final report - probability.ca

ˆ By using closeness of drift function (3.13) with µ (x), and z N
Ŵ N ⇒ W as N → ∞ such that
with ¯z N , it follows there exists a process
)
so z N = Θ
(x 0,N , Ŵ N .
z N (t) = x 0,N + h (l)
ˆ t
0
µ ( z N (u) ) du + √ 2h (l)Ŵ N (t) ,
)
– Since Θ is continuous, it follows z N = Θ
(x 0,N , Ŵ N ⇒ Θ ( z 0 , W ) = z as N → ∞.
A
Infinite Dimensional Analysis
A.1 Introduction
There is no natural analogue of the Lebesgue measure on an infinite dimensional Hilbert space. A natural substitute
is provided by Gaussian measures. The theory expounded in these notes define Gaussian measures on a finite
dimensional space, and then through an infinite product of measures, on the infinite dimensional Hilbert space H,
assumed to be separable.
Later we talk about the Cameron-Martin formula, which is an important tool discussing absolute continuity and
singularity of a Gaussian measure and its translates.
We consider first Gaussian measure on (H, B (H)), where H is a real separable Hilbert space with inner product
〈·, ·〉 and norm ‖·‖, and B (H) is the Borel sigma algebra on H. Let L (H) denote the Banach algebra of all continuous
linear operators on H, L + (H) denote the set of all T ∈ L (H) that are symmetric (〈T x, y〉 = 〈x, T y〉 , ∀x, y ∈ H)
and positive (〈T x, x〉 ≥ 0, ∀x ∈ H), and finally L + 1 (H) the set of all operators Q ∈ L+ (H) of trace class, i.e.,
operators Q such that trace (Q) := ∑ k≥1 〈Qe k, e k 〉 < ∞ for all complete orthonormal systems {e k } k≥1
∈ H.
A.1.1
Infinitely Divisible Laws
The law of a random variable X is the probability measure X # P on (E, B (E)) defined as
X # P (I) = P ( X −1 (I) ) = P (X ∈ I) , I ∈ B (E) .
The convolution ν 1 ⋆ ν 2 of two finite Borel measures ν 1 and ν 2 on R N is given by
ν 1 ⋆ ν 2 (Γ) =
ˆR N ˆ
R N I Γ (x + y) ν 1 (dx) (dy) , Γ ∈ B R N ,
and the distribution of the sum of two independent random variables is the convolution of their distributions. We
want to describe the probability measure µ that, for each n ≥ 1, can be written as the n-fold convolution power
µ ⋆n 1 of some probability measure µ 1 .
n
n
Recall that the Fourier transform takes convolution into ordinary multiplication, the Fourier formulation for
infinite divisibility involves describing those Borel probability measures on R N whose Fourier transform ˆµ has, for
each n ∈ Z + , an n th root which is again the Fourier transform of a Borel probability measure on R N .
A.2 Gaussian Measures in Hilbert Spaces
A.2.1
One-dimensional Hilbert spaces
Let us first consider the well-known case where H is a one-dimensional Hilbert space. Consider (a, λ) ∈ R 2 with
a ∈ R and λ ∈ R + . We define a probability measure N a,λ in (R, B (R)) as follows. If λ = 0, we set
where δ a is the Dirac measure at a,
N a,0 = δ a ,
δ a (B) = I {a∈B} ,
for B ∈ B (R). This is a degenerate Gaussian measure, and is not of much use to us for our applications.
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