2538 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS

Note that, since X is progressively measurable into V, this implies that BX is progressivelymeasurable into W ′. Also W (t) is a JJ∗ Wiener process on U1 in the following diagram.(W is a cylindrical Wiener process.)

U↓ Q1/2

U1 ⊇ JQ1/2U J←1−1

Q1/2U

↓ Φ

W

We will also make use of the following generalization of familiar concepts from Hilbertspace.

Lemma 76.2.1 Suppose V,W are separable Banach spaces, W also a Hilbert space suchthat V is dense in W and B ∈L (W,W ′) satisfies

⟨Bx,x⟩ ≥ 0, ⟨Bx,y⟩= ⟨By,x⟩ ,B ̸= 0.

Then there exists a countable set {ei} of vectors in V such that⟨Bei,e j

⟩= δ i j

and for each x ∈W,

⟨Bx,x⟩=∞

∑i=1|⟨Bx,ei⟩|2 ,

and also

Bx =∞

∑i=1⟨Bx,ei⟩Bei,

the series converging in W ′.

Then in the above situation, we have the following fundamental estimate.

Lemma 76.2.2 In the above situation where, off a set of measure zero, 76.2.1 holds for allt ∈ [0,T ], and X is progressively measurable into V ,

E

(sup

t∈[0,T ]⟨BX ,X⟩(t)

)< C

(||Y ||K′ , ||X ||K , ||Z||J ,∥⟨BX0,X0⟩∥L1(Ω)

)< ∞.

where ⟨BX ,X⟩(t) = ⟨B(X (t)) ,X (t)⟩ a.e. and ⟨BX ,X⟩ is progressively measurable andcontinuous in t.

J = L2([0,T ]×Ω;L2

(Q1/2U ;W

)),K ≡ Lp ([0,T ]×Ω;V ) ,

K′ ≡ Lp′ ([0,T ]×Ω;V ′).