A. The Bochner Integral
This chapter is a slight modification of Chap. A in [FK01]. Let X, be a Banach space, B(X) the Borel σ-field of X and (Ω, F,µ) a measure space with finite measure µ.
A.1. Definition of the Bochner integral
Step 1: As first step we want to define the integral for simple functions which are defined as follows. Set n E → ∈ ∈F ∈ N := f :Ω X f = xk1Ak ,xk X, Ak , 1 k n, n k=1 and define a semi-norm E on the vector space E by
f E := f dµ, f ∈E. To get that E, E is a normed vector space we consider equivalence classes with respect to E . For simplicity we will not change the notations. ∈E n For f , f = k=1 xk1Ak , Ak’s pairwise disjoint (such a representation n is called normal and always exists, because f = k=1 xk1Ak , where f(Ω) = {x1,...,xk}, xi = xj,andAk := {f = xk}) and we now define the Bochner integral to be
n f dµ := xkµ(Ak). k=1
(Exercise: This definition is independent of representations, and hence linear.) In this way we get a mapping E → int : , E X, f → f dµ which is linear and uniformly continuous since f dµ f dµ for all f ∈E. Therefore we can extend the mapping int to the abstract completion of E with respect to E which we denote by E.
105 106 A. The Bochner Integral
Step 2: We give an explicit representation of E.
Definition A.1.1. A function f :Ω→ X is called strongly measurable if it is F/B(X)-measurable and f(Ω) ⊂ X is separable.
Definition A.1.2. Let 1 p<∞. Then we define
Lp(Ω, F,µ; X):=Lp(µ; X) := f :Ω→ X f is strongly measurable with respect to F,and f p dµ<∞ and the semi-norm
1 p p p f Lp := f dµ ,f∈L (Ω, F,µ; X).
p The space of all equivalence classes in L (Ω, F,µ; X) with respect to Lp is denoted by Lp(Ω, F,µ; X):=Lp(µ; X).
Claim: L1(Ω, F,µ; X)=E. 1 Step 2.a: L (Ω, F,µ; X), L1 is complete. The proof is just a modification of the proof of the Fischer–Riesz theorem by the help of the following proposition.
Proposition A.1.3. Let (Ω, F) be a measurable space and let X be a Banach space. Then:
(i) the set of F/B(X)-measurable functions from Ω to X is closed under the formation of pointwise limits, and
(ii) the set of strongly measurable functions from Ω to X is closed under the formation of pointwise limits.
Proof. Simple exercise or see [Coh80, Proposition E.1, p. 350].
1 Step 2.b: E is a dense subset of L (Ω, F,µ; X) with respect to L1 . This can be shown by the help of the following lemma.
Lemma A.1.4. Let E be a metric space with metric d and let f :Ω→ E be strongly measurable. Then there exists a sequence fn, n ∈ N,ofsimpleE- valued functions (i.e. fn is F/B(E)-measurable and takes only a finite number of values) such that for arbitrary ω ∈ Ω the sequence d fn(ω),f(ω) , n ∈ N, is monotonely decreasing to zero. A.2. Properties of the Bochner integral 107
Proof. [DPZ92, Lemma 1.1, p. 16] Let {ek | k ∈ N} be a countable dense subset of f(Ω). For m ∈ N define dm(ω) := min d f(ω),ek k m = dist(f(ω), {ek,k m}) , km(ω) := min k m dm(ω)=d f(ω),ek ,
fm(ω):=ekm(ω).
Obviously fm, m ∈ N, are simple functions since they are F/B(E)-measurable (exercise) and
fm(Ω) ⊂{e1,e2,...,em}.
Moreover, by the density of {ek | k ∈ N}, the sequence dm (ω), m ∈ N ,is monotonically decreasing to zero for arbitrary ω ∈ Ω. Since d fm(ω),f(ω) = dm(ω) the assertion follows.
Let now f ∈ L1(µ; X). By the Lemma A.1.4 above we get the existence of a sequence of simple functions fn, n ∈ N, such that fn(ω) − f(ω) ↓ 0 for all ω ∈ Ωasn →∞.
n→∞ Hence fn −−−−→ f in L1 by Lebesgue’s dominated convergence theorem.
A.2. Properties of the Bochner integral
Proposition A.2.1 (Bochner inequality). Let f ∈ L1(Ω, F,µ; X).Then f dµ f dµ.
Proof. We know the assertion is true for f ∈E, i.e. int : E→X is linear, continuous with int f f E for all f ∈E, so the same is true for its unique continuous extension int : E = L1(µ; X) → X, i.e. for all f ∈ L1(X, µ) f dµ = intf f E = f dµ.
Proposition A.2.2. Let f ∈ L1(Ω, F,µ; X).Then L ◦ f dµ = L f dµ holds for all L ∈ L(X, Y ),whereY is another Banach space.
Proof. Simple exercise or see [Coh80, Proposition E.11, p. 356]. 108 A. The Bochner Integral
Proposition A.2.3 (Fundamental theorem of calculus). Let −∞
This chapter is identical to Chap. B in [FK01]. Let U, , U and H, , be two separable Hilbert spaces. The space of all bounded linear operators from U to H is denoted by L(U, H); for simplicity we write L(U) instead of L(U, U). If we speak of the adjoint operator of L ∈ L(U, H) we write L∗ ∈ L(H, U). An element L ∈ L(U) is called symmetric if Lu, vU = u, LvU for all u, v ∈ U. In addition, L ∈ L(U) is called nonnegative if Lu, u 0 for all u ∈ U. Definition B.0.1 (Nuclear operator). An element T ∈ L(U, H) is said to be a nuclear operator if there exists a sequence (aj)j∈N in H and a sequence (bj)j∈N in U such that ∞ Tx = ajbj,xU for all x ∈ U j=1 and aj · bj U < ∞. j∈N
The space of all nuclear operators from U to H is denoted by L1(U, H). If U = H, T ∈ L1(U, H) is nonnegative and symmetric, then T is called trace class.
Proposition B.0.2. The space L1(U, H) endowed with the norm ∞ · ∈ T L1(U,H) := inf aj bj U Tx = aj bj,x U ,x U j∈N j=1 is a Banach space. Proof. [MV92, Corollar 16.25, p. 154].
Definition B.0.3. Let T ∈ L(U) and let ek, k ∈ N, be an orthonormal basis of U. Then we define tr T := Tek,ekU k∈N if the series is convergent.
109 110 B. Nuclear and Hilbert–Schmidt Operators
One has to notice that this definition could depend on the choice of the orthonormal basis. But there is the following result concerning nuclear oper- ators.
Remark B.0.4. If T ∈ L1(U) then tr T is well-defined independently of the choice of the orthonormal basis ek, k ∈ N. Moreover we have that | | tr T T L1(U).
Proof. Let (aj)j∈N and (bj)j∈N be sequences in U such that Tx = ajbj,xU j∈N for all x ∈ U and aj U · bj U < ∞. j∈N Then we get for any orthonormal basis ek, k ∈ N,ofU that Tek,ekU = ek,ajU ·ek,bj U j∈N and therefore Tek,ekU ek,ajU ·ek,bj U k∈N j∈N k∈N 1 1 2 2 2 2 ek,aj U · ek,bjU j∈N k∈N k∈N = aj U · bj U < ∞. j∈N
This implies that we can exchange the summation to get that Tek,ekU = ek,ajU ·ek,bj U = aj ,bjU , k∈N j∈N k∈N j∈N and the assertion follows.
Definition B.0.5 (Hilbert–Schmidt operator). A bounded linear opera- tor T : U → H is called Hilbert–Schmidt if 2 Tek < ∞ k∈N where ek, k ∈ N, is an orthonormal basis of U. The space of all Hilbert–Schmidt operators from U to H is denoted by L2(U, H). B. Nuclear and Hilbert–Schmidt Operators 111
Remark B.0.6. (i) The definition of Hilbert–Schmidt operator and the number T 2 := Te 2 L2(U,H) k k∈N
does not depend on the choice of the orthonormal basis ek, k ∈ N,andwe ∗ have that T L2(U,H) = T L2(H,U). For simplicity we also write T L2
instead of T L2(U,H). (ii) T L(U,H) T L2(U,H).
(iii) Let G be another Hilbert space and S1 ∈ L(H, G),S2 ∈ L(G, U),T ∈ L2(U, H).ThenS1T ∈ L2(U, G) and TS2 ∈ L2(G, H) and
S1T L2(U,G) S1 L(H,G) T L2(U,H),
TS2 L2(G,H) T L(U,H) S2 L2(G,U).
Proof. (i) If ek, k ∈ N, is an orthonormal basis of U and fk, k ∈ N,isan orthonormal basis of H we obtain by the Parseval identity that 2 2 ∗ 2 Tek = Tek,fj = T fj U k∈N k∈N j∈N j∈N
and therefore the assertion follows.
(ii) Let x ∈ U and fk, k ∈ N, be an orthonormal basis of H. Then we get that Tx 2 = Tx,f 2 x 2 T ∗f 2 = T 2 · x 2 . k U k U L2(U,H) U k∈N k∈N
(iii) Let ek,k∈ N be an orthonormal basis of U. Then S Te 2 S 2 T 2 . 1 k G 1 L(H,G) L2(U,H) k∈N
∗ ∗ ∗ Furthermore, since (TS2) = S2 T , it follows that by the above and (i) that TS2 ∈ L2(G, H)and
∗ TS2 L2(G,H) = (TS2) L2(H,G) ∗ ∗ = S2 T L2(H,G) · S2 L(G,U) T L2(U,H). 112 B. Nuclear and Hilbert–Schmidt Operators
Proposition B.0.7. Let S, T ∈ L2(U, H) and let ek, k ∈ N, be an orthonor- mal basis of U. If we define T,S L2 := Sek,Tek k∈N we obtain that L2(U, H), , L2 is a separable Hilbert space. If fk, k ∈ N, is an orthonormal basis of H we get that fj ⊗ek := fjek, ·U , j, k ∈ N, is an orthonormal basis of L2(U, H). Proof. We have to prove the completeness and the separability.
1. L2(U, H) is complete:
Let Tn, n ∈ N, be a Cauchy sequence in L2(U, H). Then it is clear that it is also a Cauchy sequence in L(U, H). Because of the completeness of L(U, H) there exists an element T ∈ L(U, H) such that Tn − T L(U,H) −→ 0as n →∞. But by the lemma of Fatou we also have for any orthonormal basis ek, k ∈ N,ofU that − 2 − − Tn T L2 = (Tn T )ek, (Tn T )ek k∈N 2 = lim inf (Tn − Tm)ek m→∞ k∈N − 2 − 2 lim inf (Tn Tm)ek = lim inf Tn Tm L <ε m→∞ m→∞ 2 k∈N for all n ∈ N big enough. Therefore the assertion follows.
2. L2(U, H) is separable:
If we define fj ⊗ ek := fjek, ·U , j, k ∈ N, then it is clear that fj ⊗ ek ∈ L2(U, H) for all j, k ∈ N and for arbitrary T ∈ L2(U, H) we get that ⊗ · fj ek,T L2 = ek,en U fj,Ten = fj,Tek . n∈N
Therefore it is obvious that fj ⊗ ek, j, k ∈ N, is an orthonormal system. ⊗ ∈ N In addition, T =0if fj ek,T L2 = 0 for all j, k , and therefore span(fj ⊗ ek | j, k ∈ N) is a dense subspace of L2(U, H). Proposition B.0.8. Let G, , G be a further separable Hilbert space. If T ∈ L2(U, H) and S ∈ L2(H, G) then ST ∈ L1(U, G) and · ST L1(U,G) S L2 T L2 .
Proof. Let fk, k ∈ N, be an orthonormal basis of H. Then we have that STx = Tx,fkSfk,x∈ U k∈N B. Nuclear and Hilbert–Schmidt Operators 113 and therefore ∗ · ST L1(U,G) T fk U Sfk G k∈N 1 1 ∗ 2 2 · 2 2 · T fk U Sfk G = S L2 T L2 . k∈N k∈N ∈ N ∈ Remark B.0.9. Let ek, k , be an orthonormal basis of U.IfT L(U) ∞ ∈ is symmetric, nonnegative with k∈N Tek,ek U < then T L1(U). Proof. The result is obvious by the previous proposition and the fact that 1 1 1 there exists T 2 ∈ L(U) nonnegative and symmetric such that T = T 2 T 2 (see 1 Proposition 2.3.4). Then T 2 ∈ L2(U). ∗ Proposition B.0.10. Let L ∈ L(H) and B ∈ L2(U, H).Then LBB ∈ ∗ L1(H), B LB ∈ L1(U) and we have that
tr LBB∗ =trB∗LB.
Proof. We know by Remark B.0.6 (iii) and Proposition B.0.8 that LBB∗ ∈ ∗ L1(H)andB LB ∈ L1(U). Let ek, k ∈ N, be an orthonormal basis of U and let fk, k ∈ N, be an orthonormal basis of H. Then the Parseval identity implies that fk,Ben·fk,LBen k∈N n∈N 1 1 2 2 2 2 fk,Ben · fk,LBen n∈N k∈N k∈N · · 2 = Ben LBen L L(H) B L2 . n∈N
Therefore, it is allowed to interchange the sums to obtain that ∗ ∗ ∗ ∗ ∗ tr LBB = LBB fk,fk = B fk,B L fkU k∈N k∈N ∗ ∗ ∗ = B fk,enU ·B L fk,enU = fk,Ben·fk,LBen k∈N n∈N n∈N k∈N ∗ ∗ = Ben,LBen = en,B LBenU =trB LB. n∈N n∈N C. Pseudo Inverse of Linear Operators
This chapter is a slight modification of Chapter C in [FK01]. Let U, , U and H, , be two Hilbert spaces. Definition C.0.1 (Pseudo inverse). Let T ∈ L(U, H) and Ker(T ):={x ∈ U | Tx =0}. The pseudo inverse of T is defined as −1 −1 ⊥ ⊥ T := T |Ker(T )⊥ : T Ker(T ) = T (U) → Ker(T ) .
(Note that T is one-to-one on Ker(T )⊥.) Remark C.0.2. (i) There is an equivalent way of defining the pseudo in- verse of a linear operator T ∈ L(U, H).Forx ∈ T (U) one sets T −1x ∈ U to be the solution of minimal norm of the equation Ty = x, y ∈ U. (ii) If T ∈ L(U, H) then T −1 : T (U) → Ker(T )⊥ is linear and bijective. Proposition C.0.3. Let T ∈ L(U) and T −1 the pseudo inverse of T . (i) If we define an inner product on T (U) by
−1 −1 x, yT (U) := T x, T yU for all x, y ∈ T (U), then T (U), , T (U) is a Hilbert space. ⊥ (ii) Let ek, k ∈ N, be an orthonormal basis of (Ker T ) .ThenTek, k ∈ N, is an orthonormal basis of T (U), , T (U) . Proof. T : (Ker T )⊥ → T (U) is bijective and an isometry if (Ker T )⊥ is equipped with , U and T (U) with , T (U). Now we want to present a result about the images of linear operators. To this end we need the following lemma. ∈ ∈ Lemma C.0.4. Let T L(U, H). Then the set T Bc(0) (= Tu u U, u U c ), c 0, is convex and closed. Proof. Since T is linear it is obvious that the set is convex. Since a convex subset of a Hilbert space is closed (with respect to the norm) if and only if it is weakly closed, it suffices to show that T Bc(0) is weakly closed. Since T : U → H is linear and continuous (with respect to the norms
115 116 C. Pseudo Inverse of Linear Operators on U, H respectively) it is also obviously continuous with respect to the weak topologies on U, H respectively. But by the Banach–Alaoglou theorem (see e.g. [RS72, Theorem IV.21, p. 115]) closed balls in a Hilbert space are weakly compact. Hence Bc(0) is weakly compact, and so is its continuous image, i.e. T Bc(0) is weakly compact, therefore weakly closed. Proposition C.0.5. Let U1, , 1 and U2, , 2 be two Hilbert spaces. In addition, we take T1 ∈ L(U1,H) and T2 ∈ L(U2,H). Then the following statements hold.
∗ ∗ (i) If there exists a constant c 0 such that T x 1 c T x 2 for all 1 2 x ∈ H then T1u u ∈ U1, u 1 1 ⊂ T2v v ∈ U2, v 2 c .In particular, this implies that Im T1 ⊂ Im T2. ∗ ∗ ∈ −1 (ii) If T1 x 1 = T2 x 2 for all x H then Im T1 =ImT2 and T1 x 1 = −1 ∈ T2 x 2 for all x Im T1. Proof. [DPZ92, Proposition B.1, p. 407]
(i) Assume that there exists u0 ∈ U1 such that u0 1 1andT1u0 ∈/ T2v v ∈ U2, v 2 c . By Lemma C.0.4 we know that the set T2v v ∈ U2, v 2 c is closed and convex. Therefore, we get by the separation theorem (see [Alt92, 5.11 Trennungssatz, p. 166]) there exists x ∈ H, x = 0, such that
1 < x, T1u0 and x, T2v 1 for all v ∈ U2 with v 2 c. ∗ ∗ ∗ Thus T1 x 1 > 1andc T2 x 2 =sup T2 x, v 2 1, a contradiction. v2c
(ii) By (i) we know that Im T1 =ImT2. It remains to verify that −1 −1 ∈ T1 x 1 = T2 x 2 for all x Im T1. −1 −1 If x = 0 then T1 0 1 =0= T2 0 2. ⊥ ⊥ If x ∈ Im T1 \{0} then there exist u1 ∈ (Ker T1) and u2 ∈ (Ker T2) such that x = T1u1 = T2u2. We have to show that u1 1 = u2 2.
Assume that u1 1 > u2 2 > 0. Then (i) implies that x u2 = T2 u2 2 u2 2 ∈ T2v v ∈ U2, v 2 1 = T1u u ∈ U1, u 1 1 .
But x u1 u1 = T1 and > 1, u2 2 u2 2 u2 2 1 C. Pseudo Inverse of Linear Operators 117
u1 therefore, there existsu ˜1 ∈ U1, u˜1 1 1, so that foru ˜2 := ∈ u22 ⊥ (Ker T1) we have x T1u˜1 = = T1u˜2 , i.e.u ˜1 − u˜2 ∈ Ker T1. u2 2 Therefore,
− − 2 0= u˜1 u˜2, u˜2 1 = u˜1, u˜2 1 u˜2 1 − 2 − u˜1 1 u˜2 1 u˜2 1 = 1 u˜2 1 u˜2 1.
This is a contradiction.
Corollary C.0.6. Let T ∈ L(U, H) and set Q := TT∗ ∈ L(H).Thenwe have 1 − 1 −1 Im Q 2 =ImT and Q 2 x = T x U for all x ∈ Im T ,
− 1 1 where Q 2 is the pseudo inverse of Q 2 .
1 Proof. Since by Lemma 2.3.4 Q 2 is symmetric we have for all x ∈ H that 2 1 ∗ 1 2 ∗ ∗ 2 2 2 Q x = Q x = Qx, x = TT x, x = T x U .
Therefore the assertion follows by Proposition C.0.5. D. Some Tools from Real Martingale Theory
We need the following Burkholder–Davis inequality for real-valued continuous local martingales.
Proposition D.0.1. Let (Nt)t∈[0,T ] be a real-valued continuous local mar- tingale on a probability space (Ω,E,P) with respect to a normal filtration (Ft)t∈[0,T ]. Then for all stopping times τ( T )
| | 1/2 E(sup Nt ) 3E( N τ ). t∈[0,τ]
Proof. See e.g. [KS88, Theorem 3.28].
Corollary D.0.2. Let ε, δ ∈]0, ∞[. Then for N as in Proposition D.0.1
| | 3 1/2 ∧ 1/2 P (sup Nt ε) E( N T δ)+P ( N T >δ). t∈[0,T ] ε
Proof. Let { | 1/2 }∧ τ := inf t 0 N t >δ T.
Then τ( T )isanFt-stopping time. Hence by Proposition D.0.1
P sup |Nt| ε t∈[0,T ]
=P sup |Nt| ε, τ = T + P sup |Nt| ε, τ < T t∈[0,T ] t∈[0,T ] 3 1/2 | | 1/2 E( N τ )+P sup Nt ε, N T >δ ε t∈[0,T ] 3 E(N1/2 ∧ δ)+P (N1/2 >δ). ε T T
119 E. Weak and Strong Solutions: the Yamada-Watanabe Theorem
Let (Ω, F,P) be a complete probability space with normal filtration Ft,t∈ [0, ∞[. Below we shall call ((Ω, F,P,(Ft)) a stochastic basis.Letd, d1 ∈ N and let M(d × d1, R) denote the set of all real d × d1-matrices equipped with the norm (3.1.2). Let W d := C([0, ∞[ → Rd) (E.0.1) and d { ∈ d| } W0 := w W w(0) = 0 . (E.0.2) W d is equipped with metric ∞ −k d (w1,w2):= 2 ( max |w1(t) − w2(t)|∧1),w1,w2 ∈ W , (E.0.3) 0tk k=1 which makes it a Polish space. Its Borel σ-algebra is denoted by B(W d). d Let Bt(W ) denote the σ-Algebra generated by all maps πs, 0 s t, d d,d1 where πs(w):=w(s),w∈ W .LetA denote the set of all B([0, ∞[) ⊗ d d B(W )/B(M(d × d1, R))-measurable maps α :[0, ∞[ ×W → M(d × d1, R) such that for each t ∈ [0, ∞[ the map
d W w → α(t, w) ∈ M(d × d1, R)
d is Bt(W )/B(M(d × d1, R))-measurable.
E.1. The main result
Fix σ ∈Ad,d1 and b ∈Ad,1 and consider the following stochastic differential equation: dX(t)=b(t, X)dt + σ(t, X)dW (t),t∈ [0, ∞[. (E.1.1)
d Definition E.1.1. An R -valued continuous, (Ft)-adapted process X(t), t ∈ [0, ∞[, on some stochastic basis (Ω, F,P,(Ft)) is called a (weak) solution to (E.1.1), if
121 122 E. Weak and Strong Solutions: Yamada–Watanabe Theorem
(i) t |b(s, X)| ds<∞ P -a.e. for all t ∈ [0, ∞[. 0
(ii) t σ(s, X) 2 ds<∞ P -a.e. for all t ∈ [0, ∞[. 0
d1 (iii) There exists an R -valued standard (Ft)-Wiener process W (t),t ∈ [0, ∞[, on (Ω, F,P) such that P -a.e. t t X(t)=X(0)+ b(s, X)ds+ σ(s, X)dW (s),t∈ [0, ∞[. (E.1.2) 0 0
Remark E.1.2. (i) Clearly, by the measurability assumption on elements in Ad,d1 it follows that if X is a solution, then [0,t] × Ω (s, ω) → σ(s, X(ω)) is B([0,t])⊗F/B(M(d×d1, R))-measurable and σ(t, X) is Ft- measurable for t ∈ [0, ∞[. Likewise for b(·,X).The(Ft)-adaptedness for σ(·,X) and b(·,X) follows since the (Ft)-adaptiveness of X is equivalent d to the Ft/Bt(W ) measurability of X. (ii) Below we shall briefly say (X, W) in Definition E.1.1 is a (weak) solution to (E.1.1) not always mentioning explicitly the stochastic basis, that comes with it.
Definition E.1.3. We say that (weak) uniqueness holds for (E.1.1) if when- ever X and X are two (weak) solutions (with stochastic bases F F F F (Ω, ,P,( t)), (Ω , ,P , ( t)) and associated Wiener processes W (t), W (t),t∈ [0, ∞[) such that
P ◦ X(0)−1 = P ◦ X (0)−1,
(as measures on (Rd, B(Rd))), then
P ◦ X−1 = P ◦ (X )−1
(as measures on (W d, B(W d))).
Definition E.1.4. We say that pathwise uniqueness holds for (E.1.1), if whenever X and X are two (weak) solutions on the same stochastic basis (Ω, F,P,(Ft)) and with the same (Ft)-Wiener process W (t),t∈ [0, ∞[on (Ω, F,P) such that X(0) = X (0) P -a.e., then P -a.e.
X(t)=X (t),t∈ [0, ∞[.
To define strong solutions we need to introduce the following class Eˆ of maps: E.1. The main result 123
Eˆ Rd × d1 → d Let denote the set of all maps F : W0 W such that for every prob- µ⊗P W Rd B Rd B Rd ⊗B d1 B d ability measure µ on ( , ( )) there exists a ( ) (W0 ) / (W )- Rd × d1 → d ∈ Rd measurable map Fµ : W0 W such that for µ-a.e. x
W ∈ d1 F (x, w)=Fµ(x, w)forP -a.e. w W0 .
µ⊗P W B Rd ⊗B d1 B Rd ⊗B d1 Here ( ) (W0 ) denotes the completion of ( ) (W0 ) with respect to µ ⊗ P W ,andP W denotes classical Wiener measure on d1 B d1 (W0 , (W0 )). Let F ∈ Eˆ. For an F/B(Rd)-measurable map ξ :Ω→ Rd on some prob- ability space (Ω, F,P)andanRd1 -valued, standard Wiener process W (t), t ∈ [0, ∞[, on (Ω, F,P) independent of ξ,weset
F (ξ,W):=FP ◦ξ−1 (ξ,W).
Definition E.1.5. A (weak) solution X to (E.1.1) on (Ω, F,P,(Ft)) and associated Wiener process W (t),t∈ [0, ∞[, is called a strong solution if there P W ∈ Eˆ ∈ Rd → B d1 B d exists F such that for x ,w F (x, w)is t(W0 ) / t(W )- measurable for every t ∈ [0, ∞[and
X = F (X(0),W) P -a.e.,
P W B d1 W B d1 where t(W0 ) denotes the completion with respect to P in (W0 ). Definition E.1.6. Equation (E.1.1) is said to have a unique strong solution, if there exists F ∈ Eˆ satisfying the adaptiveness condition in Definition E.1.5 and such that:
d1 1. For every R -valued standard (Ft)-Wiener process W (t),t∈ [0, ∞[, on d a stochastic basis (Ω, F,P,(Ft)) and any F0/B(R )-measurable ξ :Ω→ Rd the continuous process
X := F (ξ,W)
satisfies (i), (ii) and (E.1.2) in Definition E.1.1, i.e. (F (ξ,W),W)isa (weak) solution to (E.1.1), and X(0) = ξ P-a.e.. 2. For any (weak) solution (X, W) to (E.1.1) we have
X = F (X(0),W) P -a.e..
Remark E.1.7. Since X(0) in the above definition is P -independent of W , thus P ◦ (X(0),W)−1 = µ ⊗ P W , we have that the existence of a unique strong solution for (E.1.1) implies that also (weak) uniqueness holds. 124 E. Weak and Strong Solutions: Yamada–Watanabe Theorem
Now we can formulate the main result of this section.
Theorem E.1.8. Let σ ∈Ad,d1 and b ∈Ad,1. Then equation (E.1.1) has a unique strong solution if and only if both of the following properties hold: (i) For every probability measure µ on (Rd, B(Rd)) there exists a (weak) solution (X, W) of (E.1.1) such that µ is the distribution of X(0). (ii) Pathwise uniqueness holds for (E.1.1). Proof. Suppose (E.1.1) has a unique strong solution. Then (ii) obviously holds. To show (i) one only has to take the classical Wiener space µ⊗P W d1 B d1 W Rd × d1 B Rd ⊗B d1 ⊗ W (W0 , (W0 ),P ) and consider ( W0 , ( ) (W0 ) ,µ P ) with filtration B Rd ⊗B d1 N σ( ( ) t+ε(W0 ), ),t 0, ε>0 µ⊗P W N ⊗ W B Rd ⊗B d1 where denotes all µ P -zero sets in ( ) (W0 ) .Let Rd × d1 → Rd Rd × d1 → d1 ξ : W0 and W : W0 W0 be the canonical projec- tions. Then X := F (ξ,W) is the desired weak solution in (i).
Now let us suppose that (i) and (ii) hold. The proof that then there exists a unique strong solution for (E.1.1) is quite technical. We structure it through a series of lemmas. Lemma E.1.9. Let (Ω, F) be a measurable space such that {ω}∈F for all ω ∈ Ω and such that D := {(ω, ω)|ω ∈ Ω}∈F⊗F (which is e.g. the case if Ω is a Polish space and F its Borel σ-algebra). Let P1,P2 be probability measures on (Ω, F) such that P1 ⊗ P2(D)=1.Then ∈ P1 = P2 = δω0 for some ω0 Ω. Proof. Let f :Ω→ [0, ∞[beF-measurable. Then
f(ω1)P1(dω1)= f(ω1)P1(dω1)P2(dω2)
= 1D(ω1,ω2)f(ω1)P1(dω1)P2(dω2)
= 1D(ω1,ω2)f(ω2)P1(dω1)P2(dω2)= f(ω2)P2(dω2), so P1 = P2. Furthermore,
1= 1D(ω1,ω2)P1(dω1)P2(dω2)= P1({ω2})P2(dω2),
{ } ∈ ∈ hence 1 = P1( ω2 )forP2-a.e. ω2 Ω. Therefore, P1 = δω0 for some ω0 Ω. E.1. The main result 125
Fix a probability measure µ on (Rd, B(Rd)) and let (X, W) with stochastic basis (Ω, F,P,(Ft)) be a (weak) solution to (E.1.1) with initial distribution µ. Rd× d× d1 B Rd ⊗B d ⊗B d1 Define a probability measure Pµ on ( W W0 , ( ) (W ) (W0 )), by −1 Pµ := P ◦ (X(0),X,W) .
∈ Rd ∈ d1 Lemma E.1.10. There exists a family Kµ((x, w),dw1),x ,w W0 ,of probability measures on (W d, B(W d)) having the following properties:
(i) For every A ∈B(W d) the map
Rd × d1 → W0 (x, w) Kµ((x, w),A)
B Rd ⊗B d1 is ( ) (W0 )-measurable.
B Rd ⊗B d ⊗B d1 Rd × d × (ii) For every ( ) (W ) (W0 )-measurable map f : W d1 → ∞ W0 [0, [ we have
f(x, w1,w)Pµ(dx dw1 dw) W = f(x, w1,w)Kµ((x, w),dw1)P (dw)µ(dx). Rd d1 d W0 W
d d (iii) If t ∈ [0, ∞[ and f : W → [0, ∞[ is Bt(W )-measurable, then Rd × d1 → W0 (x, w) f(w1)Kµ((x, w),dw1)
µ⊗P W µ⊗P W B Rd ⊗B d1 B Rd ⊗B d1 is ( ) t(W0 ) -measurable, where ( ) t(W0 ) de- ⊗ W B Rd ⊗B d1 notes the completion with respect to µ P in ( ) (W0 ).
Rd × d × d1 → Rd × d1 Proof. Let Π : W W0 W0 be the canonical projection. Since X(0) is F0-measurable, hence P -independent of W , it follows that
−1 −1 W Pµ ◦ Π = P ◦ (X(0),W) = µ ⊗ P .
Hence by the existence result on regular conditional distributions (cf. e.g. [IW81, Corollary to Theorem 3.3 on p. 15]), the existence of the family ∈ Rd ∈ d1 Kµ((x, w), dw1),x ,w W0 , satisfying (i) and (ii) follows. d To prove (iii) it suffices to show that for t ∈ [0, ∞[ and for all A0 ∈B(R ), ∈B d ∈B d1 A1 t(W ), A t(W0 )and { − ∈ − ∈ } A := πr1 πt B1,...,πrk πt Bk ,
d1 t r1 <... W 1A∩A (w)Kµ((x, w),A1)P (dw)µ(dx) A W d1 0 0 d · |B Rd ⊗B 1 W = 1A∩A (w)Eµ⊗P W (Kµ( ,A1) ( ) t(W0 ))P (dw)µ(dx), d1 A0 W0 (E.1.3) ∩ ∈B d1 B d1 since the system of all A A ,A t(W0 ),A as above generates (W0 ). But by part (ii) above, the left-hand side of (E.1.3) is equal to 1A0 (x)1A∩A (w)1A1 (w1)Pµ(dx dw1 dw) = 1A0 (X(0))1A1 (X)1A(W )1A (W )dP (E.1.4) |F = 1A0 (X(0))1A1 (X)1A(W )EP (1A (W ) t)dP. But 1A (W )isP -independent of Ft, since W is an (Ft)-Wiener process on (Ω, F,P), so EP (1A (W )|Ft)=EP (1A (W )). Hence the right-hand side of (E.1.4) is equal to W P (A ) 1A0 (x)1A(w)1A1 (w1)Pµ(dx dw1 dw) W W = P (A ) Kµ((x, w),A1)P (dw)µ(dx) A0 A W · |B Rd ⊗B d1 = P (A ) Eµ⊗P W (Kµ( ,A1) ( ) t(W0 ))((x, w)) A0 A P W(dw)µ(dx) d · |B Rd ⊗B 1 = 1A∩A (w)Eµ⊗P W (Kµ( ,A1) ( ) t(W0 ))((x, w)) d1 A0 W0 P W(dw)µ(dx), W B d1 since A is P -independent of t(W0 ). d For x ∈ R define a measure Qx on Rd × d × d × d1 B Rd ⊗B d ⊗B d ⊗B d1 ( W W W0 , ( ) (W ) (W ) (W0 )) by Qx(A):= 1A(z,w1,w2,w) d d1 d d R W0 W W W Kµ((z,w),dw1)Kµ((z,w),dw2)P (dw)δx(dz). E.1. The main result 127 Define the stochastic basis ˜ Rd × d × d × d1 Ω:= W W W0 Qx F˜x B Rd ⊗B d ⊗B d ⊗B d1 := ( ) (W ) (W ) (W0 ) F˜x B Rd ⊗B d ⊗B d ⊗B d1 N t := σ( ( ) t+ε(W ) t+ε(W ) t+ε(W0 ), x), ε>0 where x Nx := {N ∈ F˜ |Qx(N)=0}, and define maps d Π0 : Ω˜ → R , (x, w1,w2,w) → x, d d Πi : Ω˜ → W , (x, w1,w2,w) → wi ∈ W ,i=1, 2, ˜ → d1 → ∈ d1 Π3 : Ω W0 , (x, w1,w2,w) w W0 . Then, obviously, ◦ −1 Qx Π0 = δx (E.1.5) and ◦ −1 W ◦ −1 Qx Π3 = P (= P W ). (E.1.6) d Lemma E.1.11. There exists N0 ∈B(R ) with µ(N0)=0such that for all ∈ c F˜x ˜ F˜x x N0 we have that Π3 is an ( t )-Wiener process on (Ω, ,Qx) taking values in Rd1 . F˜x ∈ Rd Proof. By definition Π3 is ( t )-adapted for every x . Furthermore, for ∈ Rd ∈B Rd ∈B d ∈B d1 0 s d1 Rd W0 where we used Lemma E.1.10(iii) in the last step. Now the assertion follows by (E.1.6), a monotone class argument and the same reasoning as in the proof of Proposition 2.1.13. 128 E. Weak and Strong Solutions: Yamada–Watanabe Theorem d Lemma E.1.12. There exists N1 ∈B(R ), N0 ⊂ N1,withµ(N1)=0such ∈ c that for all x N1 , (Π1, Π3) and (Π2, Π3) with stochastic basis ˜ F˜x F˜x (Ω, ,Qx, ( t )) are (weak) solutions of (E.1.1) such that Π1(0) = Π2(0) = xQx-a.e., therefore, Π1 =Π2 Qx-a.e. x Proof. For i =1, 2 consider the set Ai ∈ F˜ defined by t t Ai := Πi(t) − Πi(0) = b(s, Πi)ds + σ(s, Πi)dΠ3(s) 0 0 for all t ∈ [0, ∞[ ∩{Πi(0) = Π0}. ∈BRd ⊗B d ⊗B d1 Define A ( ) (W ) (W0 ) analogously with Πi replaced by the Rd × d × d1 canonical projection from W W0 onto the second and Π0, Π3 by the canonical projection onto the first and third coordinate respectively. Then by Lemma E.1.10 (ii) for i =1, 2 1Ai (x, w1,w2,w) Rd d1 d d W0 W W W (E.1.7) Kµ((x, w),dw1)Kµ((x, w),dw2)P (dw)µ(dx) = Pµ(A)=P ({(X(0),X,W) ∈ A})=1. Since all measures in the left-hand side of (E.1.7) are probability measures, it follows that for µ-a.e. x ∈ Rd 1=Qx(Ai)=Qx(Ai,x), where for i =1, 2 t t Ai,x := Πi(t) − x = b(s, Πi)ds + σ(s, Πi)dΠ3(s), ∀t ∈ [0, ∞[ !. 0 0 Hence the first assertion follows. The second then follows by the pathwise uniqueness assumption in condition (ii) of the theorem. µ⊗P W B Rd ⊗B d1 B d Lemma E.1.13. There exists a ( ) (W0 ) / (W )-measurable map Rd × d1 → d Fµ : W0 W such that · Kµ((x, w), )=δFµ(x,w) d (= Dirac measure on B(W ) with mass in Fµ(x, w)) E.1. The main result 129 ⊗ W ∈ Rd × d1 for µ P -a.e. (x, w) W0 . Furthermore, Fµ is µ⊗P W B Rd ⊗B d1 B d ∈ ∞ ( ) t(W0 ) / t(W )-measurable for all t [0, [,where µ⊗P W B Rd ⊗B d1 ⊗ W ( ) t(W0 ) denotes the completion with respect to µ P in B Rd ⊗B d1 ( ) t(W0 ). ∈ c Proof. By Lemma E.1.12 for all x N1 ,wehave 1=Qx({Π1 =Π2}) W = 1D(w1,w2)Kµ((x, w),dw1)Kµ((x, w),dw2)P (dw), d1 d d W0 W W d d d where D := {(w1,w1) ∈ W × W |w1 ∈ W }. Hence by Lemma E.1.9 there ∈B Rd ⊗B d1 ⊗ W ∈ c exists N ( ) (W0 ) such that µ P (N) = 0 and for all (x, w) N d there exists Fµ(x, w) ∈ W such that Kµ((x, w),dw1)=δFµ(x,w)(dw1). d Set Fµ(x, w):=0,if(x, w) ∈ N.LetA ∈B(W ). Then c {Fµ ∈ A} =({Fµ ∈ A}∩N) ∪ ({Kµ(·,A)=1}∩N ) and the measurability properties of Fµ follow from Lemma E.1.10. Having defined the mapping Fµ let us check the conditions of Definition E.1.5 and Definition E.1.6. We start with condition 2. Lemma E.1.14. We have X = Fµ(X(0),W) P -a.e.. Proof. By Lemmas E.1.10 and E.1.13 we have P ({X = Fµ(X(0),W)}) W = 1{w1=Fµ(x,w)}(x, w1,w)δFµ(x,w)(dw1)P (dw)µ(dx) Rd d1 W d W0 =1. Rd1 F Now let us check condition 1. Let W be another -valued standard ( t)- F F → Rd Wiener process on a stochastic basis (Ω , ,P, ( t)) and ξ :Ω an F B Rd ◦ −1 0/ ( )-measurable map and µ := P ξ .LetFµ be as above and set X := Fµ(ξ,W ). 130 E. Weak and Strong Solutions: Yamada–Watanabe Theorem Lemma E.1.15. (X ,W ) is a (weak) solution to (E.1.1) with X (0) = ξ P -a.s.. Proof. We have P ({ξ = X (0)})=P ({ξ = Fµ(ξ,W )(0)}) ⊗ W { ∈ Rd × d1 | } = µ P ( (x, w) W0 x = Fµ(x, w) ) = P ({X(0) = Fµ(X(0),W)(0)})=1, where we used Lemma E.1.14 in the last step. To see that (X ,W ) is a (weak) solution we consider the set A ∈B(Rd) ⊗ B d ⊗B d1 (W ) (W0 ) defined in the proof of Lemma E.1.12. We have to show that P ({(X (0),X ,W ) ∈ A})=1. But since X (0) = ξ is P -independent of W ,wehave 1A(X (0),Fµ(X (0),W ),W )dP W = 1A(x, Fµ(x, w),w)P (dw)µ(dx) Rd W d1 0 W = 1A(x, w1,w)δFµ(x,w)(dw1)P (dw)µ(dx) Rd W d1 W d 0 = 1A(x, w1,w)Pµ(dx dw1 dw) =P ({(X(0),X,W) ∈ A})=1, where we used E.1.10 and E.1.11 in the second to last step. To complete the proof we still have to construct F ∈ Eˆ and to check the adaptiveness conditions on the corresponding mappings Fµ. Below we shall apply what we have obtained above now also to δx replacing µ.So,foreach ∈ Rd x we have a function Fδx . Now define ∈ Rd ∈ d1 F (x, w):=Fδx (x, w),x ,w W0 . (E.1.8) The proof of Theorem E.1.8 is then completed by the following lemma. d d Lemma E.1.16. Let µ be a probability measure on (R , B(R )) and Fµ : Rd × d1 → d ∈ Rd W0 W as constructed in Lemma E.1.13. Then for µ-a.e. x W F (x, ·)=Fµ(x, ·) P − a.e.. P W · B d1 B d ∈ Rd Furthermore, F (x, ) is t(W0 ) / t(W )-measurable for all x , P W ∈ ∞ B d1 B d1 t [0, [,where t(W0 ) denotes the completion of t(W0 ) with respect W d1 to P in B(W0 ). E.1. The main result 131 Proof. Let ¯ Rd × d × d1 Ω:= W W0 F¯ B Rd ⊗B d ⊗B d1 := ( ) (W ) (W0 ) d and fix x ∈ R . Define a measure Q¯x on (Ω¯, F¯)by W Q¯x(A):= 1A(z,w1,w)Kµ((z,w),dw1)P (dw)δx(dz) Rd d1 d W0 W ¯ F¯x ¯ F¯x with Kµ as in Lemma E.1.10. Consider the stochastic basis (Ω, , Qx, ( t )) where Q¯x F¯x := B(Rd) ⊗B(W d) ⊗B(W d1 ) , 0 F¯x B Rd ⊗B d ⊗B d1 N¯ t := σ( ( ) t+ε(W ) t+ε(W0 ), x), ε>0 x where N¯x := {N ∈ F¯ |Q¯x(N)=0}. As in the proof of Lemma E.1.12 one ¯ F¯x ¯ F¯x shows that (Π, Π3)on(Ω, , Qx, ( t )) is a (weak) solution to (E.1.1) with Π(0) = x Q¯x-a.e. Here Rd × d × d1 → Rd → Π0 : W W0 , (x, w1,w) x, Rd × d × d1 → d → Π: W W0 W , (x, w1,w) w1, Rd × d × d1 → d1 → Π3 : W W0 W0 , (x, w1,w) w. ¯ F¯x ¯ F¯x By Lemma E.1.15 (Fδx (x, Π3), Π3) on the stochastic basis (Ω, , Qx, ( t )) is a (weak) solution to (E.1.1) with Fδx (x, Π3)(0) = x. Hence by our pathwise uniqueness assumption (ii), it follows that ¯ Fδx (x, Π3)=Π Qx-a.s.. (E.1.9) ∈B Rd ⊗B d ⊗B d1 Hence for all A ( ) (W ) (W0 ) by Lemma E.1.13 and (E.1.9) W 1A(x, w1,w)δFµ(x,w)(dw1)P (dw)µ(dx) Rd W d W d1 0 = Q¯x(A)µ(dx) Rd ¯ = 1A(Π0,Fδx (x, Π3), Π3)dQxµ(dx) Rd ¯ Ω W = 1A(x, Fδ (x, w),w)P (dw)µ(dx) d x Rd W ! 0 = 1 (x, w ,w)δ (dw )P W(dw)µ(dx), A 1 Fδx (x,w) 1 Rd d1 d W0 W 132 E. Weak and Strong Solutions: Yamada–Watanabe Theorem which implies the assertion. d d Let x ∈ R ,t∈ [0, ∞[,A∈Bt(W ), and define ¯ Fδx := 1{ }× d1 Fδx . x W0 Then ¯ ⊗ W − Fδx = Fδx δx P a.e., hence W δx⊗P { ¯ ∈ }∈B Rd ⊗B d1 Fδx A ( ) t(W0 ) . (E.1.10) But { ¯ ∈ } { }×{ · ∈ }∪ Rd\{ } ×{ ∈ } Fδx A = x Fδx (x, ) A ( x ) 0 A , so by (E.1.10) it follows that P W { · ∈ }∈B d1 Fδx (x, ) A t(W0 ) . F. Strong, Mild and Weak Solutions This chapter is a short version of Chapter 2 in [FK01]. We only state the results and refer to [FK01], [DPZ92] for the proofs. As in previous chapters let (U, U ) and (H, ) be separable Hilbert spaces. We take Q = I and fix a cylindrical Q-Wiener process W (t), t 0, in U on a probability space (Ω, F,P) with a normal filtration Ft, t 0. Moreover, we fix T>0 and consider the following type of stochastic differential equations in H: dX(t)=[CX(t)+F (X(t))] dt + B(X(t)) dW (t),t∈ [0,T], (F.0.1) X(0) = ξ, where: • C : D(C) → H is the infinitesimal generator of a C0-semigroup S(t), t 0, of linear operators on H, • F : H → H is B(H)/B(H)-measurable, • B : H → L(U, H), • ξ is a H-valued, F0-measurable random variable. Definition F.0.1 (mild solution). An H-valued predictable process X(t), t ∈ [0,T], is called a mild solution of problem (F.0.1) if t X(t)=S(t)ξ + S(t − s)F (X(s)) ds 0 (F.0.2) t + S(t − s)B(X(s)) dW (s) P -a.s. 0 for each t ∈ [0,T]. In particular, the appearing integrals have to be well- defined. Definition F.0.2 (analytically strong solutions). A D(C)-valued pre- dictable process X(t), t ∈ [0,T], (i.e. (s, ω) → X(s, ω)isPT /B(H)- measurable) is called an analytically strong solution of problem (F.0.1) if t t X(t)=ξ + CX(s)+F (X(s)) ds + B(X(s)) dW (s) P -a.s. (F.0.3) 0 0 133 134 F. Strong, Mild and Weak Solutions for each t ∈ [0,T]. In particular, the integrals on the right-hand side have to be well-defined, that is, CX(t), F (X(t)), t ∈ [0,T], are P -a.s. Bochner integrable and B(X) ∈NW . Definition F.0.3 (analytically weak solution). An H-valued predictable process X(t), t ∈ [0,T], is called an analytically weak solution of problem (F.0.1) if t X(t),ζ = ξ,ζ + X(s),C∗ζ + F (X(s)),ζ ds 0 (F.0.4) t + ζ,B(X(s))dW (s) P -a.s. 0 for each t ∈ [0,T]andζ ∈ D(C∗). Here (C∗,D(C∗)) is the adjoint of (C, D(C)) on H. In particular, as in Definitions F.0.2 and F.0.1, the appearing integrals have to be well-defined. Proposition F.0.4 (analytically weak versus analytically strong so- lutions). (i) Every analytically strong solution of problem (F.0.1) is also an analyti- cally weak solution. (ii) Let X(t), t ∈ [0,T], be an analytically weak solution of problem (F.0.1) with values in D(C) such that B(X(t)) takes values in L2(U, H) for all t ∈ [0,T]. Besides we assume that T P CX(t) dt < ∞ =1 0 T P F (X(t)) dt < ∞ =1 0 T 2 ∞ P B(X(t)) L2 dt < =1. 0 Then the process is also an analytically strong solution. Proposition F.0.5 (analytically weak versus mild solutions). (i) Let X(t), t ∈ [0,T], be an analytically weak solution of problem (F.0.1) such that B(X(t)) takes values in L2(U, H) for all t ∈ [0,T]. Besides F. Strong, Mild and Weak Solutions 135 we assume that T P X(t) dt < ∞ =1 0 T P F (X(t)) dt < ∞ =1 0 T 2 ∞ P B(X(t)) L2 dt < =1. 0 Then the process is also a mild solution. (ii) Let X(t), t ∈ [0,T], be a mild solution of problem (F.0.1) such that the mappings t (t, ω) → S(t − s)F (X(s, ω)) ds 0 t (t, ω) → S(t − s)B(X(s)) dW (s)(ω) 0 have predictable versions. In addition, we require that T P ( F (X(t)) dt < ∞)=1 0 T t E( S(t − s)B(X(s)),C∗ζ 2 ds) dt < ∞ L2(U,R) 0 0 for all ζ ∈ D(C∗). Then the process is also an analytically weak solution. Remark F.0.6. The precise relation of mild and analytically weak solutions with the variational solutions from Definition 4.2.1 is obviously more difficult to describe in general. 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MR MR1033498 (91b:47002) Index Bochner inequality, 107 maximal inequality, 20 Bochner integral, 105 boundedness, 56 normal filtration, 16 Burkholder-Davis inequality, 119 operator coercivity, 56 Hilbert–Schmidt -, 110 conditional expectation, 17 nonnegative -, 109 covariance operator, 6 nuclear -, 109 symmetric -, 109 demicontinuity, 56 trace class -, 109 elementary process, 22 p-Laplacian, 65 existence predictable process, 28 - of a unique solution in finite predictable σ-field, 27 dimensions, 44 pseudo inverse, 115 fundamental theorem of calculus, quadratic variation of the stochas- 108 tic integral, 37 Gaussian Sobolev space, 62 -law,6 solution -measure,5 analytically strong -, 133 - random variable, 9 analytically weak -, 134 representation and existence - in finite dimensions, 44 of a -, 9 mild -, 133 Gelfand triple, 55 strong -, 123 weak -, 121 hemicontinuity, 56 square root of a linear operator, 25 Hilbert–Schmidt norm, 111 stochastic basis, 121 stochastic differential equation Itˆo isometry, 25 Burgers equation, 3 local weak monotonicity, 44 heat equation, 65 localization lemma, 45 - in finite dimensions, 44 localization procedure, 30 - in infinite dimensions, 55 Navier–Stokes equation, 3 martingale, 19 porous media equation, 3 141 142 Index reaction diffusion equation, 3, weak monotonicity, 56 67 Wiener process stochastic integral - as a martingale, 21 - w.r.t. a standard Wiener process, standard -, 13 22 representation and existence stochastically integrable process, 30 of a -, 13 strong measurability, 106 - with respect to a filtration, trace, 109 16 uniqueness Yamada–Watanabe pathwise -, 122 Theorem of -, 124 strong -, 123 weak -, 122 Symbols N(m, Q) Gaussian measure with mean m and covariance Q,6 W (t), t ∈ [0,T] standard Wiener process, 13 cylindrical Wiener process, 39 E(X|G) conditional expectation of X given G,18 M2 T (E) space of all continuous E-valued, square inte- grable martingales, 20 E class of all L(U, H)-valued elementary processes, 22 ΩT [0,T] × Ω, 21 dx Lebesgue measure, 21 PT dx|[0,T ] ⊗ P ,21 P T predictable σ-field on ΩT ,27 Φ(s)dW (s) stochastic integral w.r.t. W ,22 Lp(Ω, F,µ; X) set of all with respect to µp-integrable mappings from Ω to X, 106 Lp(Ω, F,µ) Lp(Ω, F,µ; R) p p F L0 L (Ω, 0,P; H) Lp([0,T]; H) Lp([0,T], B([0,T]), dx; H) Lp([0,T], dx) Lp([0,T], B([0,T]), dx; R) 2 2 P 0 T L -norm on L (ΩT , T ,PT ; L2), 25 N 2 2 P 0 W (0,T; H) L (ΩT , T ,PT ; L2), 28 N 2 N 2 W (0,T) W (0,T; H) N 2 N 2 W W (0,T; H) NW (0,T; H) space of all stochastically integrable processes, 30 NW (0,T) NW (0,T; H) L(U, H) space of all bounded and linear operators from U to H, 109 L(U) L(U, U) L1(U, H) space of all nuclear operators from U to H, 109 tr Q trace of Q, 109 L2(U, H) space of all Hilbert–Schmidt operators from U to H, 110 L2 Hilbert–Schmidt norm, 111 143 144 Symbols A∗ adjoint operator of A ∈ L(U, H) 1 Q 2 square root of Q ∈ L(U), 25 T −1 (pseudo) inverse of T ∈ L(U, H), 115 1 U0 Q 2 (U), 27 0 1 L2 L2(Q 2 (U),H), 27 − 1 − 1 u, v0 Q 2 u, Q 2 vU ,27 { | ∈ } L(U, H)0 TU0 T L(U, H) ,27 M(d × d1, R) set of all real d × d1-matrices, 43 (V,H,V ∗) Gelfand triple, 55 ∞ C0 (Λ) set of all infinitely differentiable real-valued func- tions on Λ with compact support, 62 ∞ 1,p norm on C0 (Λ), 62 1,p ∞ H0 (Λ) Sobolev space, completion of C0 (Λ) w.r.t. 1,p, 62 ∆p p-Laplacian, p 2, 65 W d C([0, ∞[→ Rd), 121 d { ∈ d | } W0 w W w(0) = 0 , 121 d d B(W ), Bt(W ) 121 Ad,d1 121 Eˆ 122 Lecture Notes in Mathematics For information about earlier volumes please contact your bookseller or Springer LNM Online archive: springerlink.com Vol. 1715: N. V. Krylov, M. Röckner, J. Zabczyk, Stochas- by Industrial Problems. Martina Franca, 1999. Editors: tic PDE’s and Kolmogorov Equations in Infinite Dimen- V. Capasso, H. Engl, J. Periaux (2000) sions. Cetraro, 1998. Editor: G. Da Prato (1999) Vol. 1740: B. Kawohl, O. Pironneau, L. Tartar, J.-P. Zole- Vol. 1716: J. Coates, R. Greenberg, K. A. Ribet, K. Ru- sio, Optimal Shape Design. Tróia, Portugal 1999. Editors: bin, Arithmetic Theory of Elliptic Curves. Cetraro, 1997. A. Cellina, A. Ornelas (2000) Editor: C. Viola (1999) Vol. 1741: E. Lombardi, Oscillatory Integrals and Phe- Vol. 1717: J. Bertoin, F. Martinelli, Y. Peres, Lectures nomena Beyond all Algebraic Orders (2000) on Probability Theory and Statistics. Saint-Flour, 1997. Vol. 1742: A. Unterberger, Quantization and Non- Editor: P. Bernard (1999) holomorphic Modular Forms (2000) Vol. 1718: A. Eberle, Uniqueness and Non-Uniqueness Vol. 1743: L. 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