Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 801812, 14 pages
doi:10.1155/2012/801812
Research Article
The Asymptotic Behavior of
a Stochastic Predator-Prey System with
Holling II Functional Response
Zhenwen Liu,1, 2 Ningzhong Shi,1 Daqing Jiang,1
and Chunyan Ji1, 3
1
School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China
School of Science, Changchun University of Science and Technology, Changchun, Jilin 130022, China
3
Department of Mathematics, Changshu Institute of Technology, Changshu, Jiangsu 215500, China
2
Correspondence should be addressed to Daqing Jiang, daqingjiang2010@hotmail.com
Received 21 October 2012; Accepted 14 December 2012
Academic Editor: Ivanka Stamova
Copyright q 2012 Zhenwen Liu et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We discuss a stochastic predator-prey system with Holling II functional response. First, we show
that this system has a unique positive solution as this is essential in any population dynamics
model. Then, we deduce the conditions that there is a stationary distribution of the system, which
implies that the system is permanent. At last, we give the conditions for the system that is going
to be extinct.
1. Introduction
One of the most popular predator-prey model is the one with Michaelis-Menten type or Holling Type II functional response 1, 2:
αyt
,
1 βxt
kαxt
ẏt yt −e ,
1 βxt
ẋt xt a − bxt −
1.1
where xt and yt are the population densities of prey and predator at time t, respectively.
The constants a, b/a, α, β, e, and k are positive constants that stand for prey intrinsic growth
rate, carrying capacity, the maximum ingestion rate, half-saturation constant, predator death
rate, and the conversion factor, respectively. This model exhibits the well-known but highly
2
Abstract and Applied Analysis
controversial “paradox of enrichment” observed by Hairston et al. 3 and by Rosenzweig 4
which is rarely reported in nature. It is very important to study the existence and asymptotical
stability of equilibria and limit cycle for autonomous predator-prey systems with Holling II
functional response. If kaαβ > aeβ2 kbα beβ, then system 1.1 has a unique limit cycle
which is stable. If akα > aeβ be, then system 1.1 has a unique positive equilibrium:
e
,
x kα − eβ
∗
kα akα − aeβ − be
y ,
2
kα − eβ
∗
1.2
which is a stable node or focus see 5.
However, countless organisms live in seasonally or diurnally forced environments.
Hence, authors considered models with periodic ecological parameters or perturbations. For
example, Liu and Chen 6 introduced periodic constant impulsive immigration of predator into system 1.1 and gave conditions for the system to be extinct and permanence,
respectively. Zhang and Chen 7 studied a Holling II functional response food chain model
with impulsive perturbations. Zhang et al. 8 further considered system 1.1 with periodic
constant impulsive immigration of predator and periodic variation in the intrinsic growth
rate of the prey.
On the other hand, the white noise is always present, and we cannot omit the influence
of the white noise to the system. May 9 pointed out that due to continuous fluctuation in
the environment, the birth rates, death rates, carrying capacity, competition coefficients, and
all other parameters involved with the model exhibit random fluctuation to a great lesser
extent, and as a result the equilibrium population distribution never attains a steady value,
but fluctuates randomly around some average value. Many authors studied the effect of the
stochastic perturbation to the predator-prey system with different functional responses, such
as 10–14. Therefore, in this paper, we also introduce stochastic perturbation system 1.1
and obtain the following stochastic system:
αyt
dt σ1 xtdB1 t,
1 βxt
kαxt
dt σ2 ytdB2 t,
dyt yt −e 1 βxt
dxt xt a − bxt −
1.3
where B1 t and B2 t are mutually independent Brownian motion with B1 0 B2 0 0,
and σ12 , σ22 are intensities of the white noise.
The aim of this paper is to discuss the long time behavior of system 1.3. As the
deterministic population models, we are also interested in the permanence and extinction
of the system. The global stability of the positive equilibrium means that the system is
permanence. But, for the stochastic system, there is no positive equilibrium. Hence, it is
impossible that the solution of system 1.3 will tend to a fixed point. In this paper, we
show that there is a stationary distribution of system 1.3 mainly according to the theory
of Has’meminskii 15, if the white noise is small. While if the white noise is large, based
on the techniques developed in 16, 17, we prove that the predator population will die out
a.s. and the prey population will either extinct or its distribution converges to a probability
measure. It does not happen that both the prey population and the predator population in
system 1.3 will die out, which is brought by large white noise, such as weather, epidemic
Abstract and Applied Analysis
3
disease. From this point, we say the stochastic model is more realistic than the deterministic
model.
The rest of this paper is organized as follows. In Section 2, we show that there is a
unique nonnegative solution of system 1.3. In Section 3, we show that there is a stationary
distribution under small white noise. While in Section 4, we consider the situation when the
white noise is large. We prove that the system will be extinct. Finally, we give an appendix
containing the stationary distribution theory used in Section 3.
2. Existence and Uniqueness of the Nonnegative Solution
To investigate the dynamical behavior, the first concern is the global existence of the solutions.
Hence in this section we show that the solution of system 1.3 is global and nonnegative. It is
not difficult to check the uniqueness and global existence of solutions if the coefficients of the
equation satisfy the linear growth condition and local Lipschitz condition cf. 18. However,
the coefficients of system 1.3 do not satisfy the linear growth condition, but locally Lipschitz
continuous, so the solution of system 1.3 may explode at a finite time. In this section, by
changing variables, we first show that system 1.3 has a local solution, then show that this
solution is global.
Theorem 2.1. For any initial value x0, y0 ∈ R2 , there is a unique solution xt, yt of
system 1.3 on t ≥ 0, and the solution will remain in R2 with probability 1.
Proof. First, consider the following system, by changing variables, xt eut , yt evt ,
σ12
αevt
ut
dt σ1 dB1 t,
− be −
dut a −
2
1 βeut
σ22
kαeut
dt σ2 dB2 t.
dvt − e −
2
1 βeut
2.1
It is clear that the coefficients of system 2.1 are locally Lipschitz continuous for the given
initial value log x0, log y0 ∈ R2 there is a unique local solution ut, vt on t ∈ 0, τe ,
where τe is the explosion time see 18. Hence, by Itô formula, we know eut , evt , t ∈
0, τe is a unique positive local solution of system 1.3. To show that this solution is global,
we need to show that τe ∞ a.s. Let m0 ≥ 1 be sufficiently large so that x0, y0 all lie
within the interval 1/m0 , m0 . For each integer m ≥ m0 , define the stopping time:
1
or max xt, yt ≥ m ,
τm inf t ∈ 0, τe : min xt, yt ≤
m
2.2
Where, throughout this paper, we set inf ∅ ∞ as usual ∅ denotes the empty set. Clearly,
τm is increasing as m → ∞. Set τ∞ limm → ∞ τm , whence τ∞ ≤ τe a.s. If we can show that
τ∞ ∞ a.s., then τe ∞ and xt, yt ∈ R2 a.s. for all t ≥ 0. In other words, to complete the
proof all we need to show is that τ∞ ∞ a.s. If this statement is false, then there is a pair of
constants T > 0 and ∈ 0, 1 such that
P {τ∞ ≤ T } > .
2.3
4
Abstract and Applied Analysis
Hence there is an integer m1 ≥ m0 such that
P {τm ≤ T } ≥ ∀m ≥ m1 .
2.4
Define a C2 -function V : R2 → R by
V x, y x
1
x − c − c log
y − 1 − log y ,
c
k
2.5
where c is a positive constant to be determined later. The nonnegativity of this function can be
seen from u − 1 − log u ≥ 0, for all u > 0. Using Itô’s formula, we get
dV : LV dt σ1 x − cdB1 t σ2 y − 1 dB2 t,
k
2.6
where
LV x − c a − bx −
αy
1 βx
cσ12 1 σ2
kαy
2
y − 1 −e 2
k
1 βx
2k
cσ12 e σ22
αcy
e
a bcx − y − bx2 2
k 2k
k
1 βx
cσ 2 e σ 2
e
− αc y.
≤ −ac 1 2 a bcx − bx2 −
2
k 2k
k
−ac 2.7
Choose c e/αk such that e/k − αc 0, then
LV ≤ −ac cσ12 e σ22
a bcx − bx2 ≤ K,
2
k 2k
2.8
where K is a positive constant. Therefore
τm ∧T
dV xt, yt ≤
0
τm ∧T
Kdt
0
τm ∧T
0
σ2 σ1 xs − cdB1 s ys − 1 dB2 s,
k
2.9
which implies that,
E V xτm ∧ T , yτm ∧ T ≤ V x0, y0 E
τm ∧T
0
Kdt ≤ V x0, y0 KT.
2.10
Abstract and Applied Analysis
5
Set Ωm {τm ≤ T } for m ≥ m1 , then by 2.4, we know that P Ωm ≥ . Note that for every
ω ∈ Ωm , there is at least one of xτm , ω and yτm , ω equals either m or 1/m, then
V xτm , yτm ≥
1
− c c logcm
m
1 1
1
∧
− 1 log m .
m − 1 − log m ∧
k
k m
m − c − c log
m
c
∧
2.11
It then follows from 2.4 and 2.10 that
V x0, y0 KT ≥ E 1Ωm ω V xτm , yτm m
1
1
≥ m − c − c log
− c c logcm ∧
∧
m − 1 − log m
c
m
k
1 1
− 1 log m ,
∧
k m
2.12
where 1Ωm ω is the indicator function of Ωm . Letting m → ∞ leads to the contradiction that
∞ > V x0, y0 KT ∞. So we must therefore have τ∞ ∞ a.s.
3. Permanence
There is no equilibrium of system 1.3. Hence we cannot show the permanence of the system
by proving the stability of the positive equilibrium as the deterministic system. In this section
we show that there is a stationary distribution of system 1.3.
Remark 3.1. Theorem 2.1 shows that there exists a unique positive solution xt, yt of
system 1.3 with any initial value x0, y0 ∈ R2 . From the proof of Theorem 2.1,
we obtain that LV ≤ K. Define V V K, then LV ≤ V , and it is clear that VR infx,y∈R2 \Dk V x, y → ∞ as k → ∞, where Dk 1/k, k × 1/k, k. Hence by Remark 2 of
Theorem 4.1 of Has’meminskii, 1980, page 86 in 15, we obtain that the solution xt, yt
is a homogeneous Markov process in R2 .
Theorem 3.2. If aeβ be < akα < aeβ bkα/β and σ1 > 0, σ2 > 0 such that σ22 < kαx∗ /1 βx∗ and
1 βx∗ l2 y∗ y∗ σ22
1
∗
∗ 2
l2 x x σ1 2
2
k
k
σ22 ∗ 2
αβy∗
αx∗
1
∗ 2 l2
,
b−
< min
y
− 2
x ,
2
1 βx∗
2 k 1 βx∗
k
3.1
where x∗ , y∗ is the positive equilibrium of system 1.1 and l2 is defined as in the proof. Then system
1.3 has a stationary ergodic solution.
6
Abstract and Applied Analysis
Proof. Since akα > aeβ be, then there is a positive equilibrium x∗ , y∗ of system 1.1, and
a bx∗ αy∗
,
1 βx∗
e
kαx∗
.
1 βx∗
3.2
Let
y
x
V1 x, y x − x∗ − x∗ log ∗ l1 y − y∗ − y∗ log ∗ ,
x
y
3.3
where l1 is a positive constant to be determined later. Let L be the generating operator of
system 1.3. Then
x∗ σ12
l1 y∗ σ22
kαx
∗
l1 y − y
−e 2
1 βx
2
x∗ σ12
αβy∗
α ∗
x − x∗ −bx − x∗ −
−
x
y − y∗ x
1 βx
2
1 βx∗ 1 βx
αy
LV1 x − x a − bx −
1 βx
∗
l1 y∗ σ22
kαx − x∗ l1 y − y∗ 2
1 βx∗ 1 βx
x∗ σ12 l1 y∗ σ22
αβy∗
l1 k
α
∗ 2
∗
∗
1
−
.
≤ − b−
−
−
x
−
x
y
−
y
x
x
1 βx∗
1 βx
1 βx∗
2
2
3.4
Choose l1 1 βx∗ /k such that 1 − l1 k/1 βx∗ 0 and yields
x∗ σ12
1 βx∗ y∗ σ22
αβy∗
∗ 2
LV1 ≤ − b −
.
−
x
x
1 βx∗
2
2k
3.5
2
1
1
∗
∗
V2 x, y .
y−y
x − x 2
k
3.6
Let
Note that
e
1
σ2
∗
∗
2
d x − x ax − bx − y dt σ1 xdB1 t ydB2 t
y−y
k
k
k
y∗ x − x∗ − x∗ y − y∗
∗ 2
−bxx − x α
dt
1 βx∗
σ1 xdB1 t σ2
ydB2 t,
k
3.7
Abstract and Applied Analysis
7
then
1
LV2 x − x y − y∗
k
∗
σ12 2 σ22 2
y∗ x − x∗ − x∗ y − y∗
−bxx − x α
x 2y
1 βx∗
2
2k
∗ 2
αy∗
by∗
b
αx∗
∗ 2
y
−
y
x − x∗ 2 − x − x∗ 2 y − ∗
1 βx
k
k
k 1 βx∗
σ12 2 σ22 2
αy∗
bx∗
αx∗
∗
∗
−
−
y
−
y
x 2y
−
x
x
1 βx∗
k
2
2k
k 1 βx∗
− bxx − x∗ 2 ≤
σ22 2
αy∗
by∗
αx∗
2
∗ 2
σ1 x − x −
y − y∗
− 2
∗
∗
1 βx
k
k
k 1 βx
αy∗
bx∗
αx∗
−
−
1 βx∗
k
k 1 βx∗
σ 2 2
x − x∗ y − y∗ σ12 x∗ 2 22 y∗ ,
k
3.8
where L is also the generating operator of system 1.3. Note that
αy∗
bx∗
αx∗
−
−
1 βx∗
k
k 1 βx∗
x − x∗ y − y∗
2
αy∗ / k 1 βx∗ − αx∗ / 1 βx∗ − bx∗ /k
≤
x − x∗ 2
2 αx∗ / k 1 βx∗ − σ22 /k2
σ22 2
αx∗
1
y − y∗
− 2
∗
2 k 1 βx
k
1
: δx − x 2
∗ 2
σ22
αx∗
−
k2
k 1 βx∗
3.9
2
y − y∗ ,
then
LV2 ≤
σ22 2
αy∗
by∗
αx∗
1
2
∗ 2
σ1 δ x − x −
y − y∗
− 2
∗
∗
1 βx
k
2 k 1 βx
k
3.10
σ 2 2
σ12 x∗ 2 22 y∗ .
k
Now define
V x, y V1 x, y l2 V2 x, y ,
3.11
8
Abstract and Applied Analysis
where l2 is a positive constant to be determined later. Then
αβy∗
by∗
αy∗
2
LV ≤ − b −
σ
−
l
δ
x − x∗ 2
2
1
1 βx∗
1 βx∗
k
3.12
σ22 1 βx∗ l2 y∗ y∗ σ22
αx∗
l2
1
∗ 2
∗
∗ 2
l2 x x σ1 .
−
y−y
− 2
2 k 1 βx∗
2
2
k
k
k
Choose l2 > 0 such that b − αβy∗ /1 βx∗ − l2 αy∗ /1 βx∗ by∗ /k σ12 δ 1/2b −
αβy∗ /1 βx∗ , then it follows from 3.12 that
σ22 2
αβy∗
αx∗
l2
1
∗ 2
b−
LV ≤ −
y − y∗
− 2
x − x −
2
1 βx∗
2 k 1 βx∗
k
1 βx∗ l2 y∗ y∗ σ22
1
l2 x∗ x∗ σ12 .
2
2
k
k
3.13
Note that
1 βx∗ l2 y∗ y∗ σ22
1
∗
∗ 2
l2 x x σ1 2
2
k
k
σ22 ∗ 2
αβy∗
αx∗
1
∗ 2 l2
,
b−
< min
y
− 2
x ,
2
1 βx∗
2 k 1 βx∗
k
3.14
then the ellipsoid
σ22 αβy∗
αx∗
1
l2
∗ 2
∗ 2
−
−
y
−
y
b−
x − x −
∗
2
2
1 βx
2 k 1 βx∗
k
1 βx∗ l2 y∗ y∗ σ22
1
∗
∗ 2
l2 x x σ1 0
2
2
k
k
3.15
lies entirely in R2 . We can take U to be a neighborhood of the ellipsoid with U ⊆ El R2 ,
so that for x, y ∈ U \ El , LV ≤ −C C is a positive constant, which implies condition
B.2 in Lemma A.1 is satisfied. Hence the solution xt, yt is recurrent in the domain U,
which together with Lemma A.3 and Remark 3.1 implies that xt, yt is recurrent in any
bounded domain D ⊂ R2 . Besides, for all D, there is an M min{σ12 x2 , σ22 y2 , x, y ∈ D} > 0
such that
2
λij ξi ξj σ12 x2 ξ12 σ22 y2 ξ22 ≥ Mξ2 all x ∈ D, ξ ∈ R2 ,
3.16
i,j1
which implies that condition B.1 is also satisfied. Therefore, system 1.3 has a stable a
stationary distribution μ· and it is ergodic.
Abstract and Applied Analysis
9
Note that
αy
1 dt pσ1 xp dB1 t p p − 1 σ12 xp dt
1 βx
2
p
≤ pxp a σ12 − bx dt pσ1 xp dB1 t,
2
dxp pxp a − bx −
3.17
then
pσ12
dExp ≤p a
Exp − bE xp1
dt
2
pσ12
≤p a
Exp − bExp p1/p .
2
3.18
Hence by comparison theorem, we get
lim supEx t ≤
p
t→∞
a pσ12 /2
b
p
3.19
,
by which together with the continuity of Exp t, we have that there exists a positive constant
K Kp such that
Exp t ≤ K p .
3.20
By Doob’s martingale inequality, together with the 3.20, for δ > 0, we have
xt
>δ
P ω : sup
n−1δ≤t≤nδ t
K p
Exp nδ
≤
≤ p p1 ,
nδp δ
n δ
p > 1.
3.21
In view of the well-known Borel-Cantelli lemma, we see that for almost all ω ∈ Ω,
xt
≤δ
n−1δ≤t≤nδ t
sup
3.22
holds for all but finitely many n. Hence there exists an n0 ω, for all ω ∈ Ω excluding a P -null
set, for which 3.22 holds whenever n ≥ n0 . Consequently, letting δ → 0, we have, for almost
all ω,
xt
0.
t→∞ t
lim
3.23
10
Abstract and Applied Analysis
By the ergodic property, for any given constant m > 0, we have
1
t→∞ t
t
lim
xp s ∧ mds p
z1 ∧ m μdz1 , dz2 ,
R2
0
a.s.
3.24
On the other hand, by dominated convergence theorem and 3.20, we get
1
E lim
t→∞ t
t
1
t→∞ t
x s ∧ mds lim
p
0
t
Exp s ∧ mds ≤ K p ,
3.25
0
which together with 3.24 implies
R2
p
z1 ∧ m μdz1 , dz2 ≤ K p .
3.26
p
z1 μdz1 , dz2 ≤ K p .
3.27
Letting m → ∞, we get
R2
p
That is to say, the function f1 z z1 is integrable with respect to the measure μ. Therefore,
by ergodicity property again, we get
1
t→∞ t
t
lim
xp sds p
R2
0
z1 μdz1 , dz2 ,
3.28
a.s.
Besides,
xt − x0 a
t
t
Let M1 t t
0
t
b
xsds −
t
0
t
α
x sds −
t
0
2
t
0
xsys
σ1
ds 1 βxs
t
t
xsdB1 s.
3.29
0
xsdB1 s which is a martingale with M1 0 0 and
M1 , M1 t
1
lim
lim sup
t
→
∞
t
t
t→∞
t
0
x12 sds
R2
z21 μdz1 , dz2 < ∞,
3.30
then by the strong law of large numbers, we get
M1 t
1
lim
lim
t→∞
t→∞ t
t
t
0
x1 sdB1 s 0,
3.31
Abstract and Applied Analysis
11
which together with 3.23 and 3.28 implies 3.29 that
1
t→∞ t
t
lim
0
xsys
a
ds 1 βxs
α
R2
z1 μdz1 , dz2 −
b
α
R2
z21 μdz1 , dz2 .
3.32
Hence from these arguments, we get the following result.
Theorem 3.3. Assume the same conditions as in Theorem 3.2. Then one has
1
t→∞ t
t
lim
1
lim
t→∞ t
t
0
xp sds 0
xsys
a
ds 1 βxs
α
R2
p
z1 μdz1 , dz2 ,
b
z1 μdz1 , dz2 −
α
R2
a.s.,
3.33
R2
z21 μdz1 , dz2 .
4. Extinction
In this section, we show the situation when system 1.3 will be extinct.
Case 1. a < σ12 /2.
Obviously,
dx ≤ xa − bxdt σ1 xdB1 t,
4.1
then when a < σ12 /2,
lim xt 0,
t→∞
a.s.
4.2
That is to say, for all 0 < 1 < e σ22 /2, there exist T1 T1 ω and a set Ω1 such that P Ω1 >
1 − 1 and kαxt ≤ 1 for t ≥ T1 and ω ∈ Ω1 . Then
−eytdt σ2 ytdB2 t ≤ dyt ≤ yt−e 1 dt σ2 ytdB2 t,
4.3
and so
lim yt 0,
t→∞
a.s.
4.4
Case 2. a > σ12 /2, e σ22 /2 > kα/β.
Note that
kα
dyt ≤ yt −e dt σ2 ytdB2 t,
β
4.5
12
Abstract and Applied Analysis
then if e σ22 /2 > kα/β, we have
lim yt 0,
t→∞
4.6
a.s.
In this situation, for all 0 < 2 < a − σ12 /2, there exist T2 T2 ω and a set Ω2 such that
P Ω2 > 1 − 2 and αyt ≤ 2 for t ≥ T2 and ω ∈ Ω2 . Then
xta − bxt − 2 dt σ1 xtdB1 t ≤ dxt ≤ xta − bxtdt σ1 xtdB1 t,
σ12
σ12
− bxt − 2 dt σ1 dB1 t ≤ dxt ≤ a −
− bxt dt σ1 dB1 t.
a−
2
2
4.7
4.8
Consider the following equation:
dΦt σ12
Φt
dt σ1 dB1 t.
− be
a−
2
4.9
If a > σ12 /2, 4.9 has the density g∗ ζ such that
1 2 σ g ζ 2 1 ∗
σ12
ζ
− b11 e g∗ ζ.
a1 −
2
4.10
Therefore from 4.8 and the arbitrary of 2 , we get that the distribution of log xt converges
weakly to the probability measure with density g∗ . Thus, from 4.10, we obtain that the
distribution of xt converges weakly to the probability measure with density f ∗ ζ 2
2
2
2
2
C0 ζ2a−σ1 /2/σ1 −1 e−2bζ/σ1 , where C0 2b/σ12 2a−σ1 /2/σ1 /Γ2a − σ12 /2/σ12 . Besides, from the
ergodic theorem and 4.10, it follows that
1
lim
t→∞ t
t
xsds ∞
0
e g∗ ζdζ ζ
−∞
∞
−∞
a − σ12 /2
a − σ12 /2
g∗ ζdζ b
b
a.s.
4.11
Therefore, by the above arguments, we obtain the following.
Theorem 4.1. Let xt, yt be the solution of system 1.3 with the initial value x0, y0 ∈ R2 .
Then,
i if a < σ12 /2, then
lim xt 0,
t→∞
lim yt 0,
t→∞
a.s.,
4.12
Abstract and Applied Analysis
13
ii if a > σ12 /2, e σ22 /2 > kα/β, then the distribution of xt converges weakly to
2
2
2
the probability measure with density f ∗ ζ C0 ζ2a−σ1 /2/σ1 −1 e−2bζ/σ1 , where C0 2
2
2b/σ12 2a−σ1 /2/σ1 /Γ2a − σ12 /2/σ12 , and
1
t→∞ t
t
lim
0
xsds a − σ12 /2
,
b
lim yt 0,
t→∞
a.s.
4.13
Appendix
For the completeness of the paper, in this section, we list some theories about stationary distribution see 15.
Let Xt be a homogeneous Markov process in El El denotes euclidean l-space
described by
dXt bXdt k
gr XdBr t.
A.1
r1
The diffusion matrix is Ax aij x, aij x k
r1
j
gri xgr x.
Assumption B. There exists a bounded domain U ⊂ El with regular boundary Γ, having the
following properties.
B.1 In the domain U and some neighbourhood thereof, the smallest eigenvalue of
the diffusion matrix Ax is bounded away from zero.
B.2 If x ∈ El \ U, the mean time τ at which a path issuing from x reaches the set
U is finite, and supx∈K Ex τ < ∞ for every compact subset K ⊂ El .
Lemma A.1 see 15. If (B) holds, then the Markov process Xt has a stationary distribution μ·.
T
Let
f· be a function integrable with respect to the measure μ. Then Px {limT → ∞ 1/T 0 fXtdt fxμdx} 1 for all x ∈ El .
El
Remark A.2. The proof is given in 15. Exactly, the existence of stationary distribution with
density is referred to Theorem 4.1, Page 119, and Lemma 9.4, Page 138, in 4. The weak
convergence and the ergodicity is obtained in Theorem 5.1, Page 121, and Theorem 7.1, Page
130, in 4.
To validate B.1, it suffices to prove that F is uniformly elliptical in any bounded
domain D, where Fu bx·ux 1/2trAxuxx ; that is, there is a positive number M such
that ki,j1 aij xξi ξj ≥ M|ξ|2 , x ∈ D, ξ ∈ Rk see Chapter 3, Page 103 of 19 and Rayleigh’s
principle in 20, Chapter 6, Page 349. To verify B.2, it is sufficient to show that there exists
some neighborhood U and a nonnegative C2 -function such that and for any El \ U, LV is
negative for details refer to 21, Page 1163.
Lemma A.3. Let Xt be a regular temporally homogeneous Markov process in El . If Xt is recurrent
relative to some bounded domain U, then it is recurrent relative to any nonempty domain in El .
14
Abstract and Applied Analysis
Acknowledgments
The work was supported by the Ministry of Education of China no. 109051, the Ph.D. Programs Foundation of Ministry of China no. 200918, NSFC of China no. 10971021, and
the Graduate Innovative Research Project of NENU no. 09SSXT117, Youth Fund of Jiangsu
Province no. BK2012208, and the Tian Yuan Special Funds of the National Natural Science
Foundation of China no. 11226205.
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