Segmentation of Stochastic Images using ... - Jacobs University

3.2 Stochastic Partial Differential Equations
domain in IR d , (Ω,A ,Π) a complete probability space, and a : Ω× ¯D → IR a stochastic function with
continuous and bounded covariance function that satisfies ∃a min ,a max ∈ (0,∞) with
P(ω ∈ Ω : a(x,ω) ∈ [a min ,a max ],∀x ∈ ¯D) = 1 , (3.8)
i.e. the diffusion coefficient is bounded away from zero and infinity for realizations ω ∈ Ω almost
sure. In addition, let f : Ω × ¯D → IR be a stochastic function that satisfies
∫ ∫
( ∫ )
f 2 (x,ω)dxdω = E f 2 (x,ω)dx < ∞ . (3.9)
Ω D
D
Then the elliptic SPDE analog to 3.7 reads
−∇ · (a(ω,·)∇u(ω,·)) = f (ω,·)
almost sure on D
u(·) = g(·) on ∂D .
Applying this concept to other PDEs yields parabolic and hyperbolic SPDEs.
3.2.1 Existence and Uniqueness of Solutions for Elliptic SPDEs
(3.10)
The proof of the existence and uniqueness of solutions for elliptic SPDEs is closely related to the
existence and uniqueness proof of the classical problem. The Lax-Milgram theorem [37] is applicable
in the stochastic context when we show continuity and coercivity of the related linear and
bilinear forms. The main difficulty of the proof is that the stochastic PDE requires the multiplication
of stochastic quantities, because the expression a∇u has to be well-defined. For this, we introduce
the Wick product [68, 155] and have to investigate conditions for its existence. Let us begin with
notation for the definition of the Wick product. The presentation is based on [150]. In the following
let {H α : α ∈ I}, where I is an index set, be an orthogonal basis of L 2 (Ω).
Definition The Wick product of two random variables f ,g : Ω → IR is the formal series
(
f g = ∑ f α g β H α+β (ξ ) = ∑ ∑ f α g β
)H γ (ξ ) , (3.11)
α,β
γ α+β=γ
whereas a random variable is expressed in the orthogonal basis via f = ∑ α f α H α (ξ ). The H α depend
on a vector ξ = (ξ 1 ,...) of basic random variables.
The Wick product is not well-defined for all second order random variables, i.e. L 2 (Ω) is not closed
under Wick multiplication (see [150]). Therefore, we introduce restrictions of the space L 2 (Ω) to
ensure a well-defined Wick multiplication.
Definition The Kondratiev-Hilbert spaces S ρ,k [85] are
{
}
(S ) ρ,k := f = ∑ α
f α H α : f α ∈ IR for α ∈ I and ‖ f ‖ ρ,k < ∞
where −1 ≤ ρ ≤ 1 and k ∈ IR. We define the norm ‖ · ‖ ρ,k via the scalar product
and the expression (2N) α via
, (3.12)
( f ,g) ρ,k := ∑ α
f α g α (α!) 1+ρ (2N) αk (3.13)
(2N) α :=
∞
∏
i=1
(2 d β (i)
1 β (i)
2
...β
(i)
d )α i
. (3.14)
The product ∏ ∞ i=1 is the product over all possible multi-indices β. Kondratiev spaces are separable
Hilbert spaces [150].
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