2554 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS

being progressively measurable, into E, where E is a Hilbert space dense in V with ∥u∥E ≥∥u∥V , then there exists a unique solution to the integral equation

Bu(t)−Bu0 +∫ t

0A(s,u,ω)ds =

∫ t

0f (s,ω)ds+B

∫ t

0ΦdW

in the sense that u is in V and there exists a set of measure zero N such that if ω /∈ N, thenthe above integral equation holds for all t.

76.4 The General CaseSuppose λB+A(ω) ,λB+A are both monotone bounded and hemicontinuous on Vω andV respectively for λ sufficiently large. Also suppose the two estimates giving boundednessand coercivity 76.3.23 - 76.3.24 of Corollary 76.3.7 above. We strengthen the assumptionthat λB+A(ω) is monotone as follows. In the usual case where B is the identity, thisconclusion is obvious, but here we need to assume it.

⟨(λB+A(ω))(u)− (λB+A(ω))(v) ,u− v⟩ ≥ δ ∥u− v∥α

U , α ≥ 1 (76.4.26)

where here U is a reflexive Banach space such that V ⊆U and the inclusion map is con-tinuous, V being dense in U . In regards to this monotonicity condition, here is a simplelemma which will be used later.

Lemma 76.4.1 Suppose un→ w weakly in Vω and that for a.e.t,un (t)→ u(t) in U. Thenw(t) = u(t) a.e.

Proof: You know that ∥un∥Lp([0,T ],V ) is bounded. Now consider φ ∈ U ′ and ψ ∈C ([0,T ]) . Then the weak convergence implies

limn→∞

∫ T

0⟨φ ,un⟩U ′,U ψdt =

∫ T

0⟨φ ,w⟩U ′,U ψdt

because it is also the case that un → w weakly in Lp ([0,T ] ,U) . However, the fact that∥un∥Lp([0,T ],V ) is bounded means that, by the assumed pointwise convergence,

limn→∞

∫ T

0⟨φ ,un⟩U ′,U ψdt =

∫ T

0⟨φ ,u⟩U ′,U ψdt

It follows that ∫ T

0⟨φ ,u−w⟩ψdt = 0

Since this is true for all ψ ∈C ([0,T ]) , there exists a set of measure zero Qφ such that fort /∈ Qφ ,

⟨φ ,u(t)−w(t)⟩= 0

Letting Q = ∪φ∈DQφ , where D is a countable dense subset of U ′, it follows that for t /∈ Q,the above holds for all φ ∈U ′. Hence u(t) = w(t) for t /∈ Q and m(Q) = 0.

Typically α = 2 and U = W .