- Informació
- Xat IA
Lecture Notes SPDEs
Stochastic Calculus (568191)
Universitat de Barcelona
Comentaris
Altres estudiants també han consultat
Vista prèvia del text
An Introductory Course on
Stochastic Partial Differential Equations
EPFL, February-May, 2016
Marta Sanz-Sol ́e, University of Barcelona
Contents
- 1 Introduction
- 2 Lecture 1: Gaussian Noise
- 2 White noise onRk
- 2 White noise as a vector measure for stochastic integrals
- 2 Examples of processes derived from white noise
- 2 More results concerning the Brownian sheet
- 3 Lecture 2: Gaussian Spatially Correlated Noise.
- 3 Spatially homogeneous Gaussian noise
- 3 Examples of covariances
- 3 Cylindrical Wiener processes
- 4 Lecture 3: Stochastic Integrals (I)
- process 4 Stochastic integral with respect to a standard cylindrical Wiener
- 4 Stochastic integral for spatially homogeneous noise
- 5 Lecture 4: Stochastic Integrals (II)
- 5 Examples of integrands.
- 5 Examples of spectral measures
- 6 Lecture 5: Stochastic Integrals (III)
- 6 Walsh stochastic integral
- 6 Walsh integral and the stochastic integral for correlated noise
- 7 Lecture 6: Linear SPDEs (I)
- 7 Notions of solution
- 7.1 Random field solution
- 7.1 Examples
- 7 A stochastic heat equation with a discontinuous solution
- linear SPDEs 8 Lecture 7: Linear SPDEs (II) and a Brief Introduction to Non-
- 8 Weak solutions to linear SPDEs
- 8 Weak solution to the heat equation on [0,T]×Rk.
- 8 Gaussian free field onRk
- 8 An Introduction to Non-linear SPDEs
- 8 An Example of Theorem of Existence and Uniqueness of Solution
- Solution 9 Lecture 8: Non-linear SPDEs. Existence and Uniqueness of
- constant coefficients. Sample paths properties 10 Lecture 9: Non-linear SPDEs. Parabolic Equations with non-
- 10 SPDEs with non-constant coefficients
- 10 Sample path properties
- 10.2 Factorisation method
- 7 Notions of solution
1 Introduction
In this course we are interested in equations of the type
Lu(y) =b(u(y)) +σ(u(y))W ̇(y), y∈O ⊂R1+k, (1)
with suitable initial conditions and with boundary conditions, if necessary. HereLis a linear differential operator,b,σ:R−→R, andW ̇ is arandom input. In many examples, the regionOwill be a subset ofR+×D, withD⊂Rk, and then we will writey= (t,x) (for time and space, respectively). Typical examples of differential operators include the following:
∂t∂ −∆. Heat (or diffussion) equation.
∂ 2 ∂t 2 +a
∂ ∂t−∆. Wave equation ifa= 0. Damped wave equation ifa 6 = 0.
- ∂
2 ∂t 2 −∆ +b,b >0. Klein-Gordon equation.
- ∂
2 ∂t 2 + ∆
- Beam equation.
∆. Laplace equation.
...
. Why a random term?
To describe dynamics with stochastic influence.
To model complex systems. For example, the motion of a elastic string in a random environment ([35]), branching mechanisms in population dy- namics ([26]).
What is the meaning of Equation(1)?
In general, it is not possible to interpret (1) in theclassical sense, except ifW ̇ issmooth. The interesting cases correspond to irregular noises. Therefore, the meaning to (1) will be inspired by the notion ofdistribution solutions. Consider the case where the regionOis the whole space. The fundamental solution corresponding to the operatorLalways exist (see e. [40, Theorem 10.2]). By definition, this is a distributionGsuch that
LG=δ{ 0 }.
In some examples,Gis a smooth function. This is the case for instance for the heat operator. In others, it is not a function, for example ifLis the wave operator in dimensionk≥3. Consider the non-homogeneous PDE
Lu=f,
wherefis a distribution with compact support. Then, a solution is given by
u=G∗f,
where the symbol∗denotes de convolution operator. Indeed, this is obtained by the following computations:
L(G∗f) =LG∗f=δ{ 0 }∗f=f.
Consider the situation whenO=R+×Rk. The preceding discussion provides us with arguments to state that a “solution” to (1) will be a stochasticprocess satisfying
u(t,x) =
∫t
0
∫
Rk
G(t−s,x−y)σ(u(s,y))dW(s,y), (1)
for any (t,x)∈R+×Rk, a., where for the sake of simplicity we takeb≡0.
A close analysis to (1) raises the following questions:
What is a noise (W(s,y))?
What type of noises will be appropriate to develop a theory of stochastic integration?
What type of stochastic integrals can be used in the right-hand side of (1)?
What about results on existence and uniqueness of solutions to (1)?
What properties on the sample paths of the solutions are expected?
Does a solutionu(t,x) at a fixed point (t,x) possess a density?
These are just a small sample of fundamental questions. Throughout the lectures, some of them will be addressed.
- IfAn↑Aandν(A)<+∞,thenW(An)→W(A) inL 2 (Ω,F,P). Indeed,
E((W(A)−W(An)) 2 ) =ν(A)+ν(An)− 2 ν(A∩An) =ν(A)−ν(An)−→ n→∞
0 ,
sinceν(An)↑ν(A).
White noise as a measure
The mappingA7→W(A) fromBf(A)→L 2 (Ω,F,P) is aσ-additive vector- valued measure. However, for fixedω∈Ω,A7→W(A)(ω) isnota real-valued signed measure. Indeed, consider the casek= 1 and letW be a white noise onRbased on the measureν(dx) = 1R+(x)dx. Observe that for anyt≥0, the functions 1 ]−∞,t]and 1[0,t]are equalν–a. By defining
Bt=W(]−∞,t]) =W([0,t]), t≥ 0 , (2)
we obtain the Brownian motion. Hence, the well-known results on its quadratic variation yield
lim n→∞
∑ 2 n
j=
(
W
([
j− 1 2 n
,
j 2 n
])) 2
= lim n→∞
∑ 2 n
j=
(
Bj 2 n −B[j− 1 2 n
) 2
= 1 a.
This implies,
lim n→∞
∑ 2 n
j=
∣
∣
∣
∣W
([
j− 1 2 n
,
j 2 n
])∣∣
∣
∣= +∞.
Therefore, ifW were a random measure, it could not beσ-finite (in fact, the total variation measure of every compact set would be infinite).
2 White noise as a vector measure for stochastic inte-
grals
Starting fromL 2 (Rk,ν), which is the Ito-Wiener stochastic integral of deter- ministic functions with respect to the white noise. ForA∈Bf(Rk), set W(1A) :=W(A).
Fixc 1 ,...,ck∈RandA 1 ,...,Ak∈ Bf(Rk) pairwise disjoint sets. Leth= ∑k j=1cj 1 Aj. We define
W(h) =W
∑k
j=
cj 1 Aj
:=
∑k
j=
cjW(Aj).
It is easy to check that this definition is legitimate. That is, ifhcan also be written as
∑m ℓ=1dℓ 1 Bℓ, withd 1 ,...,dm∈RandB 1 ,...,Bm∈Bf(R
k) pairwise
disjoint, then
∑k j=1cjW(Aj) =
∑m ℓ=1dℓW(Bℓ).
Since theAjare pairwise disjoint, and by the properties of white noise, we have that
∥ ∥ ∥ ∥ ∥ ∥
W
∑k
j=
cj 1 Aj
∥ ∥ ∥ ∥ ∥ ∥
2
L 2 (Ω)
=E
∑k
j=
cjW(Aj)
2
=
∑k
j=
c 2 jE(W(Aj) 2 ) =
∑k
j=
c 2 jν(Aj)
=
∫
Rk
∑k
j=
cj 1 Aj(x)
2 ν(dx).
Hence, on the set of simple functions of the formh=
∑k j=1cj 1 Aj, the mapping h7→W(h) is an isometry fromL 2 (Rk,ν) toL 2 (Ω). The set of simple functions is dense inL 2 (Rk,ν). Therefore, by classical results of functional analysis this isometry admits a unique extension fromL 2 (Rk,ν) toL 2 (Ω) given as follows. For a fixedh∈L 2 (Rk, ν), let (hn) be a sequence of simple functions such that‖h−hn‖L 2 (Rk,ν)→0. Then
W(h) := lim n→∞ W(hn),
where the limit is inL 2 (Ω,F,P). The above isometry is known as theWiener’s isometry. By definition, the random variableW(h) is the Wiener integral ofhwith respect to the white noiseW:
W(h) =
∫
Rk
h(x)W(dx). (2)
Proposition 2.4 family(W(h), h∈L 2 (Rk,ν))consists of Gaussian, zero mean, random variables andE(W(h 1 )(W(h 2 ))) =〈h 1 ,h 2 〉.
Proof follows directly from the definition thatW(h) is Gaussian with mean zero andE(W(h) 2 ) =‖h‖ 2 L 2 (Rk,ν)We now check by using polarisation that
E(W(h 1 )W(h 2 )) =
∫
Rk
h 1 (x)h 2 (x)ν(dx).
Indeed,
W(h 1 )W(h 2 ) =
1
4
(W(h 1 ) +W(h 2 )) 2 −
1
4
(W(h 1 )−W(h 2 )) 2
=
1
4
(W(h 1 +h 2 )) 2 −
1
4
(W(h 1 −h 2 )) 2 ,
so, taking expectations,
E(W(h 1 )W(h 2 )) =
1
4
‖h 1 +h 2 ‖ 2 L 2 (Rk,ν)−
1
4
‖h 1 −h 2 ‖ 2 L 2 (Rk,ν)
=
∫
Rk
h 1 (x)h 2 (x)ν(dx).
we obtain the Brownian motion.
Brownian sheet
Consider the casek= 2 and let (W(A), A∈Bb(R 2 )) be a white noise based on the measureν(dx) = 1R 2 +(x)dx. We define a two-parameter Gaussian process in a manner similar to that used in (2) to derive the Brownian motion from a white noise based on 1R+(x)dx. Indeed, for (t 1 ,t 2 )∈R 2 ,set
Wt 1 ,t 2 =W(]−∞,t 1 ]×]−∞,t 2 ]) =
{
W([0,t 1 ]×[0,t 2 ]), if (t 1 ,t 2 )∈R 2 +, 0 otherwise.
Proposition 2.(Wt 1 ,t 2 ,(t 1 ,t 2 )∈R 2 +)is aBrownian sheet, that is, a mean- zero Gaussian process such thatE(Wt 1 ,t 2 Ws 1 ,s 2 ) = (t 1 ∧s 1 )(t 2 ∧s 2 ),(s 1 ,s 2 )∈ R 2 +,(t 1 ,t 2 )∈R 2 +.
Proof definition, this process is clearly Gaussian,E(Wt 1 ,t 2 ) = 0 and more- over,
E(Wt 1 ,t 2 Ws 1 ,s 2 ) =E(W([0,t 1 ]×[0,t 2 ])W([0,s 1 ]×[0,s 2 ])) =ν(([0,t 1 ]×[0,t 2 ])∩[0,s 1 ]×[0,s 2 ]) =ν([0,t 1 ∧s 1 ]×[0,t 2 ∧s 2 ]) = (t 1 ∧s 1 )(t 2 ∧s 2 ).
2 More results concerning the Brownian sheet
Let us go back for a moment to the isonormal process based onν(dx) = 1R 2 +dx,
(W(h),h ∈L(R 2 ,ν)). By restricting the indices toS(R 2 ), we have now a random linear functional (W(φ),φ∈ S(R 2 )), in the sense thatφ7→W(φ) is linear a. (the a. may depend onφ). We also have the following property:
Ifφn→φinS(R 2 )) (and therefore inL 2 (R 2 )), thenW(φn)→W(φ) in L 2 (R 2 ). This implies, by a result from Itˆo (see [67]), the existence ofaversion ofW, denoted byW ̃, with values inS′(R 2 ), i.
∀φ,W ̃(φ) =W(φ), a.
a.ω,φ7→ ̃(ω)(φ) is an element ofS′(R 2 ).
In the rest of the lecture we will identify this version with thesecond cross- derivative of the Brownian sheet. The Brownian sheet satisfies the following property:
lim sup t 1 +t 2 ↑∞
Wt 1 ,t 2 |t 1 +t 2 | 2
= 0, a.
Hence for a.ω, (t 1 ,t 2 )→W(ω)(t 1 ,t 2 ) is slowly growing, i.e(ω)∈S′(R 2 +), a.
The Brownian sheet as a convolution
Denote by ∫
R 2 +
h(t 1 ,t 2 )dWt 1 ,t 2
the r.v(h) (Wiener isometry),h∈L 2 (R 2 +,dx).
Since 1]−∞,t 1 ]×]−∞,t 2 ](s 1 ,s 2 ) = 1R 2 +(t 1 −s 1 ,t 2 −s 2 ),
Wt 1 ,t 2 =W(]−∞,t 1 ]×]−∞,t 2 ]) =W(1]−∞,t 1 ]×]−∞,t 2 ])
=W(1R 2 +(t 1 −·,t 2 −·)) =
∫
R 2 +
1 R 2 +(t 1 −s 1 ,t 2 −s 2 )dWs 1 ,s 2. (2)
The last integral is a particular example ofstochastic convolution—a notion that will be introduced later in the course. The terminology is suggested by the following formula concerning the version with values inS′(R 2 ) of the the white noiseWbased onν(dx) = 1R 2 +(x)dx:
1 R 2 +∗W= ((u 1 ,u 2 )7→Wu 1 ,u 2 ) a. (2)
We now prove this formula. Takeφ∈S(R 2 ). By the definition of convolution of distributions, we have
〈 1 R 2 +∗W,φ〉=〈W, ̃ 1 R 2 +∗φ〉,
where ̃ 1 R 2 +(x) = 1R 2 +(−x) and〈W, ̃ 1 R 2 +∗φ〉=W( ̃ 1 R 2 +∗φ) a. Computing the
function ̃ 1 R 2 +∗φ, we obtain
〈W, ̃ 1 R 2 +∗φ〉=W
(∫∞
·
du 1
∫∞
·
du 2 φ(u 1 ,u 2 )
)
=
∫
R 2 +
(∫∞
t 1
du 1
∫∞
t 2
du 2 φ(u 1 ,u 2 )
)
W(dt 1 ,dt 2 )
=
∫
R 2 +
du 1 du 2 φ(u 1 ,u 2 )
(∫
R 2 +
1 R 2 +(u 1 −t 1 ,u 2 −t 2 )dWt 1 ,t 2
)
=
∫
R 2 +
du 1 du 2 φ(u 1 ,u 2 )Wu 1 ,u 2 a., (2)
where we have used a stochastic Fubini’s theorem. SinceS(R 2 ) admits a countable dense subset (is separable), (2) yields (2).
Recovering white noise from Brownian sheet
White noise inR 2 +can be seen as the second cross-derivative of the Brownian sheet (Wt 1 ,t 2 ,(t 1 ,t 2 )∈R 2 +). Indeed, for anyφ∈ S(R 2 ), by definition of the
3 Lecture 2: Gaussian Spatially Correlated Noise.
In this lecture, we introduce Gaussian noises that are more general than the white noise introduced in Lecture 2. The indices of the noise will be of the type (t,φ), wheret∈R+stands for “time” andφis a “space” argument belonging to some functional space.
3 Spatially homogeneous Gaussian noise
We consider a family of mean zero Gaussian random variables
W={W(φ), φ∈C 0 ∞(Rk+1)},
whereC 0 ∞(Rk+1) denotes the space of infinitely differentiable functions with compact support, with covariance
E(W(φ)W(ψ)) =
∫∞
0
dt
∫
Rk
Λ(dx) (φ(t)∗ψ ̃(t))(x), (3)
where “∗” denotes convolution in the spatial variable andψ ̃(t,x) :=ψ(t,−x) (reflection in thexvariable).
- Λ is a non-negative definite measure onRk. That means ∫
Rk
Λ(dx) (φ∗φ ̃)(x)≥ 0.
Λ is symmetric. This implies thatE(W(φ)W(ψ)) =E(W(ψ)W(φ)) (sym- metry).
By Bochner-Schwartz theorem, Λ being non-negative definite , itis the Fourier transform of a non-negative tempered measureμonRk:Fμ= Λ (actually the two statements are equivalent).
Moreover, since Λ is symmetric,μis symmetric. In particular,
∀f:Rk→R,
∫
Rk
f(ξ)μ(dξ) =
1
2
∫
Rk
[f(ξ) +f(−ξ)]μ(dξ),
and iffis odd,
∫
Rkf(ξ)μ(dξ) = 0. The measureμis called thespectral measure
- By definition of the Fourier transform on the spaceS′(Rk) of tempered distributions, for allφ∈S(Rk) (space of rapidly decreasingC∞functions), ∫
Rk
φ(x) Λ(dx) =
∫
Rk
Fφ(ξ)μ(dξ), (3)
and there is an integerm≥1 such that ∫
Rk
(1 +|ξ| 2 )−mμ(dξ)<∞. (3)
- μtempered implies that Λ is also tempered.
The results on distributions used in the statements before can be found in several references. For example in Gelfand-Vilenkin, Chap II, Theory of Distributions; L. Schwartz, Th ́eorie des distributions.
Some important remarks/formulas
- The covariance (3) can also be written, using (3), as
E(W(φ)W(ψ)) =
∫∞
0
dt
∫
Rd
μ(dξ)Fφ(t)(ξ)Fψ(t)(ξ).
- Assume that Λ(dx) =f(x)dxthen formula (3) becomes ∫∞
0
dt
∫
Rk
dx
∫
Rk
dy φ(t,x)f(x−y)ψ(t,y),
which makes clear the spatially homogeneous (or translation invariant) character of the noise. In fact, any nonnegative definite, translation-invariant, bilinear hermitian form, continuous in each argument,Q:C∞ 0 (Rk)×C 0 ∞(Rk)−→Ris of the form Q(φ 1 ,φ 2 ) =〈Λ ̄,φ 1 ∗φ ̃ 2 〉, for some distributionΛ which is the Fourier transform of a non-negative, ̄ tempered measure.
A Hilbert space derived from the covariance
Definition 3.1 the completion of the Schwartz spaceC 0 ∞(Rk), after iden- tifyingφ 1 ,φ 2 such that‖φ 1 −φ 2 ‖U= 0, endowed with the semi-inner product
〈φ,ψ〉U=
∫
Rk
Λ(dx)(φ∗ψ ̃)(x) =
∫
Rk
μ(dξ)Fφ(ξ)Fψ(ξ), (3)
φ,ψ∈C 0 ∞(Rk), and associated semi-norm‖·‖U.
Semi-inner means not required to be strictly positive:‖f‖≥0, and‖f‖= 0, if and only iff= 0.
Remarks
We will see later thatC 0 ∞(Rk) is dense inS(Rk) for‖·‖U. HenceUis isomorphic to the completion ofS(Rk) with respect to‖·‖U.
Uis isomorphic to the bidual ofC 0 ∞(Rk) (with respect to the norm‖·‖U). This is not a very useful description here. An interesting issueis to identify elements of this set. Ifμis absolutely continuous, one can prove the following: Assume thatμis absolutely continuous with respect to the Lebesgue mea- sure. Then the set
Sμ′(Rk) =
{
Γ∈S′(Rk) :FΓis a function and
∫
Rk
μ(dξ)|FΓ(ξ)| 2 <∞
}
is included inU. The same conclusion holds under the slightly different assumptions:
because Λ is a tempered measure and|φ 2 | ∗ |φ ̃ 2 |decreases rapidly, together with dominated convergence, one checks that there is a sequence (φn 2 )n ⊂ C 0 ∞(Rk) such that limn→∞‖φ 2 −φn 2 ‖U = 0. Then, by the very definition of the norm inUT, one easily proves that limn→∞‖φ 1 φ 2 −φ 1 φn 2 ‖UT= 0. There- fore,φ 1 (·)φ 2 (⋆)∈UT.
Step 2. Elements ofUTof the formφ 1 (·)φ 2 (⋆),φ 1 ∈L 2 ([0,T];R)andφ 2 ∈ S(Rk). belong toC. Indeed, let (φn 1 )n∈ C 0 ∞(R+) be such that, for alln, the support ofφn 1 is contained in [0,T] andφn 1 →φ 1 inL 2 ([0,T];R). Thenφn 1 φ 2 ∈Cby Step 1, and one checks thatφn 1 φ 2 converges, asntends to infinity, toφ 1 φ 2 inUT. Therefore,φ 1 (·)φ 2 (⋆)∈C.
Step 3. In general, suppose thatφ∈UT. We show thatφ∈C. Indeed, let (ej)jbe a complete orthonormal basis ofUwithej∈S(Rk), for allj. Then, sinceφ(s)∈Ufor anys∈[0,T],
‖φ‖ 2 UT=
∫T
0
‖φ(s)‖ 2 Uds=
∑∞
j=
∫T
0
〈φ(s),ej〉 2 Uds.
In particular, for anyj≥1, the functions7→〈φ(s),ej〉Ubelongs toL 2 ([0,T];R). Thus, it follows from Step 2 that
φn(·) :=
∑n
j=
〈φ(·),ej〉Uej
belongs toC. Moreover, well-known results on Hilbert spaces tell us that‖φ− φn‖ 2 UT→0 asn→∞. This shows thatφ∈C.
Conclusion
A direct calculation using (3) shows that:
- the generalized Gaussian random field{W(φ), φ∈ C 0 ∞([0,T]×Rk)}is a random linear functional, in the sense given in Lecture 1. That is,
W(aφ+bψ) =aW(φ) +bW(ψ),a.
- the mappingφ7→W(φ) is an isometry from (C 0 ∞([0,T]×Rk),‖·‖UT) into L 2 (Ω,F,P).
Hence, taking into account the above lemma,W(φ) can be defined for allφ∈UT following the standard method for extending an isometry. This establishes the following property.
Proposition 3. (a)(W(φ),φ∈UT)is an isonormal process onUT.
(b) Fort≥ 0 andφ∈U, setWt(φ) =W(10,tφ(⋆)). Then the process W= (Wt(φ), t≥ 0 , φ∈U)has the following two properties.
For anyφ∈U,{Wt(φ), t≥ 0 }is a Brownian motion with variance t‖φ‖ 2 U.
For alls,t≥ 0 andφ,ψ∈U,E(Wt(φ)Ws(ψ)) = (s∧t)〈φ,ψ〉U.
In the sequel the class of isonormal processes (W(φ),φ∈UT) will be termed white in time, spatially homogeneous Gaussian noises, or justspatially homoge- neous Gaussian noisesif there is no confusion.
3 Examples of covariances
We give here a list of examples of covariance measures Λ.
Space-time white noise inRk
Let Λ(dx) =δ 0 (dx). Its spectral measure is the Lebesgue measure:μ(dx) = dx. One checks that in this particular case the formula (3) becomes
E(W(φ)W(ψ)) =
∫∞
0
dt
∫
Rk
dxφ(t,x)ψ(t,x) =〈φ,ψ〉L 2 (R+×Rk,dt×dx). (3)
Hence,UT=L 2 (R+×Rk,dt×dx).
Riesz kernels
This is the case where Λ(dx) =rβ(x)dx, whererβ(x) =|x|−βand 0< β < k. The functionrβ(x) is also called a Riesz potential. It is the convolution
kernel of the fractional Laplacian (−∆)−
k−β 2. More precisely, up to the factor γ(β 2 ) γ(k− 2 β)2k−βπk 2
, on the right-hand side,
(−∆)−
k−β 2 f=f∗rβ.
Here we have denoted byγthe Euler gamma function. The spectral measure isμ(dx) =cd,βrk−β(x)dx, with a positive constant cd,βwhich depends only onkandβ(see [29]). For certain values ofkandβ, the associated Hilbert spaceUmay contain Schwartz distributions that are not functions (see [14]). This class of spatial covariances have bee widely used in the literature on SPDEs (see e. [20] and references herein). An extension can be obtained by multiplying the Riesz kernel by a smooth functionφsuch that Λ(dx) = φ(x)rβ(x)dxis a covariance measure. For example, one can take
φ= exp(−σ 2 |x| 2 /2),
and thenf=F(rk−β∗ψ), with
ψ(x) = (2πσ 2 )−
32 exp(−|x| 2 /(2σ 2 )),
and the spectral measure isμ(dx) =cd,β(F− 1 f∗rk−β)(x)dx.
Bessel kernels
This is the case where Λ(dx) =fα(x)dx, with
fα(x) =
∫∞
0
w(α−k−2)/ 2 e−w−(|x|
2 )/(4w) dw, α > 0.
An interesting example isμ(dx) =e−|x|
2 , which corresponds to a Gaussian type densityf. Let (W(φ),φ ∈ C∞ 0 (Rk+1)) be a white in time, spatially homogeneous Gaussian noise with spectral measureμ. There exists a stochastic process (Bt(x),(t,x)∈[0,∞]×Rk) satisfying the following properties
for eachx∈Rk, (Bt(x),t≥0) is a one–dimensional Brownian motion; 2(Bt(x)Bs(y)) = (t∧s)f(x−y), for anys,t≥0,x,y∈Rk;
for anyφ∈C∞ 0 (Rk)),
W
(
1 0,tφ(∗)
)
=
∫
Rk
dxφ(x)B(t,x),
in the sense that both expressions define a Gaussian process with the same mean and covariance functions.
Indeed, fixψ∈C 0 ∞(Rk) such thatψ≥0, the support ofψis contained in the unit ball ofRkand
∫
Rkψ(x)dx= 1. Forn≥1, setψn(x) =n
kψ(nx). Then,ψn→δ 0
inS′(Rk) and|Fψn|≤1. The properties offensure limn→∞f∗ψn =f(x). Define Bt(x) = lim n→∞
W
(
1 0,tψn(x+∗)
)
, t∈[0,T], (3)
where we consider the evaluation ofWonUTand the limit is inL 2 (Ω). Let us check first that the limit exists. Indeed, forn,m≥0,
E
∣
∣W
(
1 0,t[ψn(x+∗)−ψm(x+∗)]
)∣
∣ 2
∫t
0
∫
Rk
dy
∫
Rk
dz(ψn−ψm)(x+y)f(y−z)(ψn−ψm)(x+z) ∫t
0
∫
Rk
μ(dξ)|F(ψn−ψm)(ξ)| 2.
Since limn,m→∞F(ψn−ψm)(ξ) = 0 pointwise andμis a finite measure, by Lebesgue bounded convergence, we see thatW
(
1 0,tψn(x+∗)
)
is a Cauchy sequence inL 2 (Ω), for anyt≥0. Thus, (3) is well defined and claim 1. above holds. The proof of 2. follows from the following computations. E
(
W
(
1 0,sψn(x+∗)
)
W
(
1 0,tψn(y+∗)
))
= (s∧t)
∫
Rk
dz
∫
Rk
dw ψn(x+z)f(z−w)ψn(y+w).
Sincefis continuous and bounded, we can pass to the limit the last expression and obtain
lim n→∞
E
(
W
(
1 0,sψn(x+∗)
)
W
(
1 0,tψn(y+∗)
))
= (t∧s)f(x−y).
For the proof of 3, we observe that owing to property 2. above,
E
((∫
Rk
dxφ(x)B(t,x)
)(∫
Rk
dy φ(y)B(s,y)
))
= (t∧s)
(∫
Rk
dx
∫
Rk
dy φ(x)φ(y)f(x−y)
)
,
and this coincides with
E
(
W
(
1 0,sφ(x+∗)
)
W
(
1 0,tφ(y+∗)
))
.
Assume in addition thatμ(dx) =h(x)dx, withh≥0 andh∈L 1 (Rk). Clearly
√
h∈L 2 (Rk) and therefore,g:=F(
√
h)∈L 2 (Rk). With a space-time white noiseF, we define
Bt(x) =
∫t
0
∫
Rk
F(ds,dy)g(x−y).
Observe that this provides a family (Bt(x),t≥ 0 ,x∈Rk) of correlated Brownian motions, with
E(Bt(x)Bs(y)) = (t∧s)(g∗g ̃)(x−y) = (t∧s)f(x−y), (3)
where in the last equality we have used thatFh=f. The process (W(φ),φ∈C 0 ∞(Rk+1)) defined by
W(φ) =
∫
Rk
dx
∫T
0
B(ds,x)φ(s,x),
is a white in time, spatially homogeneous Gaussian noise, with covariance mea- suref(x)dx. Indeed,
E(W(φ)W(ψ))
=
∫
Rk
dx
∫
Rk
dyE
((∫
T
0
B(ds,x)φ(s,x)
)(∫
T
0
B(ds,y)ψ(s,y)
))
=
∫
Rk
dx
∫
Rk
dy
∫T
0
φ(r,x)ψ(r,y)d〈B(·,x)B(·,y)〉r,
where〈B(·,x)B(·,y)〉rstands for the quadratic variation process. From (3) it follows that〈B(·,x)B(·,y)〉r=rf(x−y). Thus the proof of the claim is complete. Notice that we can write
W
(
1 0,tφ(∗)
)
=
∫
Rk
dxφ(x)
∫t
0
B(ds,x)) =
∫
Rk
dxφ(x)B(t,x)
=
∫
Rk
dxφ(x)
∫t
0
∫
Rk
F(ds,dy)g(x−y),
which shows asmoothcharacter ofWin the spatial variable. Examples of finite spectral measures have been considered in [56] and [27].
3 Cylindrical Wiener processes
The noise introduced in the preceding section is homogeneous in space. In this section, we present a more general class. Fix a separable Hilbert spaceVwith inner product〈·,·〉V. Following [55, 52], we define the general notion ofcylindrical Wiener process inV. This is a family of Brownian motions indexed byV, as we now define.
Lecture Notes SPDEs
Assignatura: Stochastic Calculus (568191)
Universitat: Universitat de Barcelona
És una previsualització
Accedir a tots els documents
Aconsegueix descàrregues il·limitades
Aconsegueix notes més altes
