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) < ; ∞ 3. For every Borel set B , λ(B + x) = λ(B); (translational
invariance) , further
4. λ has inner, outer regularity .
In fact 1), 2) 3) nearly characterize the Lebesgue measure
(modulo a multiplicative constant). Does the Lebesgue measure make sense in infinite dimensions ? The answer is negative. To convince ourselves :
1 H : a separable Hilbert space with an orthonormal basis h ,h , ,ν { 1 2 ···} : a Borel measure in H
1 Let B 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 (h1)) = ν(B 1 (h2) = < , 2 2 ······ ∞ but
ν(B2(0)) ν(B 1 (hn)= violates 2). ≥ n 2 ∞ P Gaussian measure in Rn is absolutely continuous with respect to the Lebesgue measure in n dimensions with Radon-Nikodym derivative (density)
1 1 −1 2 (S (x m),x m)) n Φ(x)= e− − − , x R (2π)n/2√DetS ∈ m : Mean vector , S : Covariance matrix (symmetric, positive definite) ;
2 Standard Gaussian Measure : m = 0 , S = I 2 −→ 1 kxk n Φ(x)= e− 2 x R . (2π)n/2 ∈
Another way of outlook at the density : random element in Rn
(Ω, ,P ) and a measurable mapping X :Ω Rn induces a ∃ F → prob. measure µ in Rn : (assumed to be absolutely continuous)
1 µ(B)= P (X− (B)) = Φ(x)dx ; B B ∈Bn R Standard Gaussian measure in finite dimensions is rotationally invariant : A−1B Φ(x)dx = B Φ(x)dx. R R Gauss distribution in finite dimensions can also be given by its
Fourier transformation (characteristic functional) :
χ(f)= ei(f,x)Φ(x)dx, f Rn , Rn ∈ Z
i(f,m) 1 (Sf,f) n χ(f)= e − 2 f R ∈
2 2 imt σ t For n =1 ,f = t ,χ(t)= e − 2 .
3 The char. fn of random variable (f,x), f,x Rn): ∈
2 t f it(f,x) it(f,m) 2 (Sf,f) χ (t) = n e Φ(x)dx = χ(tf) = e − , ( ) R ∗ thus GaussR (in one dimension), converse is also true, i.e. if (f,.) is one dimensional Gauss for all f Rn, then the measure in ∈ Rn 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 µˆ(f) is the
Fourier transform of a finite measure µˆ(0) = µ(Rn), ⇐⇒ continuous and positive definite.
For infinite dimensional Hilbert spaces these are not sufficient ,
1 x 2 e.g. e− 2 k k satisfy the conditions but not the Fourier trans- form µˆ(x) = ei(x,y)dµ(y) x H of any finite Borel H ∈ measure on theR Hilbert space . (Otherwise for an orthonormal basis
− 1 h , ei(hn,y)dµ(y) = e 2 which is not compatible with (h ,y) 0 { n} H n → R as (y,h )2 = y 2 < ) . n n k k ∞ DefiniteP answer
Theorem 2 (Minlos - Sazonov). Let φ be a positive- definite fnl on H , then the following are equivalent :
5 i) φ is the Fourier transform of a finite Borel measure on H , ii) ǫ> 0 , a symmetric, trace class operator S , such that ∀ ∃ ǫ (S x,x) < 1= Re(φ(0) φ(x) <ǫ, 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 . k k∗ k k∗ k k
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 x Rdµ(x) < , it always exists. On the other hand : H k k ∞ R
6 Lemma 1. If µ is a finite Borel measure, then
x 2 dµ(x) < a positive, symmetric, linear, trace H k k ∞ ⇐⇒∃ Rclass operator , called S-Operator, such that x,y H ∀ ∈
(Sx,y)= H (x,z)(y,z)dµ(z) . . If further Rµ is a probability measure then B, Bx = Sx − (m,x)x satisfies
(Bx,y) = (z m,x)(z m,y)dµ(z) is the covariance H − − operator. R
A Gaussian probability on H (i.e. all (x,.) are one-dimensional
Gaussian r.v.s ) always satisfies the above conditions, thus m and B always exists.
7 In fact ;
Theorem 3. A Borel probability measure µ on H is Gaussian
Its Fourier transform can be expressed as ⇐⇒ 1 µˆ(x) = exp i(m,x) (Bx,x) , m,x H. { − 2 } ∀ ∈
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 ∈ .
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. →
<< Sf,g >>=
Operator S has properties akin to operators S : H H e.g. → : i) Symmetry : << Sf,g >>=<< Sg,f >> ; ii) Positivity : << Sf,f >> 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
(X) : all symmetric, non-negative , bounded linear mappings H
X∗ X∗∗ ; → (X) : the class of covariance operators of distributions on H1 X ;
(X) : the classs of covariance operators of all Gaussian dis- H2 tributions on X. ( (X) (X) (X)) H2 ⊂H1 ⊂H If X is separable and reflexive then (X)= (X) . Under the H1 H same conditions,for S (X) , nuclearity in the first sense is ∈ H sufficient and the nuclearity in the second sense is necessary for
S to be in , (For nuclearity of different senses c.f. [Vakha- H2 nia]).
10 If we consider for simplicity m =0 ,
1 χ(f; µ) = exp << Sf,f >> f X∗ (√) {−2 } ∀ ∈ 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
1 f 2 χ(f) = e− 2 k k 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 (X∗): be the class of all finite dimensional subspaces of F
X∗ ;
For any K (X∗) call ∈ F D = x X : (< x,y >,< x,y >, ,< x,y >) E { ∈ 1 2 ··· n ∈ } a cylinder set based on K if E , y , ,y K. ∈Bn 1 ··· n ∈
Let (X) = K (X∗) (K) , where (K) is the σ-algebra ℜ ∈F C C generated by cylinderS sets with base K.
(X) is only an algebra , but for a separable Banach space ℜ X, σ( (X)) = (the Borel σ-alg. in X). ℜ BX A non-negative set fct µ on (X) is called a ”cylinder prob- ℜ ability measure” on X, if µ(X)=1 and a measure when restricted to any (K) for K (X∗) . A cylinder probability C ∈ F measure is necessarily compatible.
A real (complex)-valued fct on X is a cylinder fct if it is mea- surable w.t. (K) for some K. C i
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 k k than . If for any ǫ > 0 π ( the set of all finite | | ∃ ǫ ∈ P dimensional orthogonal projections on H ), such that for any
π (π π ) one has µ x H : πx >ǫ <ǫ then . ∈P ⊥ ǫ { ∈ k k } k k 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 (X∗) (H → ≃ F ⊂ F
1 x 2 If µ is a cylinder measure on H and µˆ(x) = e− 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- k k measurable norm, then the lifting µ∗ of µ to X can be ex- tended to a Borel measure on X, called Standard Gaussian . measure on X . µ∗(C) = µ(C H) , C (X)) . ∩ ∈R (X,H,µ) is called an abstract Wiener space
Conversely let X be a separable Banach space and let µ be a zero mean Gaussian measure on X , i.e. for any α X∗ ∈ and ω X, <α,ω> is a one-dimensional zero mean Gauss ∈ random variable (or for any n and α X∗ < α ,ω >,i = i ∈ { i 1, 2, n is a zero mean Gaussian random vector) .Then there ··· } exists a dense Hilbert sub-space H X such that (X,H,µ) ⊂ is an abstract Wiener space.
H is called the Cameron-Martin space.
14 Classical Wiener space is an example of an abstract Wiener space :
Banach space X : C0[0, 1] with the sup norm,
For ω C [0, 1] , t [0, 1] coordinate functional on C [0, 1] ∈ 0 ∈ 0 is Wt(ω)= ω(t) .
N. Wiener : There exists a unique probability µ on such BC0 that the map (t, ω) W (ω) is a Wiener process (Brownian −→ t motion).
H : absolutely continuous functions in C0[0, 1] with square integrable derivatives.
t ˜ ˜ 2 Then h H h = 0 h(s)ds, h L [0, 1] . ∈ ⇒ ∈ 1 Rt ˜ t ˜ 2 h C0 = sup0 t 1 0 h(s)ds sup0 t 1 t 0 h(s)ds k k ≤1 ≤ | | ≤ ≤ ≤ | ≤ t 2 t 1 . h˜(s) 2ds =( R h˙ 2) 2 = h . (defn of normR in H). 0 | | 0 | | | |H R R
15 With this norm h˜ h is a continuous , linear injection from → H into C , such that its range is dense in C . . is weaker 0 0 k k than . when restricted to H. | |
If (X,H,µ) is an abstract Wiener space , considering that the cylindrical measure is on H, we want to construct a process
But since X∗ H,h may not be in X∗ . However X∗ is ⊂
∗densely imbedded in H. Indicate the injection X∗ ֒ H → ≃ ˆ H by ( . ), α X∗ αˆ H. Given h H there exists ∈ → ∈ ∈
α X∗ s.t. αˆ h in H. For ω X the Gaussian n ∈ n −→ ∈ p sequence < αn,ω > is Cauchy in L , denote the limit by
W (ω) , (δh(ω) in some sources ), which is N(0 , h 2 ) and h | |H also E(W W )=(h,g) ,h,g H. h g H ∈
Thus we have another model which is called Gaussian prob- ability space by Malliavin. Namely :
16 (Ω, ,µ) : a complete probability space, H : a real, separable F Hilbert space, W , h H is a family of zero mean Gaussian { h ∈ } r.v.’s with E[WhWg]=(h,g)H .
(Ω, ,µ; H) A Gaussian probability space. The classical and F abstract Wiener spaces are examples of Gaussian probability spaces. A third example is the White noise space : H =
2 L (R); S(R), S∗(R) : Schwartz spaces of rapidly decreasing
C∞ functions and tempered distributions respectively. ( S(R) ⊂ 2 L (R) S∗(R) is called a Gelfand Triplet). Then by ⊂ the Minlos theorem there exists a unique Gaussian measure µ
i<ω,ξ> on ∗ R such that ξ S(R) : e dµ(ω) = BS ( ) ∀ ∈ S∗(R) 1 ξ 2 e− 2 k kH ; <ω,ξ > is the canonical bilinearR form on S∗(R) × S(R) . Then W (ω) =<ω,ξ> ξ W can be ex- ξ → ξ 2 2 tended to a linear isometry L (R) L (S∗(R), ,µ) , thus −→ F 2 (S∗(R), ,µ; L (R)) is a Gaussian probability space. F
17 In an abstract Wiener space or a Gaussian probability space ,
W (ω), h H is like a stochastic process with the index set { h ∈ } H.
Some Elements of the Malliavin Calculus
In an abstract Wiener space (or a Gaussian probability space)
F : X R(C) is called a Wiener functional. →
Some examples ; Ito stochastic integrals, solutions of stochastic differential equations. As they are in fact equivalence classes , may not be differentiable in the Fr´echet sense, even not contin- uous. Paul Malliavin , using the quasi-invariance of the Gaus- sian measure, initiated a kind of weak differential calculus, so that such functionals became smooth in his sense . The quasi- invariance is translational invariance of the Gaussian measure in the directions of the Cameron-Martin space H.
18 The new kind of differentiation was obtained by perturbing the
Wiener paths in the directions of vectors in H, thus taking the name of ”stochastic calculus of variation” or as popularly known the Malliavin calculus.
If F : X R an attempt to define the derivative by −→ F (ω +∆ω) F (ω) lim ∆ω 0 − will fail since the quotient is not k k→ ∆ω k k even well-defined as a random variable, (equivalence classes).
This is remedied by the Cameron-Martin Formula (Theorem):
1 2 Wh h E[F (ω + h)] = E[F (ω)e − 2 | |H ]
If in the above quotient the perturbation ∆ω = h is taken in the Camerion Martin space, (i.e. h H) then the limit will be ∈ well-defined .
Take two functionals in the same equivalence class :
19 F = G µ a.s. = µ ω F (ω+h) = G(ω+h) = Eµ I ω F (ω+h)=G(ω+h) − ⇒ { | 6 } { | 6 } 1 2 Wh 2 h = (Cameron-Martin thm.) Eµ I ω F (ω)=G(ω) e − | |H =0 . { | 6 } n o This allows to define weak differential (Sobolev derivative) start- ing from cylindrical functionals.
Let (X,H,µ) be an abstract Wiener space.
F (ω)= f (W (ω), ,W (ω)) , ω X,f S(Rn) h1 ······ hn ∈ ∈ is called a smooth cylindrical functional, its class is de- noted by SM . Similarly depending on f being a polynomial or Lp function we have a functional in or in Lp(µ) . The P following inclusions hold
S Lp P ⊂ M ⊂ and is dense in Lp . Noticing that W (ω +λh)= W (ω)+ P hj hj
λ(hj,h)H we have the following definition of the weak direc- tional derivative (Sobolev derivative) :
20 d n ∂f D F (ω)= F (ω+λh) = (W (ω), ,W (ω))(h ,h) ; h H. h dλ |λ=0 ∂x h1 ··· hn j H ∈ i=1 j X For fixed ω,h D (ω) is linear and continuous , hence it → h determines (by the Riesz representation thm) an element of
H∗ H that we denote by DF (the gradient operator) and ≃
(DF,h)H = DhF and also E[(Df,h)] = E[F (Wh))] .
21 Also F (ω) DF (ω) is a linear operator from the cylindri- −→ cal functionals into the space of H-valued Wiener functionals
Lp(µ; H) ,
1 (the norm being defined by DF (ω) = DF (ω) p dµ(ω) p ) . k kp X | |H R Note. If G Lp(µ) and F = G (µ a.s.) then by the ∈ − Cameron-Martin thm. G(ω + λh) = F (ω + λh) (µ a.s.) , − thus DF depends only on the equivalence class that F belongs to.
We want to extend the above operator to a larger class of Wiener functionals. In fact we have
Theorem 5. D is a closable operator from Lp(µ) into Lp(µ; H) .
22 Definition 2. Dp,1 is the set of equivalence classes of Wiener functionals defined by :
F D a sequence of cylindrical fcts F converging to ∈ p,1 ⇔ ∃ n F in Lp(µ) such that DF , n N is Cauchy in Lp(µ; H). { n ∈ }
In this case denote limn DFn = DF . →∞
(DF is independent of the choice of the sequence Fn) .
p p D is a Banach space under the norm F = F p + p,1 k kp,1 k kL (µ) p DF p . k kL (µ;H)
Generalization to E valued functionals (E is any separable Hilbert space ):
DhF =(DF,h) takes the form : d [(F (ω + λh),e)E] λ=0 = (DF (ω),h e)H E h H,e dλ | ⊗ ⊗ ∈ ∈ E.
DF Lp(µ; H E) . Corresponding Sobolev space is denoted ∈ ⊗ by Dp,1(E) .
23 Higher order derivatives and Sobolev spaces are defined in an inductive manner:
2 p k k 1 p (k) D F = D(DF ) L (H H E) ,D F = D(D − F ) L (H⊗ E) . ∈ ⊗ ⊗ ······ ∈ ⊗ where denotes the (completed) Hilbert tensor product. Sim- ⊗ k 1 (k 1) ilarly F D if D − F D (H⊗ − E) and the norms ∈ p,k ∈ p,1 ⊗
k 1/p F = F p + DjF p . k kp,k k kp k kp j=1 ! X Since the gradient DF is an H-valued functional, H may be considered as tangent space. Hence H-valued functionals are vector fields and the adjoint operator δ is the divergence of vector fields.
24 For any smooth vector field V S (H) , its divergence ∈ M δV S (R) S is determined by ∈ M ≡ M
E[G.δV ]= E[(DG,V ) ] , G S . H ∀ ∈ M
More generally if V S (H E) , its divergence in S (E) is ∈ M ⊗ M defined by
E[(G,δV )E]= E[(DG,V )H E] , G SM (E) . ⊗ ∀ ∈
Explicit expression for the smooth vector fields V S (H) : ∈ M
∞ δV = [(V,h ) W (D V,h ) ] , h : complete, orthonormal basis in H i H hi − hi i H { i} i=1 X δV is also a closable operator.
For a feeling of the divergence operator consider the case n =1.
Example. One dimensional Gaussian space (R, ,γ) ; D = B d d d2 d ; D∗ δ = + u. ; and D∗D = δD = 2 + u . For du ≡ −du −du du φ and ψ real polynomials :
(Dφ,ψ)L2(R,γ) =(φ,δψ)L2(R,γ) .
25 D,δ and δD can be extended to closed operators in L2(R,γ) such that D and δ are mutually adjoint and δD is self adjoint
, number operator in one dimension . Hermite polynomials are eigen functions of the number operator, i.e.
δDHn = nHn
(Hermite polynomials the coefficients of t in the expansion exp tu t2/2 = { − } 2 ∞ n 2 n − u t n u /2 d 2 H (u) ; t,u R , H (u)=( 1) e n e , n N .) n=0 n! n ∈ n − du ∈ 0 P
= δD is the Ornstein-Uhlenbeck operator. H = L − L n nH . − n
In infinite dimensions : H = H (W (ω)) ( h is any α j αj hj { j} orthonormal base in H) , whereQα = α ∞ has only a finite { j}j=1 number of non-zero indices . All such indices form a set Γ . and Λ = α Λ: α = n . n { ∈ | | }
26 1 2 Then (α!)− 2 H : α Λ constitute a base of L (µ) . Let { α ∈ } R . For n 1 , let be the closed subspace generated H0 ≡ ≥ Hn by H : α Λ . Then the infinite direct sum decomposition { α ∈ n}
∞ L2(µ)= . Hn n=0 M The decomposition is independent of the choice of base in H .
Also isomorphic to the symmetric Fock space (Boson Fock space) over H :
2 L (µ) ∼= Γ(H). : Wiener chaos of order n . J : orthogonal projection onto Hn n . Hn ∞ = δD = nJ ; H = α H . L − − n L α −| | α n=1 X For F = f(W , ,W ) S we explicitly have h1 ··· hn ∈ M n F = ∂ ∂ f(W , ,W )(h ,h ) ∂ f(W , ,W )W . L k j h1 ··· hn k j − j h1 ··· hn hj j=1 j,kX=1 X
27 Multiple Wiener-Ito Integral representation
In abstract Wiener space let H = L2(T, , λ) . where (T, ) B B is a measurable space, λ : non-atomic, σ-finite, (covers clas- . sical Wiener space and the white-noise space). Let W (A) =
W ; A , (i.e. h = I H) . We have IA ∈B A ∈
W (A) N(0, λ(A)) ; E[(W (A)W (B)] = λ(A B) . ∼ ∩
Then for disjoint A ,W ( A ) = W (A ) (L2(µ)- { j} n n n n convergence). S P
W (a random set function) : Gaussian orthogonal random measure ,Wh = T h(t)dW (t) .
In : Multiple Wiener-ItoR integral is constructed like multiple
Lebesgue integral using this random measure:
I = n f(t ,t , ,t ) dW (t )dW (t ) dW (t ). n T 1 2 ··· n 1 2 ··· n R Relation between Hermite polynomials and multiple Wiener-Ito integrals :
28 n Hn(Wh)= In(h⊗ ) . For n =1 , H1(u)= u then it reduces to
T h(t)dW (t)= H1(Wh)= Wh .
RMore generally if hj j N is a base of H we have Hα = { } ∈ ˆ ˆ ˆ αj I α (hα) , α Λ and hα jhj⊗ . | | ∀ ∈ ≡ N 2 Also F L (µ) has a unique decomposition F = ∞ I (f ) ∈ n=0 n n ˆ n 2 2 2 where f H⊗ and F =[E( F ) + ∞ nP! f n ∈ k k | | n=0 k nk ( f stands for the norm in L2(T n, , λPn)) . k nk Bn
Another interpretation of the Ornstein-Uhlenbeck
Operator = δD L −
Ornstein-Uhlenbeck process satisfies the stochastic d.e. dX = X dt+√2 dB ,X = x Rn (B : Brownian motion) t − t t 0 ∈ t
t t (t s) has the solution X = e− x + √2 e− − dB t R . X t 0 s ∈ + t t 2t is a Gaussian process X N(eR− x, 1 e− ) . Define the t ∼ −
Ornstein-Uhlenbeck semi-group Tt which is a contraction (even e hypercontractive) semi-group.
29 . E t 2t (Ttφ)(x) = [φ(Xt)] = φ(e− x+ 1 e− y) dµ(y); φ SM X − ∈ Z p (µ : standard normal distribution) . Semi-group property follows from the Markov property of diffusion processes and it turns out that the infinitesimal generator of this semi-group is . There- L n n t t 0 fore T = e L = ∞ . Using I = J , = t n=0 nL! L ≡ n n L nJ , 2 = ( P nJ )( mJ ) = n2PJ , (the or- n n L n − n m − m n n Pthogonality of chaosP projections)P and by inductionP m = ( 1)mnmJ L n − n yielding for the representation of the O-U semigroupPTt =
nt n∞=0 e− Jn . P As each Wiener chaos belongs to the eigen-space of , we can L − define non-integer powers, e.g., (I )a = I a + a(a 1) 2 = −L − L 2 L −··· − − J + a nJ + (a(a 1) n2J + = (1 + an + a(a 1) n2 + )J = n n 2 n ··· 2 ··· n
P Pa P n(1 + n) Jn ; P
30 In particular for p> 1 , a = 1 , the completion of polynomial −2 cylindrical functionals with respect to the norm
. k p F ∼ = (I ) 2 F p is dense in L and denoted by D∼ . k kp,k k −L kL p,k Now the important Meyer inequalities relating Sobolev and Lp norms
p> 1, k N c , c˜ , such that F (E) : ∀ ∀ ∈ ∃ p,k p,k ∀ ∈P
k k c (I ) 2 F p F c˜ (I ) 2 F p . p,kk −L kL (µ;E) ≤ k kp,k ≤ p,kk −L kL (µ;E)
By these inequalities for k N , . . ∼ , then we omit ∈ k k ∼ k k sign, but whenever k is non-integer we should know that ∼
D D∼ . Using Meyer inequalities we show (for E valued p,k ≡ p,k functionals) :
31 i) Dp,k(E) has Dq, k(E) its continuous dual , 1/p +1/q =1 , − ii) D has a continuous extension from Dp,k(E) into Dp,k 1(E − ⊗ H) for any p> 1 , iii) δ has a continuous extension from Dp,k(E H) into Dp,k 1(E) ⊗ − for any p> 1 .
Generalized Functions
Similar to the test functions-Schwartz distributions dual-pair
( , ∗) we have : D D
D∞(E)= Dp,k(E) , D−∞(E)= Dp, k(E) . − k>0 1 D∞ is equipped with the projective limit topology and D−∞ is equipped with the inductive limit topology ; the latter is the Meyer-Watanabe distributions. As an application Donsker’s Delta Function is δ (W (t)) x ∈ D−∞ . 32 Using Meyer inequalities one can show that : D uniquely extends to a continuous operator D∞(E) • −→ D∞(E H) , ⊗ D uniquely extends to an operator D−∞(E) D−∞(E • −→ ⊗ H) , Similar extensions for δ = D∗ , e.g. δ : D−∞(E H) • ⊗ −→ D−∞(E) , extends uniquely to an operator D−∞(E) D−∞(E) • L −→ such that p (1, ) , k R ∀ ∈ ∞ ∈ : D (E) D (E) is continuous, in particular : D∞(E) D∞(E) L p,k+2 −→ p,k L → is continuous. 33 Densities of Non-degenerate Functionals One of the main concerns of the Malliavin calculus is the investi- gation of existence, regularity (smoothness) and other properties of the Wiener (Brownian) functionals. Let F be an Rm-valued functional (i.e. an m-dimensional ran- 1 dom vector). µ F − defines a probability measure on . ◦ Bm Under which conditions it is absolutely continuous with respect to the Lebesgue meaure λm ? Malliavin Covariance Matrix . Let F =(F ,F , ,F ) D (Rm) and σ =(DF ,DF ) ; 1 2 ··· m ∈ 1,1 ij i j H 1 i, j m. ≤ ≤ Malliavin Matrix : Σ(ω)= σ (ω) . h ij i 1 If DetΣ(ω) > 0 a.s. and satisfies [DetΣ(ω)]− L∞− ∈ ≡ p 1 34 Lemma 2. Let ν be a σ-finite measure on . If for j = Bm m 1, 2, ,m c such that φ C∞(R ) : ··· ∃ j ∀ ∈ 0 ∂jφ(x)dν(x) cj φ | Rm |≤ k k∞ Z m m∗ m then ν λ . When m> 1 , ν has density ρ L (R ) , m∗ = ≺≺ ∈ m m 1 . − Note : Clear for m =1 : take φ the cumulative D.F. of the uniform U([a,b]) r.v. It is not in C∞ . But φ C∞ such that φ φ, φ′ 0 ∃ n ∈ 0 n → n → ′ ′ 1 φ , φ = , φ ∞ = 1 . Then the condition yields ν([a,b]) b−a k k ≤ c (b a)= c λ([a,b]) implying absolute continuity . 1 − 1 Lemma 2 is utilized to prove D Rm D Rm Theorem 6. Let F =(F1,F2, ,Fm) 2∞( ) 1 35 Smoothness of Densities The density of F can be formally expressed as E[δ F ] , (δ : x ◦ x Dirac function with singularity at x ). : (To see this heuristically consider a delta sequence δ δ , then n,x → x δ (F (ω))dµ(ω)= m δ (y)p(y)dy p(x) .) Ω n,x R n,x → RHowever following WatanabeR we should give a rigorous meaning to the composition of a Schwartz distribution with a functional. m m If φ S(R ) and F D∞(R ) , then the composite func- ∈ ∈ tional φ F D∞ . For fixed F,φ φ F is a linear map ◦ ∈ → ◦ m S(R ) D∞ . Watanabe’s method is to extend it to a linear → m and (in some sense) continuous map from S∗(R ) D−∞ . → He showed that in this way every Schwartz distribution T can be lifted (pulled-back) to a generalized Wiener functional T F. ◦ (Note that S S∗ and because of Dp,k Dp, k we have ⊂ ⊂ − D∞ D−∞ ). Then δ F can be interpreted as a generalized ⊂ x ◦ functional. Using Watanabe’s approach we prove m Theorem 7. If F D∞(R ) is non-degenerate, then F has ∈ density ρ which is infinitely differentiable. 36 Hypoellipticity and H¨ormander’s condition Consider the second order partial differential operator 1 m m L = aij(.)∂ ∂ + bi(.)∂ . 2 i j i i,j=1 i=1 X X and the Cauchy problem for the heat equation : ∂ u(t,x)= Lu(t,x), t> 0,x Rm ; t ∈ u(0,x)= φ(x) . It is known that if φ C2(Rm) , then ∈ b u (t,x) E[φ(X(x,t,ω))] φ ≡ is the solution of the Cauchy problem . X is the solution of the Ito stochastic differential equation X = x + t b(X )ds + t σ(X ).dB , t 0 t 0 s 0 s s ≥ R R where x Rm,b : Rm Rm,σ : Rm Rm Rd . ∈ → → ⊗ 1 If X is non-degenerate ([detΣ]− L∞−), then the transition t ∈ 1 probability P (t,x,ω) = µ X(x,t,ω)− of diffusion process ◦ X has C∞ density p(t,x,y) = E[δy(X(x,t,ω))] which is the fundamental solution of the above Cauchy problem (the heat kernel : uφ(t,x)= Rd p(t,x,y)φ(y)dy). R 37 From the theory of p.d.e. if matrix a(x) is uniformly positive definite , i.e. η > 0 such that a(.) ηI , then the conclusion ∃ ≥ is true. H¨ormander obtained a much weaker condition through the hy- poellipticity of the differential operators, namely the H¨ormander condition. To state H¨ormander’s theorem write L in form of vec- tor fields : A (.) σi (.)∂ , k =1, ,d, k ≡ k i ··· A (.) [bi(.) 1 d σj (.)∂ σi (.)]∂ (with Einstein’s sum- 0 ≡ − 2 k=1 k j k i mation convention)P , we have : 1 d 2 L = 2 k=1 Ak + A0 . H¨ormander’sP Theorem for Hypoellipticity If for every x Rm the Lie algebra generated by vector fields ∈ A , [A ,A ],k = 1, ,d) has dimension m (H¨ormander { k 0 k ··· } condition) , then L is hypoelliptic , that is , for any open set m U R and any Schwartz distribution u ∗ if Lu ∈ ∈ D |U ∈ C∞(U) , then u C∞(U) |U ∈ If the H¨ormander’s condition holds, then (in the elliptic case) the above smooth fundamental solution exists. 38 ([ ., . ] is the Lie bracket : Given two C1 vector fields V and W on Rm , [V,W ](x)= DV (x)W (x) DW (x)V (x)) . − The key step in the probabilistic proof of the H¨ormander’ the- orem is to show that under the H¨ormander’s condition the cor- responding Malliavin covariance matrix is non-degenerate. 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