Chapter 77

Stochastic Inclusions77.1 The General Context

The situation is as follows. There are spaces V ⊆W where V,W are reflexive separableBanach spaces. It is assumed that V is dense in W. Define the space for p > 1

V ≡ Lp ([0,T ] ;V )

where in each case, the σ algebra of measurable sets will be B ([0,T ]) the Borel measurablesets. Thus, from the Riesz representation theorem,

V ′ = Lp′ ([0,T ] ;V ′) ,We also assume (Ω,F ,P) is a complete probability space. That is, if P(E) = 0 and F ⊆ E,then F ∈F . Also

V ⊆W, W ′ ⊆V ′

B(ω) will be a linear operator, B(ω) : W →W ′ which satisfies

1. ⟨B(ω)x,y⟩= ⟨B(ω)y,x⟩

2. ⟨B(ω)x,x⟩ ≥ 0 and equals 0 if and only if x = 0.

3. ω → B(ω) is a measurable L (W,W ′) valued function.

In the above formulae, ⟨·, ·⟩ denotes the duality pairing of the Banach space W, with itsdual space. We will use this notation in the present paper, the exact specification of whichBanach space being determined by the context in which this notation occurs.

For example, you could simply take W = H = H ′ and B the identity and consider astandard Gelfand triple where H is a Hilbert space and B equal to the identity. An interest-ing feature is the requirement that B(ω) be one to one. It would be interesting to includethe case of degenerate B, but B one to one includes the case of most interest just mentioned.Also a more general set of assumptions will allow the inclusion of this case of degenerateB(ω) also.

We assume always that the norm on the various reflexive Banach spaces is strictlyconvex.

77.2 Some Fundamental TheoremsThe following fundamental result will be very useful. It says essentially that if (Bu)′ ∈Lp′ (0,T ;V ′) and u ∈ Lp (0,T ;V ) then the map u→ Bu(t) is continuous as a map from

X ≡{

u ∈ Lp ([0,T ] ;V ) : (Bu)′ ∈ Lp′ ([0,T ] ;V ′)}having norm equal to

∥u∥X ≡ ∥u∥Lp(0,T,V )+∥∥(Bu)′

∥∥Lp′ (0,T ;V ′)

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