GAUSSIAN MEASURE vs LEBESGUE MEASURE AND ELEMENTS web.iku.edu.tr/ias/documents/ آ  GAUSSIAN...

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Transcript of GAUSSIAN MEASURE vs LEBESGUE MEASURE AND ELEMENTS web.iku.edu.tr/ias/documents/ آ  GAUSSIAN...

  • 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