Chapter 76

Implicit Stochastic Equations76.1 Introduction

In this chapter, implicit evolution equations are considered. These are of the form

Bu(t,ω)−Bu0 (ω)+∫ t

0A(s,u(t,ω) ,ω)ds =

∫ t

0f (s)ds+B

∫ t

0ΦdW

the term on the end being a stochastic integral. The novelty is in allowing B to be anoperator which could vanish or have other interesting features. Thus the integral equationcould degenerate to a non stochastic elliptic equation. This generalization of evolutionequations has proven useful in the study of deterministic evolution equations and we givesome interesting examples which indicate that this may be true in the case of stochasticequations also. In any case, it is an interesting generalization and equations of the usualform are recovered by using a Gelfand triple in which B = I.

Like deterministic equations, there are many ways to consider stochastic equations.Here it is based on an approach due to Bardos and Brezis [14] which avoids the consider-ation of finite dimensional problems. A generalized Ito formula is summarized in the nextsection. It is Theorem 76.2.3.

76.2 Preliminary ResultsLet X have values in W and satisfy the following

BX (t) = BX0 +∫ t

0Y (s)ds+B

∫ t

0Z (s)dW (s) , (76.2.1)

X0 ∈ L2 (Ω;W ) and is F0 measurable, where Z is L2(Q1/2U,W

)progressively measurable

and∥Z∥L2([0,T ]×Ω,L2(Q1/2U,W)) < ∞.

This is what is needed to define the stochastic integral in the above formula. Here Q isa nonnegative self adjoint operator defined on a separable real Hilbert space U . In whatfollows, J will denote a one to one Hilbert Schmidt operator mapping Q1/2U into anotherseparable Hilbert space U1. For more explanation on this situation see [108].

Assume X ,Y satisfy

X ∈ K ≡ Lp ([0,T ]×Ω;V ) ,Y ∈ K′ = Lp′ ([0,T ]×Ω;V ′)

where 1/p′+1/p = 1, p > 1, and X ,Y are progressively measurable into V and V ′ respec-tively.

The sense in which the equation 76.2.1 holds is as follows. For a.e. ω, the equationholds in V ′ for all t ∈ [0,T ]. Assume that

X ∈ L2 ([0,T ]×Ω,W ) ,

BX ∈ L2 ([0,T ]×Ω,B ([0,T ])×F ,W ′), X ∈ Lp ([0,T ]×Ω,B ([0,T ])×F ,V )

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