Homework Assignment #6
1: Although our treatment of Wiener’s homogeneous chaos decomposition makes no use of it,
Wiener’s own treatment rested on the connection between the decomposition of L2 (W; R) into
spaces of homogenious chaos and Hermite polynomials. In order to understand this connection,
x2
x2
recall the Hermite polynomials Hm (x) = (−1)m e 2 ∂xm e− 2 , which form an orthonormal basis in
L2 (γ0,1 ; R). Next, given L ≥ 1 and µ ∈ NL , define Hµ : RL −→ R so that
Hµ x1 , . . . , xL =
L
Y
Hµ` (x` ).
`=1
(i) Let L ≥ 1, and show that
kHµ k2L2 (γ0,I ;R)
= µ! ≡
L
Y
µj !
j=1
and that {Hµ : µ ∈ NL } is an orthogonal basis in L2 (γ0,I ; R).
(ii) Given ξ ∈ RL , show that
1
2
e(ξ,x)RL − 2 |ξ| =
X ξµ
Hµ (x),
µ!
L
µ∈N
where ξ µ ≡
QL
µ`
`=1 ξ` .
(iii) Suppose that {f1 , . . . , fL } is an orthonormal subset of L2 [0, ∞); RM , and, given µ ∈
PL
NL with m = kµk ≡ `=1 µ` , show that
(m)
I˜ ⊗µ1
f1
⊗µ
⊗···⊗fL L
(∞) = Hµ If1 (∞), . . . , IfL (∞) .
(iv) Let {gj : j ≥ 1} be an orthonormal basis in L2 [0, ∞); R , and, for α ∈ A = {α ∈ Nx :
kαk < ∞}, set S(α) = {j : αj ≥ 1} and
Y
1
.
Hα,j ≡ √
Hα` Ig(1)
j
α! j∈S(α)
show that, for each m ≥ 1, {Hα : kαk = m} is an orthonormal basis for Z (m) .