R E S E A R C H
Open Access
A note on stability of SPDEs driven by
α
-stable noises
Jiying Wang
*and Yulei Rao
*Correspondence:
[email protected] Business School, Central South University, Changsha, Hunan 410083, China
Abstract
In this paper, by the Minkovski inequality and the semigroup method we discuss the stability of mild solutions for a class of SPDEs driven by
α-stable noise, and the
methods are also generalized to deal with the stability of SPDEs driven by subordinated cylindrical Brownian motion and fractional Brownian motion, respectively.MSC: 60H15; 60G15
Keywords: stochastic partial differential equation;
α-stable process; exponential
stability; subordinated cylindrical Brownian motion; fractional Brownian motion1 Introduction
The stability of stochastic partial differential equations (SPDEs) driven by Brownian mo-tions or Lévy processes has been well established; see,e.g., Bao and Yuan [], Baoet al.
[, ], Chow [], Liu [] and Yuan and Bao [], to name a few, where the noise pro-cesses are assumed to be square integrable. However, such restriction clearly rules out the interestingα-stable processes since Wiener noise and Poisson-jump noise have arbi-trary finite moments, whileα-stable noise only has finitepth moment forp∈(,α) with
α< . Recently, stochastic equations driven byα-stable processes, which have plenty of applications in physics due to the fact that theα-stable noise exhibits the heavy tailed phenomenon,e.g., Solomonet al.[], receives great attention. For example, Priola and Zabczyk [] gave a proper starting point on the investigation of structural properties of SPDEs driven by an additive cylindrical stable noise, Donget al.[] studied the ergodicity of stochastic Burgers equations driven byα/-subordinated cylindrical Brownian motion withα∈(, ), and Zhang [] established the Bismut-Elworthy-Li derivative formula for stochastic differential equations (SDEs) driven byα-stable noise. For finite-dimensional cases, Wang [] derived a gradient estimate for linear SDEs driven by α-stable noise, and Wang [] established the functional inequalities for Ornstein-Uhlenbeck processes driven byα-stable noise by the sharp estimates of density function for rotationally invari-ant symmetricα-stable Lévy processes. However, there are few papers on the asymptotic behavior of mild solution of SPDEs driven byα-stable processes. In this note, we shall discuss the stability property of mild solutions of a class of SPDEs driven byα-stable pro-cesses to close the gap. Due to the fact thatα-stable noise only has finitepth moment for
p∈(,α) and that the stochastic evolution does not admit a stochastic differential, which leads to the Itô formula being unavailable, then some new tricks need to be put forward to overcome the difficulties brought about byα-stable noise.
This note is organized as follows: in Section we apply the Minkovski inequality and the semigroup method to investigate the exponential stability of mild solutions for a class of SPDEs driven byα-stable processes, and these tricks are extended to cope with the stability of mild solutions of SPDEs driven by subordinated cylindrical Brownian motion and fractional Brownian motion, respectively, in Section .
2 Stability of SPDEs driven by
α
-stable noiseWe firstly give an overview of stable processes. AnR-valued random variableηis said to be stable with stability indexα∈(, ), scale parameterσ∈(,∞), skewness parameter
β ∈[–, ], and location parameterμ∈(–∞,∞) if it has characteristic function of the
form
φη(u) =Eexp(iuη) =exp
–σα|u|α –iβsgn(u)+iμu, u∈R,
where =tan(π α/) forα= and= –(/π)log|u| forα= . We callηis strictlyα -stable wheneverμ= , and if, in addition,β= ,ηis said to be symmetricα-stable. We refer to,e.g., Applebaum [] and Sato [], for more details of stable distributions. Let (H,·,·, · H) be a real separable Hilbert space, andZ(t) a cylindricalα-stable process, defined by
Z(t) :=
∞
k=
βkHZk(t)ek. (.)
Here{ek}k≥is an orthonormal basis ofH,{Zk(t)}k≥are independent,R-valued, normal-ized, symmetricα-stable Lévy processes defined on the stochastic basis ( ,F,{Ft}t≥,P),
and{βk}k≥is a sequence of positive numbers.
In this section, we shall consider an SPDE driven byα-stable process inH
dX(t) =AX(t) +bt,X(t)dt+σ(t) dZ(t), t> , (.)
with initial valueX() =x.
In this section we shall assume the following.
(A) (A,D(A))is a self-adjoint compact operator onHsuch that–Ahas discrete spectrum <λ<λ<· · ·<λk<· · ·,limk→∞λk=∞with corresponding
eigenbasis{ek}k≥ofH. In this case, by [, Theorem ., p.] and [, Theorem ., p.],Agenerates aC-semigroupetA,t≥, such that etA ≤e–λt, where · denotes the usual operator norm.
(A) b: [,∞)×H→Hand there existL> andL> such that
b(t,x) –b(t,y)H≤Lx–yH, t≥,x,y∈H
and
b(t,x)H≤L
γ(t) +xH, t≥,x∈H,
(A) σ: [,∞)→R+is a continuous function such that
Under (A)-(A), equation (.) has a unique mild solution, see,e.g., [, Formula .], that is, there exists a predictableH-valued stochastic processX(t) such that
X(t) = etAx+
Next we recall the following Minkowski inequality, which plays a key role in revealing the stability property of SPDEs driven byα-stable processes.
Lemma . Let(S,μ)and(S,μ)be two measurable spaces and F :S×S→Rbe
We now can state our main results in this section.
Theorem . Let(A)and(A)hold and assume further that
δ:=sup
That is,the solution is exponentially stable in the pth moment with the Lyapunov exponent
By (A), it is readily seen that
I(t)≤e–λtx
H. (.)
Next, applying Lemma . and using (A) and (A) yield
I(t)≤
and (A), a direct calculation shows
I(t) =
Then, by (.) and following an argument similar to that of [, Theorem .], the stochas-tic evolutionI(t) has the estimation
wherecp> is some constant. Substituting (.)-(.) into (.) gives
Thus, by (.) and (.), we arrive at
In view of the Gronwall inequality,
eλtEX(t)p
In what follows, we establish an example to demonstrate that the condition (.) holds in many practical situations.
(A) and (.) are satisfied. There are plenty of examples such that the condition∞k=β
α
Remark . To reveal the stability property of the mild solution of (.), we replace (A) by a little bit strong condition (.), although (.) admits a unique mild solution in finite-time horizon under (A)-(A).
whereI(t) is defined as in (.). Then under appropriate conditions, we can also discuss the stability property of the mild solution of (.).
Remark . Due to the fact that stochastic evolution does not admit stochastic differen-tial, we apply the semigroup method, not the Itô formula, to study the stability property of the mild solution of (.). Comparing with the current technique, we generally adopt the following procedure to discuss the exponential stability of (.): Forp∈(,α), by the inequality (a+b+c)p≤p–(ap+bp+cp),a,b,c> , (A), and the Hölder inequality
Thus, carrying out a similar argument to that of Theorem ., we conclude by the pre-vious method that the Lyapunov exponent is dependent on p–,λ–p
andL. However, by the technique introduced in the argument of Theorem ., we find that the Lyapunov exponent is –(λ–L), which only is dependent onλandL.
3 Extension to SPDEs driven by subordinated cylindrical Brownian motions and fractional Brownian motions
In the last section, we introduce some tricks to study the stability of mild solutions for a class of SPDEs driven byα-stable noise. In this section, following these tricks, we pro-ceed to investigate the stability of SPDEs driven by subordinated cylindrical Brownian motion and fractional Brownian motion, respectively. To begin with, we recall some no-tions. For α ∈(, ), let S(t) be an α/-stable subordinator defined on the probability space ( ,F,{Ft}t≥,P), i.e., an increasingR-valued Lévy process; see,e.g., Applebaum
[, pp.-] and Sato [, Chapter ], with Laplace transform
Ee–uS(t)= e–t|u|α/, u> ,
see, e.g., [, Example .., p.]. Let {Wk(t),t≥}
k∈N be a sequence of
whereS(t) is anα/-stable subordinator independent of{Wk(t)}
k≥,{βk}k≥is a sequence of real numbers, and {ek}≥ is an orthonormal basis ofH. Note thatZk(t) :=Wk(S(t)), which satisfiesZk(t)(ω) :=Wk(S(t)(ω))(ω) for eachω∈ , is a Lévy process by [,
Theo-rem .., p.] and anα-stable process due to [, Proposition .., p.].
In this section, we shall consider an SPDE driven by the subordinated cylindrical Brow-nian motionL(t), defined by (.), onH
dX(t) =AX(t) +bt,X(t)dt+σ(t) dL(t), t> , (.)
with initial valueX=x, whereσ: [,∞)→Ris local integrable continuous function. Let (A) and (A) hold and assume further that
¯
δ(t) :=
∞
k=
βk
t
e–λk(t–s)σ(s) ds, t≥,
is locally bounded. Then (.) admits a unique mild solution in a finite-time interval; see,
e.g., [, Proposition .], that is, there exists a predictableH-valued stochastic process
X(t) such that
X(t) = etAx+
t
e(t–s)Abs,X(s)ds+ t
e(t–s)Aσ(s) dL(s).
One of our main results in this section is as follows.
Theorem . Let(A), (A),and(.)hold and assume further that
δ:=sup
t≥
eλt
∞
k=
βk
t
e–λk(t–s)σ(s) ds
<∞. (.)
Then,forα∈(, ),p∈(,α)and L∈(,λ),there exist c> andγ ∈(,λ–L)such that
EX(t)pH≤ce–pγt.
In other words,the solution is exponentially stable in the pth moment,where the Lyapunov exponent is–γ.
Proof Also by the fact that (E · pH)/p,p∈(,α), is norm, we deduce from (.) that
EX(t)pH/p≤I(t) +I(t) +
EI(t)
p H
/p
, (.)
whereI(t) andI(t) are defined as in (.), andI(t) is defined by
I(t) :=
t
e(t–s)Aσ(s) dZ(s) =
∞
k=
t
βke–λk(t–s)σ(s) dWk
S(s)ek.
By the Gaussian formula, following an argument like that of [, Proposition .] and (.), we get
EI(t)pH/p≤C
∞
k= t
βke–λk(t–s)σ(s) ds /
whereC> is some constant. In view of (.), (.), and (.), it follows from (.) that
eλtEX(t)p H
/p≤
xH+L t
eλsγ(s) +EX(s)p H
/p
ds
+C/peλt
∞
k=
t
βke–λk(t–s)σ(s) ds /
tα–.
Then, for arbitraryε> sufficiently small, one has
e(λ–ε)tEX(t)p H
/p
≤ xH+L t
eλsγ(s) ds+L
t
e(λ–ε)sEX(s) H
p/p
ds
+C/peλt
∞
k= t
βke–λk(t–s)σ(s) ds /
tα–e–εt.
Thanks to (.), (.), and the uniform boundedness oftα–e–εt byα∈(, ) andε> ,
there existsC¯ > such that
e(λ–ε)tEX(t)p H
/p
≤ ¯C+L t
e(λ–ε)sEX(s) H
p/p
ds.
This, together with the Gronwall inequality, leads to
e(λ–ε)tEX(t)p H
/p
≤ ¯CeLt.
Then, byL∈(,λ), we takeε> such thatλ–L–ε> and the desired assertion
follows immediately.
Remark . We remark that Example . still satisfies the condition (.).
In what follows, we further apply the technique adopted in the argument of Theorem . to study the stability for a class of SPDEs driven by fractional Brownian motion. We also need to recall some details of fractional Brownian motion. LetBH(t) be anH-valued
cylin-drical fractional Brownian motion defined by
BH(t) :=
∞
k=
βkH(t)ek, t≥. (.)
Here, {βkH}k≥, H∈(, ), areR-valued independent fractional Brownian motions with
Hurst indexH,i.e., for eachk> ,βkHis anR-valued centered Gaussian process starting from zero, defined on the probability space ( ,F,{Ft}t≥,P), with covariance function
R(t,s) :=EβkH(t)βkH(s)=tH+sH–|t–s|H/, t,s≥,
and{ek}k≥ is an orthonormal basis ofH. {βkH}k≥ are standard Brownian motions for
H=; see,e.g., [, Chapter ]. To the best of our knowledge, there are essentially two dif-ferent approaches to construct stochastic integrals with respect to the fractional Brownian motion,i.e., a path-wise approach and the Malliavin calculus; see,e.g., the monograph []. For anyT > ,H> andσ : [,∞)→L(H,H), the family of linear operators fromH
toH, the stochastic integral with respect to the cylindrical fractional Brownian motion is defined by
t
σ(s) dBH(s) :=
∞
k=
t
σ(s)ekdβkH(s) =
∞
k=
t
KH∗(σek)
(s) dWk(s), t∈[,T]
provided that∞k=TKH∗(σek)(t)Hdt<∞. Here, for eachk≥,βkHadmits the Wiener
integral representation:
βkH(t) =
t
KH(t,s) dWk(s),
whereWkis anR-valued Wiener process, and the kernelKHis given by
KH(t,s) :=cHs/–H
t
s
(u–s)H–/uH–/du, s<t
for some positive constantcH, and, forφ: [,T]→H,
KH∗φ(t) :=
T
t
φ(s)∂KH
∂s (s,t) ds, t∈[,T].
For more details of the stochastic integration with respect to the infinite-dimensional frac-tional Brownian motion, we refer to,e.g., Tindelet al.[].
In the following section, we consider an SPDE driven by the cylindrical fractional Brow-nian motionBH(t), defined by (.), onH
dX(t) =AX(t) +bt,X(t)dt+σ(t) dBH(t), t> (.)
with initial valueX() =x, whereAandbare defined as in (.) andσ: [, )→LHS(H,H),
the space of all Hilbert-Schmidt operators, such that
T
σ(t)HSdt<∞ for anyT> , (.)
where · HSdenotes the usual Hilbert-Schmidt norm,i.e.,THS:= (
∞
i=TeiH)/for
a Hilbert-Schmidt operator. Under (A), (A), and (.), (.) has a unique mild solution; see,e.g., [, Theorem ], that is, there exists a predictableH-valued stochastic process
X(t) such that
X(t) = etAx+
t
e(t–s)AbX(s)ds+ t
e(t–s)Aσ(s) dBH(s).
Note that the solution of (.) is even not a semi-martingale so that the Itô formula is unavailable.
Theorem . Let(A), (A),and(.)hold and assume further that
˜
δ:=sup
t≥
t
esλσ(s) HSds
<∞. (.)
Then,for H>
and L∈(,λ),there exist c> andγ ∈(,λ–L)such that
EX(t)H≤ce–γt.
In other words,the solution is exponentially stable in mean square,where the Lyapunov exponent is–γ.
Proof We here only outline the argument of Theorem . since it is similar to that of Theorem .. Recall from [, Lemma ] that
E t
e(t–s)Aσ(s) dBH(s)
H≤
cH(H– )tH–
t
e(t–s)Aσ(s)HSds
≤cH(H– )tH–
t
e–(t–s)λσ(s)
HSds (.)
for some constant c> , due to the fact that etA is a bounded linear operator andσ is
Hilbert-Schmidt. Following the argument of Theorem ., for anyε> sufficiently small, we have
e(λ–ε)tEX(t)
H
/
≤ xH+L
t
eλsγ(s) ds+L
t
e(λ–ε)sEX(s) H
/ ds
+cH(H– )t(H–)/e–εt
t
esλσ(s) HSds
/ .
Taking (.), (.), and the uniform boundedness of t(H–)/e–εt into account and thus
applying the Gronwall inequality yields the desired assertion.
Remark . By the approach introduced in the previous section, we discuss the mean square exponential stability of (.) under the conditionL∈(,λ). However, Caraballo
et al.[] investigated the same problem underL∈(, –/λ). In other words, we have improved some existing results in certain sense.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The main idea of this paper was proposed by JYW and YLR. JYW prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
Acknowledgements
This research is supported by the Natural Science Foundation of China under Grant no. 71372063.
Received: 24 December 2013 Accepted: 18 March 2014 Published:28 Mar 2014 References
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