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GAUSSIAN MEASURE vs LEBESGUE

MEASURE AND

ELEMENTS OF MALLIAVIN CALCULUS

λ : Lebesgue measure has the following properties :

1. For every non-empty open set U , λ(U) > 0 ;

2. For every bounded Borel set K , λ(K)

H : a separable Hilbert space with an orthonormal basis {h1, h2, · · · } , ν

: a Borel measure in H

Let B 1 2

(hn) be the open ball of radius half, centered at hn ,

similarly B = B2(0) ;

By 1), 2) and 3) 0 < ν(B 1 2

(h1)) = ν(B 1 2

(h2) = · · · · · · < ∞ ,

but

ν(B2(0)) ≥ ∑

n ν(B 1 2

(hn) = ∞ violates 2).

Gaussian measure in Rn is absolutely continuous with respect

to the Lebesgue measure in n dimensions with Radon-Nikodym

derivative (density)

Φ(x) = 1

(2π)n/2 √ DetS

e− 1

2(S−1(x−m),x−m)), x ∈ Rn

m : Mean vector , S : Covariance matrix (symmetric, positive

definite) ;

2

Standard Gaussian Measure : m = 0 , S = I −→

Φ(x) = 1 (2π)n/2

e− ‖x‖2

2 x ∈ Rn .

Another way of outlook at the density : random element in Rn

∃(Ω,F , P ) and a measurable mapping X : Ω → Rn induces a

prob. measure µ in Rn : (assumed to be absolutely continuous)

µ(B) = P (X−1(B)) = ∫

B Φ(x)dx ; B ∈ Bn

Standard Gaussian measure in finite dimensions is rotationally

invariant : ∫

A−1B Φ(x)dx = ∫

B Φ(x)dx .

Gauss distribution in finite dimensions can also be given by its

Fourier transformation (characteristic functional) :

χ(f) =

∫

Rn

ei(f,x)Φ(x)dx, f ∈ Rn ,

χ(f) = ei(f,m)− 1

2 (Sf,f) f ∈ Rn

For n = 1 , f = t , χ(t) = eimt− σ2t2

2 .

3

The char. fn of random variable (f, x), f, x ∈ Rn) :

χf(t) = ∫

Rn eit(f,x)Φ(x)dx = χ(tf) = eit(f,m)−

t2

2 (Sf,f) , (∗)

thus Gauss (in one dimension), converse is also true, i.e. if (f, .)

is one dimensional Gauss for all f ∈ Rn, then the measure in

R n is Gaussian (put t = 1 in (*) ). Hence

µ ∈ Rn is Gaussian ⇐⇒ (f, .) has one dmnl Gaussian dist

forall f ∈ Rn

Take this point of view to define Gaussian measure in infinite

dimensions :

-A measure on a Hilbert space H is Gaussian ⇐⇒ (x, .) is a

Gaussian r.v. ∀x ∈ H

-A measure on a Banach space X is Gaussian ⇐⇒ (f, .) is a

Gaussian r.v. ∀f ∈ X∗ .

4

Two Problems : A) How to characterize the Fourier trans-

form of a finite measure

B) How to characterize the Fourier transform of Gaussian mea-

sures ?

Theorem 1 (Bochner) In Rn a functional µ̂(f) is the

Fourier transform of a finite measure ⇐⇒ µ̂(0) = µ(Rn),

continuous and positive definite.

For infinite dimensional Hilbert spaces these are not sufficient ,

e.g. e− 1

2 ‖x‖2 satisfy the conditions but not the Fourier trans-

form µ̂(x) = ∫

H e i(x,y)dµ(y) x ∈ H of any finite Borel

measure on the Hilbert space . (Otherwise for an orthonormal basis

{hn} , ∫

H ei(hn,y)dµ(y) = e−

1

2 which is not compatible with (hn, y) → 0

as ∑

n(y, hn) 2 = ‖y‖2

i) φ is the Fourier transform of a finite Borel measure on H ,

ii) ∀ǫ > 0 , ∃ a symmetric, trace class operator Sǫ , such that

(Sǫx, x) < 1 =⇒ Re(φ(0)− φ(x) < ǫ ,

iii) ∃ a symmetric, trace class operator S on H , such that

φ is continuous (or continuous at x = 0) w.t. the norm

‖ . ‖∗ defined by ‖x‖∗ .= (Sx, x)1/2 = ‖S1/2x‖.

If µ is a Borel prob. measure then the mean vector is a vector

m ∈ H satisfying

(m, x) = ∫

H(x, z)dµ(z) .

If ∫

H ‖x‖ dµ(x)

Lemma 1. If µ is a finite Borel measure, then

∫

H ‖x‖2 dµ(x)

In fact ;

Theorem 3. A Borel probability measure µ on H is Gaussian

⇐⇒ Its Fourier transform can be expressed as

µ̂(x) = exp {i(m, x)− 1 2 (Bx, x)} , ∀m, x ∈ H .

In a Banach space X the definition of the mean vector is the

same. If we use the random element outlook, the mean vector

m ∈ X can be given by a Pettis integral

< m, f > . = ∫

Ω < f, x(ω) > dP (ω) , ∀f ∈ X∗ . (†)

Thus a necessary condition for the existence of m ∈ X is that

< f, x(.) >∈ L1(Ω,F , P ) ∀f ∈ X∗ . If the necessary condi-

tion is satisfied then the r.h.s. of (†) defines a linear , continuous

function of f , i.e. an element of X∗∗.

8

Hence if m exists m ∈ X ∩ X∗∗, (X →֒ X∗∗) . Therefore for

the reflexive spaces , the necessary condition is also sufficient.

In the non-reflexive case there are extra sufficient conditions.

Leaving them out we know that for Gaussian distribution m

always exists.

The covariance operator in X is defined through an S-operator

S : X∗ → X∗∗ , i.e.

>= ∫

X < f, z >< g, z > dµ(z), f, g ∈ X∗ and

Bf = Sf− < m, f > m (Covariance operator).

Operator S has properties akin to operators S : H → H e.g.

:

i) Symmetry : >=> ;

ii) Positivity : >≥ 0, ∀f ∈ X∗ .

9

The existence of the covariance operator in a Banach space is

characterized in terms of the nuclearity of S or B but a mod-

ified definition of nuclearity is needed in Banach spaces. Let

H(X) : all symmetric, non-negative , bounded linear mappings

X∗ → X∗∗ ;

H1(X) : the class of covariance operators of distributions on

X ;

H2(X) : the classs of covariance operators of all Gaussian dis-

tributions on X. (H2(X) ⊂ H1(X) ⊂ H(X))

If X is separable and reflexive then H1(X) = H(X) . Under the

same conditions,for S ∈ H(X) , nuclearity in the first sense is

sufficient and the nuclearity in the second sense is necessary for

S to be in H2 , (For nuclearity of different senses c.f. [Vakha-

nia]).

10

If we consider for simplicity m = 0 ,

χ(f ;µ) = exp {−12 >} ∀f ∈ X∗ ( √ )

is the Fourier transform of some Gaussian measure in X

⇐⇒ S ∈ H2 .

Standard Gaussian Cylinder Measures

The quasi-invariance property which is so important in the dif-

ferential analysis in infinite dimensions is possessed by the stan-

dard Gaussian measure where m = 0 and S = I. However

S = I does not make sense in ( √ ) as an operator X∗ → X∗∗ ,

and in the Hilbert case I is not nuclear (or trace class ) and

χ(f) = e− 1

2 ‖f‖2 obtained by B = I can not be the charac-

teristic functional of a standard type Gaussian distribution in

X. Therefore an approach via finite dimensional subspaces is

essential.

11

Let F(X∗) : be the class of all finite dimensional subspaces of

X∗ ;

For any K ∈ F(X∗) call

D = {x ∈ X : (< x, y1 >,< x, y2 >, · · · , < x, yn >) ∈ E}

a cylinder set based on K if E ∈ Bn, y1, · · · , yn ∈ K.

Let ℜ(X) = ⋃

K∈F(X∗) C(K) , where C(K) is the σ-algebra

generated by cylinder sets with base K.

ℜ(X) is only an algebra , but for a separable Banach space

X, σ(ℜ(X)) = BX (the Borel σ-alg. in X).

A non-negative set fct µ on ℜ(X) is called a ”cylinder prob-

ability measure” on X , if µ(X) = 1 and a measure when

restricted to any C(K) for K ∈ F(X∗) . A cylinder probability

measure is necessarily compatible.

A real (complex)-valued fct on X is a cylinder fct if it is mea-

surable w.t. C(K) for some K.

µ̂(f) = ∫

X e idµ(x) , f ∈ X∗ : the characteristic fnl of

the cylinder measure µ .

12

Question : What kind of cylinder measures can be extended

to BX ?

Answer in a Special Case (Important for Malliavin calculus)

X is the completion of some Hilbert space H w.t. a weaker

norm and the cylinder measure on X is lifted from that on H.

Definition 1. (Gross) Let (H, | . |) be a Hilbert space , µ

a cylindrical measure on H, ‖ . ‖ another norm on H weaker

than | . | If for any ǫ > 0 ∃ πǫ ∈ P ( the set of all finite

dimensional orthogonal projections on H ), such that for any

π ∈ P (π ⊥ πǫ) one has µ{x ∈ H : ‖πx‖ > ǫ} < ǫ then ‖ . ‖

is said to be a measurable norm with respect to µ.

(Cylinder measures can also be defined on H : since

X∗ →֒ H∗ ≃ H we have F(X∗) ⊂ F(H) )

If µ is a cylinder measure on H and µ̂(x) = e− 1

2 |x|2 , x ∈ H

then µ is called a ”standard Gaussian cylinder measure” on H.

13

Theorem 4.(Gross) Suppose the triplet (X,H, µ) is as above.

If µ is a Gaussian cylinder measure on H and ‖ . ‖ is a µ-

measurable norm, then the lifting µ∗ of µ to X can be ex-

tended to a Borel measure on X , called Standard Gaussian