2584 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS

Lemma 76.7.1 Let Φ∈ L∞([0,T ]×Ω,L2

(Q1/2U,H

))then for any γ < 1/2, the stochas-

tic integral∫ t

0 ΦdW is Holder continuous with exponent γ .

To begin with, we consider a stochastic inclusion. Suppose, in the context of Theorem76.4.7, that V is a closed subspace of W σ ,p (U) ,σ > 1 which contains C∞

c (U) where U isan open bounded set in Rn, different than the Hilbert space U . (In case the matrix A whichfollows equals 0, it suffices to take σ ≥ 1.) Let

∑i, j

ai, j (x)ξ iξ j ≥ 0, ai j = a ji

where the ai, j ∈C1 (Ū). Denoting by A the matrix whose i jth entry is ai j, let

W ≡{

u ∈ L2 (U) :(

u, A1/2∇u)∈ L2 (U)n+1

}with a norm given by

∥u∥W ≡

(∫U

(uv+∑

i, jai j (x)∂iu∂ jv

)dx

)1/2

B : W →W ′ be given by

⟨Bu,v⟩ ≡∫

U

(uv+∑

i, jai j (x)∂iu∂ jv

)dx

so that B is the Riesz map for this space. The case where the ai j could vanish is allowed.Thus B is a positive self adjoint operator and is therefore, included in the above discussion.In this example, it will be significant that B is one to one and does not vanish.

This operator maps onto L2 (U) because of basic considerations concerning maximalmonotone operators. This is because

⟨Du,v⟩ ≡∫

U∑i, j

ai j (x)∂iu∂ jvdx

can be obtained as a subgradient of a convex lower semicontinuous and proper functionaldefined on L2 (U). Therefore, the operator is maximal monotone on L2 (U) which meansthat I +D is onto. The domain of D consists of all u ∈ L2 (U) such that

Du =−∑i, j

∂ j (ai j (x)∂iu) ∈ L2 (U)

along with suitable boundary conditions determined by the choice of V . It follows that ifu+Du = Bu = f ∈ H = L2 (U) , then

u−∑i, j

∂ j (ai j∂iu) = f