
(n+m)σ
nσ
Z
D
D
P(n+m)σ−s
(φf )(x)S J,(1) (ds, dx).
Since
−λ1 (n+m)σ
Mt := e
Z
=
Z
t
nσ
Z
D
D
P(n+m)σ−s
(φf )(x)S J,(1) (ds, dx)
t
nσ
Fe−λ1 (n+m)σ P D
(n+m)σ−·
(1)
(φf ) (s, ν)(Nφ
b (1) )(ds, dν),
−N
φ
is a martingale with quadratic variation
Z t
Fe−λ1 (n+m)σ P D
(φf ) (s, ν)
(n+m)σ−·
nσ
2
t ∈ [nσ, (n + m)σ]
b (1) (ds, dν),
N
φ
we have
2
Pµ H(n+m)σ (f ) − Pµ (H(n+m)σ (f )Fnσ )
Z (n+m)σ
2 b (1)
Fe−λ1 (n+m)σ P D
= Pµ
(φf ) (s, ν) Nφ (ds, dν)
(n+m)σ−·
nσX
2
Fe−λ1 (n+m)σ P D
= Pµ
(φf ) (s, ∆Xs ) ,
(2.17)
(n+m)σ−·
(1)
s∈J˜n,m
(1)
(1) T
[nσ, (n + m)σ]. Note that for any f ∈ Bb (D), kPtφ f k∞ ≤ kf k∞ for all t ≥ 0,
where J˜n,m = Jφ
which is equivalent to
PtD (φf )(y) ≤ kf k∞ eλ1 t φ(y),
11
∀ t ≥ 0, y ∈ D.
(2.18)
Using (2.7) and (2.11), we obtain
X
Fe−λ1 (n+m)σ P D
Pµ
2
(n+m)σ−·
(φf ) (s, ∆Xs )
s∈Jen,m
Z
(n+m)σ
nσ
≤
Fe2−λ1 (n+m)σ P D
Xs (dx)
0
D
Z
(n+m)σ
ds
= Pµ
= e
eλ1 s
Z
Z
−2λ1 (n+m)σ
kf k2∞
Z
Z
(n+m)σ−·
Z
p (s, y, x)dx
D
D
e
−2λ1 s
Z
ds
nσ
(s, rφ(x)−1 δx )nφ (x, dr)
eλ1 s
Z
D
µ(dy)
ds
nσ
(n+m)σ
(φf )
Z
pD (s, y, x)dx
µ(dy)
D
0
Z eλ1 s
D
D
[P(n+m)σ−s
(φf )(x)φ(x)−1 ]2 r2 nφ (x, dr)
r2 nφ (x, dr),
0
where in the second equality we used the fact that
Fe−λ1 (n+m)σ P D
(n+m)σ−·
−1
(φf ) (s, rφ(x) δx )
D
= re−λ1 (n+m)σ φ−1 (x)P(n+m)σ−s
(φf )(x)
(2.19)
and in the last inequality we used (2.18). It follows from (1.2) that there is a constant C > 0 such
that
e
pD (s, y, x) ≤ Ceλ1 s φ(y)φ(x),
∀ s > σ, x, y ∈ D.
(2.20)
Thus
Pµ
X
Fe−λ1 (n+m)σ P D
(n+m)σ−·
(φf ) (s, ∆Xs )
2
s∈Jen,m
≤
Ckf k2∞ hφ, µi
Z
∞
Z
e
φ(x)dx
e
D
−λ1 s
eλ1 s
Z
r2 nφ (x, dr).
ds
nσ
0
Summing over n, we get
∞
X
n=1
≤
∞
X
Pµ
X
Fe−λ1 (n+m)σ P D
(n+m)σ−·
(φf ) (s, ∆Xs )
s∈Jen,m
Ckf k2∞ hφ, µi
Ckf k2∞ hφ, µi
Z
Z
Z
e−λ1 s ds
Z
nσ
∞
Z
e
φ(x)dx
D
∞
e
φ(x)dx
D
n=1
≤
2
r2 nφ (x, dr)
0
Z
∞
dt
0
eλ1 s
e
tσ
−λ1 s
Z
ds
eλ1 s
r2 nφ (x, dr)
0
Z
Z ∞
Z eλ1 s
C
2
−λ1 s
e
=
kf k∞ hφ, µi
φ(x)dx
se
ds
r2 nφ (x, dr)
σ
D
0
0
Z
Z ∞
Z ∞
C
2
2 φ
e
≤
kf k∞ hφ, µi
φ(x)dx
r n (x, dr)
se−λ1 s ds
σ
λ−1
ln
r
D
1
1
Z
Z 1
Z ∞
C
e
+
kf k2∞ hφ, µi
φ(x)dx
r2 nφ (x, dr)
se−λ1 s ds
σ
D
0
0
=: I + II.
12
(2.21)
Using (1.5) we immediately get that II < ∞. On the other hand,
Z
Z ∞
C
e
I = 2 kf k2∞ hφ, µi
φ(x)dx
r(ln r + 1)nφ (x, dr).
λ1 σ
D
1
R
e
Now we can use D l(x)φ(x)dx
< ∞ and (1.3) to get that I < ∞. The proof of (2.15) is now
complete. For any ε > 0, using (2.15) and Chebyshev’s inequality we have
∞
X
Pµ H(n+m)σ (f ) − Pµ [H(n+m)σ (f )Fnσ ] > ε
n=1
≤ ε
−2
∞
X
2
Pµ H(n+m)σ (f ) − Pµ (H(n+m)σ (f )Fnσ )
n=1
< ∞.
2
Then (2.16) follows easily from the Borel-Cantelli Lemma.
Lemma 2.4 If
have
R
D
e
l(x)φ(x)dx
< ∞, then for any m ∈ N, σ > 0, µ ∈ MF (D)0 and f ∈ Bb+ (D) we
lim L(n+m)σ (f ) − Pµ L(n+m)σ (f )Fnσ = 0,
n→∞
Pµ −a.s.
(2.22)
Proof: It is easy to see that
Pµ
L(n+m)σ (f )Fnσ = e−λ1 (n+m)σ
Z
nσ
Z
0
D
D
P(n+m)σ−s
(φf )(x)S J,(2) (ds, dx).
Therefore,
L(n+m)σ (f ) − Pµ L(n+m)σ (f )Fnσ = e−λ1 (n+m)σ
(n+m)σ
Z
nσ
Z
D
D
P(n+m)σ−s
(φf )(x)S J,(2) (ds, dx).
(2.23)
It follows from Lemma 2.2 that, almost surely, the support of the measure
consists of finitely
many points. Hence almost surely there exists N0 ∈ N such that for any n > N0 ,
Z (n+m)σ Z
−λ1 (n+m)σ
D
e
P(n+m)σ−s
(φf )(x)S J,(2) (ds, dx)
(2)
Nφ
nσ
−λ1 (n+m)σ
Z
D
(n+m)σ
Z
= −e
−λ1 (n+m)σ
nσ
(n+m)σ
MF (D)
Z
= −e
(n+m)σ−·
Z
ds
nσ
FP D
b (2)
(φf )(·) (s, ν)Nφ (ds, dν)
−1
Xs (dx)φ(x)
D
D
P(n+m)σ−s
(φf )(x)
Z
∞
rnφ (x, dr). (2.24)
eλ1 s
where in the last equality we used (2.8) and (2.19). Using (2.18) we get
Z (n+m)σ Z
−λ1 (n+m)σ
D
J,(2)
P(n+m)σ−s (φf )(x)S
(ds, dx)
e
nσ
D
Z ∞
Z
Z ∞
≤ kf k∞
e−λ1 s ds
Xs (dx)
rnφ (x, dr).
nσ
eλ1 s
D
13
(2.25)
On the other hand, by (2.20), we have
Z ∞
Z
Z ∞
−λ1 s
φ
e
ds
Xs (dx)
Pµ
rn (x, dr)
nσ
D
eλ1 s
Z ∞
Z
Z
Z ∞
−λ1 s
D
=
e
ds
µ(dy)
p (s, y, x)dx
rnφ (x, dr)
eλ1 s
nσ
D
D
Z ∞
Z ∞ Z
e
ds
φ(x)dx
rnφ (x, dr)
≤ Chφ, µi
nσ
eλ1 s
D
Z
Z
≤ Chφ, µi
e
φ(x)dx
C
hφ, µi
λ1
Z
φ
λ−1
1 ln r
ds
rn (x, dr)
λ nσ
Ze 1∞
D
=
∞
Z
e
φ(x)dx
D
0
r ln rnφ (x, dr).
eλ1 nσ
Applying the dominated convergence theorem, we obtain that in L1 (Pµ ),
Z ∞
Z
Z ∞
−λ1 s
lim
e
ds
Xs (dx)
rnφ (x, dr) = 0.
n→∞ nσ
R∞
eλ1 s
D
R∞
Since nσ e−λ1 s ds D Xs (dx) eλ1 s rnφ (x, dr) is decreasing in n, the above limit holds almost surely
as well. Therefore, by (2.25), we have
lim e
R
−λ1 (n+m)σ
n→∞
Z
(n+m)σ
Z
nσ
D
D
(P(n+m)σ−s
φf )(x)S J,(2) (ds, dx) = 0,
Pµ −a.s.
(2.26)
2
Now (2.22) follows from (2.23) and (2.26). The proof is complete.
For any m, n ∈ N, σ > 0, set
−λ1 (n+m)σ
Z
(n+m)σ
Z
C(n+m)σ (f ) := e
0
D
D
(P(n+m)σ−s
φf )(x)S C (ds, dx),
f ∈ Bb+ (D).
Then {Cnσ (f )}n∈N is a martingale with respect to (Fnσ ) and
Z nσ Z
−λ1 (n+m)σ
D
(P(n+m)σ−s
φf )(x)S C (ds, dx).
Pµ (C(n+m)σ (f ) Fnσ ) = e
0
D
Lemma 2.5 For any m ∈ N, σ > 0, µ ∈ MF (D)0 and f ∈ Bb+ (D) we have
lim C(n+m)σ (f ) − Pµ [C(n+m)σ (f )Fnσ ] = 0,
n→∞
14
Pµ −a.s.
(2.27)
Proof: From the quadratic variation formula (1.13),
2
Pµ H(n+m)σ (f ) − Pµ (H(n+m)σ (f )Fnσ )
#2
"
Z (n+m)σ Z
D
φf )(x)S C (ds, dx)
(P(n+m)σ−s
=Pµ e−λ1 (n+m)σ
nσ
Z
(n+m)σ
=
D
−2λ1 (n+m)σ
e
nσ
(n+m)σ
Z
−2λ1 s
ds
µ(dx)
D
Z
Z
Z
(n+m)σ
e−λ1 s ds
2
D
φf (y)dy
pD (s, x, y) P(n+m)σ−s
D
2
p (s, x, y)φ (y)
µ(dx)
ds
D
D
nσ
≤kf k2∞
Z
D
e
=
Z
Z
φ
P(n+m)σ−s
f
2
(2.28)
(y)dy
φ(x)µ(dx)Psφ φ(y)dy
D
nσ
1
≤ kφk∞ kf k2∞ hφ, µie−λ1 nσ .
λ1
Therefore, we have
∞
X
2
Pµ C(n+m)σ (f ) − Pµ (C(n+m)σ (f )Fnσ ) < ∞.
(2.29)
n=1
2
By the Borel-Cantelli lemma, we get (2.27).
Combining the three lemmas above, we have the following result.
R
e
< ∞, then for any m ∈ N, σ > 0, µ ∈ MF (D)0 and f ∈ Bb+ (D) we
Lemma 2.6 If D l(x)φ(x)dx
have
h
i
lim e−λ1 (n+m)σ hφf, X(n+m)σ i − Pµ e−λ1 (n+m)σ hφf, X(n+m)σ iFnσ = 0,
Pµ −a.s.
(2.30)
n→∞
Proof: From (1.14), we know that e−λ1 (n+m)σ hφf, X(n+m)σ i can be decomposed into three
parts:
e−λ1 (n+m)σ hφf, X(n+m)σ i
= e
−λ1 (n+m)σ
D
hP(n+m)σ
(φf ), µi
+e
−λ1 (n+m)σ
Z
0
(n+m)σ
Z
D
D
P(n+m)σ−s
(φf )(x)S(ds, dx)
D
= e−λ1 (n+m)σ hP(n+m)σ
(φf ), µi + H(n+m)σ (f ) + L(n+m)σ (f ) + C(n+m)σ (f ).
Therefore,
h
i
e−λ1 (n+m)σ hφf, X(n+m)σ i − Pµ e−λ1 (n+m)σ hφf, X(n+m)σ iFnσ
= H(n+m)σ (f ) − Pµ H(n+m)σ (f )|Fnσ + L(n+m)σ (f ) − Pµ L(n+m)σ (f )|Fnσ
+C(n+m)σ (f ) − Pµ C(n+m)σ (f )|Fnσ .
Now the conclusion of this lemma follows immediately from Lemma 2.3, Lemma 2.4, and Lemma
2.5.
2
15
Theorem 2.7 If
e
l(x)φ(x)dx
< ∞, then for any σ > 0, µ ∈ MF (D)0 and f ∈ Bb+ (D) we have
Z
e
lim e−λ1 nσ hφf, Xnσ i = M∞ (φ)
φ(z)φ(z)f
(z)dz,
Pµ −a.s.
R
D
n→∞
D
Proof: By (2.5) and the Markov property of super-processes we have
h
i
D
Pµ e−λ1 (n+m)σ hφf, X(n+m)σ iFnσ = e−λ1 nσ he−λ1 mσ Pmσ
(φf ), Xnσ i.
It follows from (1.2) that there exist constants c > 0 and ν > 0 such that
−λ mσ D
Z
Z
e 1 Pmσ (φf )(x)
−νmσ
e
e
−
φ(z)φ(z)f (z)dz ≤ ce
φ(z)φ(z)f
(z)dz,
φ(x)
D
D
which is equivalent to
e−λ1 mσ P D (φf )(x)
mσ
− 1 ≤ ce−νmσ .
R
e
φ(x)
φ(z)φ(z)f
(z)dz
D
Thus there exist positive constants km ≤ 1 and Km ≥ 1 such that
Z
Z
D
e
e
km φ(x)
φ(z)φ(z)f
(z)dz ≤ e−λ1 mσ Pmσ
(φf )(x) ≤ Km φ(x)
φ(z)φ(z)f
(z)dz,
D
D
and that limm→∞ km = limm→∞ Km = 1. Hence,
e
−λ1 nσ
he
−λ1 mσ
D
Pmσ
(φf ), Xnσ i
≥ km e
Z
−λ1 nσ
e
hφ, Xnσ i
φ(z)φ(z)f
(z)dz
D
Z
e
= km Mnσ (φ)
φ(z)φ(z)f
(z)dz,
Pµ −a.s.
D
and
Z
D
e
e−λ1 nσ he−λ1 mσ Pmσ
(φf ), Xnσ i ≤ Km e−λ1 nσ hφ, Xnσ i
φ(z)φ(z)f
(z)dz
D
Z
e
= Km Mnσ (φ)
φ(z)φ(z)f
(z)dz,
Pµ −a.s.
D
These two inequalities and Lemma 2.6 imply that
lim sup e−λ1 nσ hφf, Xnσ i = lim sup e−λ1 (n+m)σ hφf, X(n+m)σ i
n→∞
n→∞
h
i
= lim sup Pµ e−λ1 (n+m)σ hφf, X(n+m)σ iFnσ
n→∞
D
= lim sup e−λ1 nσ he−λ1 mσ Pmσ
(φf ), Xnσ i
n→∞
Z
e
≤ lim sup Km Mnσ (φ)
φ(z)φ(z)f
(z)dz
n→∞
D
Z
e
= Km M∞ (φ)
φ(z)φ(z)f
(z)dz,
Pµ −a.s.
D
16
(2.31)
and that
lim inf e
−λ1 nσ
n→∞
Z
hφf, Xnσ i ≥ km M∞ (φ)
Pµ −a.s.
e
φ(z)φ(z)f
(z)dz,
D
Letting m → ∞, we get
−λ1 nσ
lim e
n→∞
Z
hφf, Xnσ i = M∞ (φ)
e
φ(z)φ(z)f
(z)dz,
Pµ −a.s.
D
2
The proof is now complete.
We are now ready to give the proof of Theorem 2.1.
Proof of Theorem 2.1 Put ∆σ (f ) := sup0≤t≤σ kPtφ f − f k∞ . Then for t ∈ [nσ, (n + 1)σ],
D
E
D
E
−λ1 t
φ
φ
−λ1 t
−λ1 t
e
φP
f,
X
−
e
hφf,
X
i
≤
e
φ|P
f
−
f
|,
X
t
t t ≤ Mt (φ)∆σ (f ). (2.32)
(n+1)σ−t
(n+1)σ−t
By the strong continuity of the semigroup (Ptφ ) in L∞ (D), we have limσ→0 ∆σ (f ) = 0. Thus,
D
E
−λ1 t
φ
−λ1 t
lim lim
sup
φP(n+1)σ−t f, Xt − e
hφf, Xt i = 0, Pµ −a.s.
(2.33)
e
σ→0 n→∞ t∈[nσ,(n+1)σ]
Therefore, to prove Theorem 2.1, we only need to show that
Z
φ
−λ1 t
e
lim lim
sup
e
hφP(n+1)σ−t f, Xt i = M∞ (φ)
φ(x)φ(x)f
(x)dx,
σ→0 n→∞ t∈[nσ, (n+1)σ]
Pµ −a.s.
(2.34)
D
For any n ∈ N and σ > 0, (Xt , t ∈ [nσ, (n + 1)σ], Pµ (·|Fnσ )) can be regarded as a (ξ D , ψ(λ) − βλ)super-diffusion with initial value Xnσ . Thus, for arbitrary g ∈ Bb+ (D), we have by (1.14)
D
e−λ1 t hφg, Xt i = e−λ1 t hPt−nσ
(φg), Xnσ i + e−λ1 t
t
Z
Z
D
Pt−s
(φg)(x)S(ds, dx),
t ∈ [nσ, (n + 1)σ].
nσ
φ
Taking g(x) = P(n+1)σ−t
f (x) in the above identity and using (1.1), we get
−λ1 t
e
D
φ
φP(n+1)σ−t
f, Xt
E
=e
−λ1 nσ
D
φPσφ f, Xnσ
E
Z
t
+
e
−λ1 s
nσ
Z
φ
(φP(n+1)σ−s
f )(x)S(ds, dx). (2.35)
e is the invariant probability density of the semigroup (P φ ), we have by Lemma 2.7,
Since φφ
t
Z
φ
e
lim e−λ1 nσ hφPσφ f, Xnσ i = M∞ (φ)
φ(x)φ(x)P
σ f (x)dx
n→∞
ZD
e
= M∞ (φ)
φ(x)φ(x)f
(x)dx.
(2.36)
D
Hence, by (2.35) and (2.36), to prove (2.34) it suffices to show that
Z t
Z
φ
−λ1 s
lim lim
sup
e
(φP(n+1)σ−s
f )(x)S(ds, dx) = 0,
σ→0 n→∞ t∈[nσ,(n+1)σ] nσ
17
Pµ −a.s.
(2.37)
Since S(ds, dx) = S J (ds, dx) + S C (ds, dx) = S J,(1) (ds, dx) + S J,(2) (ds, dx) + S C (ds, dx), we have
Z
Z t
φ
f )(x)S(ds, dx)
e−λ1 s (φP(n+1)σ−s
nσ
Z
Z
Z t
Z t
φ
φ
J,(1)
−λ1 s
−λ1 s
(φP(n+1)σ−s
f )(x)S J,(2) (ds, dx)
(φP(n+1)σ−s f )(x)S
(ds, dx) +
e
=
e
nσ
nσ
Z
Z t
φ
e−λ1 s (φP(n+1)σ−s f )(x)S C (ds, dx)
+
nσ
σ
σ
=: Hn,t
(f ) + Lσn,t (f ) + Cn,t
(f ).
Thus we only need to prove that
lim lim
sup
σ
Hn,t
(f ) = 0,
Pµ −a.s.,
(2.38)
lim lim
sup
Lσn,t (f ) = 0,
Pµ −a.s.,
(2.39)
lim lim
sup
σ
Cn,t
(f ) = 0,
Pµ −a.s.
(2.40)
σ→0 n→∞ t∈[nσ,(n+1)σ]
σ→0 n→∞ t∈[nσ,(n+1)σ]
and
σ→0 n→∞ t∈[nσ,(n+1)σ]
It follows from Chebyshev’s inequality that, for any ε > 0, we have
!
!2
Z Z t
σ
1
φ
J,(1)
−λ
s
Hn,t (f ) > ε ≤ Pµ
Pµ
sup
φP(n+1)σ−s f (x)S
(ds, dx) (2.41)
.
e 1
sup
ε2
t∈[nσ,(n+1)σ]
t∈[nσ,(n+1)σ] nσ
σ (f ); t ∈ [nσ, (n + 1)σ]) is a martingale with respect to (F )
Since the process (Hn,t
t t∈[nσ,(n+1)σ] ,
σ
applying Burkholder-Davis-Gundy inequality to Hn,t (f ) and using an argument similar to the one
in the proof of Lemma 2.3, we obtain
Z
Z t
2
φ
−λ1 s
(φP(n+1)σ−s
f )(x)S J,(1) (ds, dx)
e
Pµ
sup
t∈[nσ,(n+1)σ] nσ
≤
C1 P µ
Z
=
Z
(n+1)σ
nσ
(n+1)σ
C1
nσ
(n+1)σ
C1
2
φ
φ(x)P(n+1)σ−s
f (x)S J,(1) (ds, dx)
Z
µ(dy)
D
e
−λ1 s
D
Z
C2 (σ)kf k2∞ hφ, µi
Z
Z
2
φ
dxpD (s, y, x) P(n+1)σ−s
f (x)
dxp (s, y, x)φ(x)
D
e−λ1 s ds
Z
nσ
≤
C2 (σ)kf k2∞ hφ, µi
"Z
Z
(n+1)σ
−λ1 s
e
Z
ds
(n+1)σ
+
nσ
=:
e−λ1 s ds
D
Z
Z
D
C2 (σ)kf k2∞ hφ, µi(I1 (n)
φ
P(n+1)σ−s
f
2
Z
(x)
eλ1 s
r2 nφ (x, dr)
0
r2 nφ (x, dr)
0
Z
e
φ(x)dx 0
1
r n (x, dr)
2 φ
∞
#
eλ1 s
e
φ(x)dx
r2 nφ (x, dr)
eλ1 s
e
dxφ(x)
D
nσ
Z
−1
φ
φ(y)µ(dy)
D
(n+1)σ
eλ1 s
0
Z
ds
nσ
≤
Z
Z
e−2λ1 s ds
Z
=
e−λ1 s
r2 nφ (x, dr)
1
+ I2 (n)),
(2.42)
18
where C1 and C2 (σ) are positive constants independent of n. (1.3) implies that k
∞. Thus
∞
X
I1 (n) < ∞.
R1
0
r2 nφ (·, dr)k∞ <
(2.43)
n=1
Using Fubini’s theorem, we have
∞
X
∞ Z
X
I2 (n) =
(n+1)σ
e−λ1 s ds
Z
∞
e−λ1 s ds
Zσ
Z
e
≤
φ(x)dx
Z
D
≤
Z
D
∞
1
Z
2 φ
r n (x, dr)
λ−1
1
Z
D
∞
λ−1
1 ln r
e−λ1 s ds
∞
rnφ (x, dr)
e
φ(x)dx
D
r2 nφ (x, dr)
e
φ(x)dx
Z
r2 nφ (x, dr)
eλ1 s
1
λ−1
1
eλ1 s
1
Z
=
=
Z
e
φ(x)dx
D
n=1 nσ
n=1
Z
1
Z
e
φ(x)dx 1
∞
rn (x, dr)
φ
< ∞.
Combining (2.41), (2.42), (2.43) and (2.44), we get that, for any ε > 0,
!
∞
X
σ
Hn,t (f ) > ε < ∞.
Pµ
sup
n=1
(2.44)
∞
(2.45)
t∈[nσ,(n+1)σ]
Thus by the Borel-Cantelli lemma we have, for any σ > 0,
lim
sup
n→∞ t∈[nσ,(n+1)σ]
σ
Hn,t
(f ) = 0,
Pµ −a.s.
(2.46)
Pµ −a.s.,
(2.47)
Therefore (2.38) is valid.
Similarly, we can prove that
lim
sup
n→∞ t∈[nσ,(n+1)σ]
σ
|Cn,t
(f )| = 0,
and then we get (2.40). The details are omitted here.
Using an argument similar to (2.24) we can see that almost surely there exists N0 ∈ N such
that when n > N0 ,
Z t
Z
Z ∞
σ
φ
−λ1 s
Ln,t (f ) =
e
ds
P(n+1)σ−s f (x)Xs (dx)
rnφ (x, dr)
nσ
Z
eλ1 s
D
(n+1)σ
≤ kf k∞
nσ
Z
≤ σkf k∞ 1
Z
∞
e−λ1 s Xs (dx)φ(x)
rnφ (x, dr)
λ
s
e 1
∞
rnφ (x, dr)
sup
Ms (φ).
(2.48)
∞ s∈[nσ,(n+1)σ]
Therefore (2.39) holds. The proof of Theorem 2.1 is now complete.
The following result strengthens Theorem 2.1 in the sense that the exceptional does not depend
on f and µ.
19
Theorem 2.8 Suppose that Assumption 1 holds, λ1 > 0 and that X is a (ξ D , ψ(λ) − βλ)−superdiffusion. Then there exists Ω0 ⊂ Ω of probability one (that is, Pµ (Ω0 ) = 1 for every µ ∈ MF (D)0 )
such that, for every ω ∈ Ω0 and for every bounded Borel measurable function f on Rd with compact
support whose set of discontinuous points has zero Lebesgue measure, we have
Z
−λ1 t
e
lim e
hf, Xt i = M∞ (φ)
φ(y)f
(y)dy.
(2.49)
t→∞
D
Proof: Note that there exists a countable base U of open sets {Uk , k ≥ 1} that is closed under
finite unions. Define
Z
−λ1 t
e
φ(y)φ(y)dy for every k ≥ 1 .
Ω0 := ω ∈ Ω : lim e
hIUk φ, Xt i = M∞ (φ)
t→∞
Uk
By Theorem 2.1, for any µ ∈ MF (D)0 , Pµ (Ω0 ) = 1. For any open set U , there exists a sequence of
S
increasing open sets {Unk ; k ≥ 1} in U so that ∞
k Unk = U . Then for every ω ∈ Ω0 ,
Z
−λ1 t
−λ1 t
e
φ(y)φ(y)dy
for every k ≥ 1.
lim inf e
hIU φ, Xt i ≥ lim e
hIUnk φ, Xt i ≥ M∞ (φ)
t→∞
t→∞
U nk
Letting k → ∞ yields
lim inf e
−λ1 t
t→∞
Z
hIU φ, Xt i ≥ M∞ (φ)
e
φ(y)φ(y)dy,
Pµ −a.s. for any µ ∈ MF (D)0 .
(2.50)
U
We now consider (2.49) on {M∞ (φ) > 0}. For each ω ∈ Ω0 ∩ {M∞ (φ) > 0} and t ≥ 0, we define
two probability measures νt and ν on D respectively by
Z
eλ1 t hIA φ, Xt i(ω)
e
νt (A)(ω) =
, and ν(A) =
φ(y)φ(y)dy,
A ∈ B(D).
M∞ (φ)(ω)
A
Note that the measure νt is well-defined for every t ≥ 0. (2.50) tells us that νt converges weakly
to ν as t → ∞. Since φ is strictly positive and continuous on D, for every function f on D with
compact support on E whose discontinuity set has zero Lebesgue-measure (equivalently zero νmeasure), g := f /φ is a bounded function with compact support with the same set of discontinuity.
We thus have
Z
Z
g(x)νt (dx) =
g(x)ν(dx),
D
D
which is equivalent to say
−λ1 t
lim e
t→∞
Z
hf, Xt i = M∞ (φ)
e
φ(y)f
(y)dy,
for every ω ∈ Ω0 ∩ {M∞ (φ) > 0}.
D
Since
e−λ1 t |hf, Xt i| ≤ e−λ1 t h|f | , Xt i = e−λ1 t h|gφ| , Xt i ≤ kgk∞ M∞ (φ).
(2.51) holds automatically on {M∞ (φ) = 0}. This completes the proof of the theorem.
20
(2.51)
Lemma 2.9 For any f ∈ Bb (D), µ ∈ MF (D)0 ,
−λ1 t
lim e
t→∞
Z
Pµ hf, Xt i = hφ, µi
e
f (y)φ(y)dy.
(2.52)
D
Proof: It follows from (2.5) that
−λ1 t
e
Z
µ(dx)e−λ1 t PtD f (x)
ZD
Z
=
µ(dx)
e−λ1 t pD (t, x, y)f (y)dy
D
D
Z
Z
f (y)
=
µ(dx)φ(x)
pφ (t, x, y)
dy.
φ(y)
D
D
Pµ hf, Xt i =
Using (1.2) and the dominated convergence theorem, we get (2.52).
Proof of Theorem 1.3 Theorem 1.3 is simply a combination of Theorem 2.8 and Lemma
2.9.
2
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Rong-Li Liu: Department of Mathematics, Nanjing University, Nanjing, 210093, P. R. China; and
LAREMA, Département de Mathématiques, Université d’Angers, 2, Bd Lavoisier - 49045, Angers
Cedex 01, E-mail: [email protected]
Yan-Xia Ren: LMAM School of Mathematical Sciences, Center for Statistical Science, Peking
University, Beijing, 100871, P. R. China, E-mail: [email protected]
Renming Song: Department of Mathematics, University of Illinois, Urbana, IL 61801 U.S.A.,
E-mail: [email protected]
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