Let ¯µ be a compensated Poisson measure on [0, T ] × Z and define U0 :=
L2(Z, Z, m). Let U be a separable Hilbert space such that the embedding U0 ,→ U is dense and Hilbert-Schmidt.
Theorem 2.24. The compensated Poisson random measure ¯µ on [0, T ] × Z can be identified with a square integrable martingale Mµ¯ on U such that
U0 = Q
1 2
Mµ¯(U ), where QMµ¯ is the covariance operator of Mµ¯. Furthermore,
Mµ¯ is a L´evy process.
Proof. See [PZ07, Theorem 7.28].
The following Proposition identifies the Poisson integral as a stochastic in- tegral with respect to a square integrable martingale .
Proposition 2.25. Let Φ ∈ Nµ¯2(T, Z; H) and Int(Φ)(t), t ∈ [0, T ], be the stochastic integral in (2.7). Define
IΦH(t)(ϕ) := Z Z Φ(t, z)ϕ(z) m(dz), ϕ ∈ L2(Z, Z, m). Then IΦH ∈ L2 ¯
µ,T(H) (cf. Definition in Theorem 2.8) and
Int(Φ)(t) = Z t
0
IΦH(s) dMµ¯(s), t ∈ [0, T ].
Proposition 2.25 can be deduced from the real-valued case: Proposition 2.26. Let Φ ∈ Nµ¯2(T, Z; R). Then IR
Φ ∈ L2µ,T¯ (R) (where IΦR is
defined in Proposition 2.25) and Int(Φ)(t) =
Z t 0
IR
Φ(s) dMµ¯(s), t ∈ [0, T ].
Proof. See [PZ07, Proposition 8.24].
Proof of Proposition 2.25. Let {en}n∈N be a orthonormal basis of U0 =
L2(Z, Z, m). Then, by the Parseval equality IΦH L2(U0,H)= ∞ X n=1 IΦH(en)) 2 H = ∞ X n=1 Z Z Φ(·, z)en(z)m(dz) 2 H ≤ ∞ X n=1 Z Z kΦ(·, z)kH|en(z)| m(dz) = ∞ X n=1 hkΦkH, eniL2(Z,Z,m) = kΦk2L2(Z,Z,m;H)< ∞ P ⊗ dt-a.s.. (2.9)
Furthermore, from (2.9) it follows that
E Z T 0 IΦH(s) L2(U0,H) ds ≤ E Z T 0 Z Z kΦ(s, z)k2H m(dz) ds < ∞, since Φ ∈ N2 ¯
µ(T, Z, H). Since IΦH is predictable, we have proved that
IH
Φ ∈ L2µ,T¯ (H).
Now, let {en}n∈N be an orthonormal basis of H. Then for Φ ∈ Nµ¯2(T, Z, H),
we have hΦ, eniH ∈ Nµ¯2(T, Z; R) for all n ∈ N. Thus, by Proposition 2.26,
Int(hΦ, eniH)(t) =
Z t
0
IR
Using this together with Proposition 2.5(v), Proposition 2.20 and the dom- inated convergence theorem, we deduce
Int(Φ)(t) = Z t 0 Z Z Φ(s, z) ¯µ(ds, dz) = Z t 0 Z Z ∞ X n=1 hΦ(s, z), eniHenµ(ds, dz)¯ = ∞ X n=1 Z t 0 Z Z hΦ(s, z), eniH µ(ds, dz)¯ ! en = ∞ X n=1 Z t 0 IR hΦ,eniH(s) dMµ¯(s) ! en = ∞ X n=1 Z Z Z t 0 hΦ(s, z), eniH(s) dMµ¯(s) m(dz) ! en = Z Z Z t 0 Φ(s, z) dMµ¯(s) m(dz) = Z t 0 IΦH(s) dMµ¯(s).
Chapter 3
Maximal Monotone
Operators on Banach Spaces
In this chapter, we introduce the general analytic framework needed to study multivalued differential equations with maximal monotone drift. We are go- ing to define maximal monotone operators on Banach spaces, introduce its Yosida approximation and prove some necessary properties. Furthermore, we consider the measurability of a multivalued operator and the measura- bility of the Yosida approximation of a maximal monotone operator. This chapter is mainly based upon [Bar93] and [Bar10].
General notions for multivalued maps are gathered in Appendix A. For all unexplained concepts in the theory of nonlinear operators on Banach spaces, we refer to Appendix B. Throughout this chapter, let X be a Banach space and X∗ its dual space. Let G(A) denote the graph of the operator A. Definition 3.1. i. A multivalued operator A : X → 2X∗ is said to be
monotone if
X∗hy1− y2, x1− x2iX ≥ 0, ∀[xi, yi] ∈ G(A), i = 1, 2. (3.1)
ii. A monotone operator A : X → 2X∗ is said to be maximal monotone if there exists no other proper monotone extension ˜A of A, i.e.
G(A) ( G( ˜A).
Proposition 3.2. Let A be maximal monotone. Then:
i. A is weakly-strongly closed in X × X∗, i.e. if [xn, yn] ∈ G(A), xn→ x
weakly in X and yn→ y strongly in X∗, then [x, y] ∈ G(A).
ii. A−1 is maximal monotone in X∗× X.
Proof. See [Bar93, Section 2.1, Proposition 1.1] .
Under certain circumstances, a maximal monotone operator is even weakly- weakly closed as the following proposition shows.
Proposition 3.3. Let X be a reflexive Banach space and let A : X → 2X∗ be a maximal monotone operator. Let [un, vn] ∈ G(A), n ∈ N, be such that
un* u, vn* v, and either lim sup n,m→∞ X ∗hvn− vm, un− umiX ≤ 0 or lim sup n→∞ X ∗hvn, uniX ≤ X∗hv, uiX.
Then [u, v] ∈ G(A).
Proof. See [Bar10, Lemma 2.3, p.38] and [Bar10, Corollary 2.4, p.41]. We will make use of the following characterizations of maximal monotonicity. Theorem 3.4. Let X be a reflexive Banach space and let A : X → X∗ be a (single-valued) monotone hemicontinuous operator. Then A is maximal monotone in X × X∗.
Proof. See [Bar93, Section 2.1, Theorem 1.3].
Theorem 3.5. Let X be a reflexive Banach space and let A and B be maximal monotone operators from X to 2X∗ such that
(int D(A)) ∩ D(B) 6= ∅. Then A + B is maximal monotone in X × X∗. Proof. See [Bar93, Section 2.1, Theorem 1.5].
Corollary 3.6. Let X be a reflexive Banach space, B be a monotone hemi- continuous operator from X to X∗and A : X → 2X∗ be a maximal monotone operator. Then A + B is maximal monotone.
Proof. Apply Theorem 3.4 and Theorem 3.5.
Proposition 3.7. Let X be a reflexive Banach space and let A be a coer- cive, maximal monotone operator from X to X∗. Then A is surjective, i.e. R(A) = X∗.
Proof. See [Bar93, Section 2.1, Corollary 1.2].
We are especially interested in the selection of a maximal monotone operator with respect to its minimal norm:
Definition 3.8. The minimal selection A0: D(A) ⊂ X → 2X∗ of a maximal monotone operator A is defined by
A0(x) := y ∈ A(x) kyk = min z∈A(x) kzk , x ∈ D(A).
Remark 3.9. If X is strictly convex, then A0 is single-valued.
Proof. Let x ∈ D(A). Assume that y1, y2 ∈ A0(x), y1 6= y2. Define δ :=
A0(x)
. If δ = 0, then y1 = y2 = 0. Thus, δ > 0. By Proposition
3.2.iii), A(x) is a closed, convex set. Hence, 12(y1+ y2) ∈ A(x) which implies
12(y1+ y2)
≥ δ. On the other hand, for ˜y1 := 1δy1 and 1δy2, we have k˜y1k = k˜y2k = 1. Since X is strictly convex, it follows that 1 > 12k˜y1+ ˜y2k =
1
2δky1+ y2k , which is a contradiction. Hence, y1 = y2.
3.1
The Duality Mapping
The duality mapping as a map from X to X∗ represents an important aux- iliary tool in the theory of maximal monotone operators on Banach spaces. Definition 3.10. The duality mapping J : X → 2X∗ is defined by
J (x) := {x∗∈ X∗|X∗hx∗, xiX = kxk2 = kx∗k2} ∀x ∈ X.
Remark 3.11. By the Hahn-Banach theorem, for every x ∈ X there exists x∗0 ∈ X∗ such that kx∗
0k = 1 and X∗hx∗0, xiX = kxk. Setting u := kxk x∗0,
it follows that X∗hu, xiX = kxk2 = kx∗0k kxk = kuk2. Therefore, u ∈ J (x)
and, indeed, D(J ) = X.
The properties of the duality mapping are closely related to the convexity of the underlying space. In general, the duality mapping is multivalued. But the following theorem is valid:
Theorem 3.12. Let X be a Banach space. If X∗ is strictly convex, then the duality mapping J : X → X∗ is single-valued.
Proof. See [Bar93, Chapter 1, Theorem 1.2].
Now, we want to state some features of the duality mapping. Proposition 3.13. Let X and X∗ be uniformly convex. Then:
i. The duality map J : X → X∗ is linearly bounded, 2-coercive, continu- ous and odd.
ii. The operator J is bijective and if we identify X∗∗ with X, the inverse operator
J−1 : X∗→ X
iii. J is strictly monotone, i.e. it is monotone and
X∗hJ u − J v, u − viX = 0 ⇒ u = v.
Proof. See [Zei90b, Proposition 32.22].
The following fundamental result in the theory of maximal monotone opera- tors due to G. Minty and F. Browder provides a very useful characterization of maximal monotonicity.
Theorem 3.14. Let X and X∗be reflexive and strictly convex. Let A : X → 2X∗ be a monotone operator and let J : X → X∗ be the duality mapping of X. Then A is maximal monotone if and only if, for any λ > 0 (equivalently, for some λ > 0),
R(A + λJ ) = X∗. Proof. See [Bar93, Section 2.1, Theorem 1.2].
Corollary 3.15. If A is maximal monotone, then µA is maximal monotone for all µ > 0.
Proof. For a fixed µ > 0, we set Aµ := µA and take x, y ∈ D(A). For vµ ∈
Aµ(x), there exists v ∈ A(x) such that µv = vµ. Since A is monotone, for x, y ∈ D(A) and vµ ∈ Aµ(x), wµ = Aµ(y) we have
X∗hvµ− wµ, x − yiX =
µ2
X∗hv − w, x − yiX ≥ 0. By the maximal monotonicity of A and Theorem
3.14 with λ := 1, we conclude that R(µA + µJ ) = µR(A + J ) = X∗. Remark 3.16. Let us emphasize that every uniformly convex Banach space is automatically strictly convex and reflexive (cf. Remark B.8 and Proposi- tion B.9). Consequently, all results above do also hold for uniformly convex Banach spaces.