Gaussian Measures in Infinite Dimen- Sion
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Lecture 2 Gaussian measures in infinite dimen- sion In this lecture, after recalling a few notions about σ-algebras in Banach spaces, we intro- duce Gaussian measures in separable Banach spaces and we prove the Fernique Theorem. This is a powerful tool to get precise estimates on the Gaussian integrals. In most of the course, our framework is an infinite dimensional separable real Banach space that we denote by X, with norm or when there is risk of confusion. The open (resp. k · k k · kX closed) ball with centre x X and radius r>0 will be denoted by B(x, r)(resp. B(x, r)). 2 We denote by X , with norm , the topological dual of X consisting of all linear ⇤ k · kX⇤ continuous functions f : X R. Sometimes, we shall discuss the case of a separable ! Hilbert space in order to highlight some special features. The only example that is not a Banach space is RI (see Lecture 3), but we prefer to describe this as a particular case rather than to present a more general abstract theory. 2.1 σ-algebras in infinite dimensional spaces and characteristic functions In order to present further properties of measures in X, and in particular approximation of measures and functions, it is useful to start with a discussion on the relevant underly- ing σ-algebras. Besides the Borel σ-algebra, to take advantage of the finite dimensional reductions we shall frequently encounter in the sequel, we introduce the σ-algebra E (X) generated by the cylindrical sets, i.e, the sets of the form C = x X :(f (x),...,f (x)) C , 2 1 n 2 0 n o n where f1,...,fn X⇤ and C0 B(R ), called a base of C. According to Definition 2 2 1.1.8, E (X)=E (X, X⇤). The following important result holds. We do not present its most general version, but we do not even confine ourselves to Banach spaces because 13 14 we shall apply it to R1, which is a Fr´echet space. We recall that a Fr´echet space is a complete metrisable locally convex topological vector space, i.e., a vector space endowed with a sequence of seminorms that generate a metrisable topology such that the space is complete. Theorem 2.1.1. If X is a separable Fr´echet space, then E (X)=B(X). Moreover, there is a countable family F X separating the points in X (i.e., such that for every pair of ⇢ ⇤ points x = y X there is f F such that f(x) = f(y)) such that E (X)=E (X, F). 6 2 2 6 Proof. Let (xn)beasequencedenseinX, and denote by (pk) a family of seminorms which defines the topology of X. By the Hahn-Banach theorem for every n and k there is ` X such that p (x )=` (x ) and sup ` (x): p (x) 1 = 1. As a n,k 2 ⇤ k n n,k n { n,k k } consequence, for every x X and k N we have pk(x)=sup `n,k(x) . Therefore, for 2 2 n{ } every r>0 B (x, r):= y X : p (y x) r = y X : ` (y x) r E (X). k { 2 k − } { 2 n,k − } 2 n N \2 As X is separable, there is a countable base of its topology. For instance,we may take the sets Bk(xn,r) with rational r, see [DS1, I.6.2 and I.6.12]. These sets are in E (X), hence the inclusion B(X) E (X) holds. The converse inclusion is trivial. ⇢ To prove the last statement, just take F = `n,k,n,k N . It is obviously a countable { 2 } family; let us show that it separates points. If x = y,thereisk N such that pk(x y)= 6 2 − sup `n,k(x y) > 0 and therefore there isn ¯ N such that `n,k¯ (x y) > 0. n − 2 − In the discussion of the properties of Gaussian measures in Banach spaces, as in Rd, the characteristic functions, defined by µˆ(f):= exp if(x) µ(dx),fX⇤, { } 2 ZX play an important role. The properties of characteristic functions seen in Lecture 1 can be extended to the present context. We discuss in detail only the extension of property (iii) (the injectivity), which is the most important for our purposes. In the following proposi- tion, we use the coincidence criterion for measures agreeing on a system of generators of the σ-algebra, see e.g. [D, Theorem 3.1.10]. Proposition 2.1.2. Let µ1,µ2 be two probability measures on (X, B(X)).Ifµ1 = µ2 then µ1 = µ2. b b Proof. It is enough to show that ifµ ˆ =0thenµ = 0 and in particular, by Theorem 2.1.1, d that µ(C)=0whenC is a cylinder with base C0 B(R ). Let beµ ˆ = 0, consider 21 d F = span f1,...,fd X⇤ and define µF = µ PF− ,wherePF : X R is given by { } ⇢ d ◦ ! PF (x)=(f1(x),...,fd(x)). Then for any ⇠ R 2 µ (⇠)= exp i⇠ y µ (dy)= exp i⇠ P (x) µ(dx)= exp iP ⇤ ⇠(x) µ(dx)=0, F { · } F { · F } { F } ZF ZX ZX b 15 d where P ⇤ : R X⇤ is the adjoint map F ! d PF⇤ (⇠)= ⇠ifi. Xi=1 It follows that µF = 0 and therefore the restriction of µ to the σ-algebra E (X, F)isthe null measure. 2.2 Gaussian measures in infinite dimensional spaces Measure theory in infinite dimensional spaces is far from being a trivial issue, because there is no equivalent of the Lebesgue measure, i.e., there is no nontrivial measure invariant by translations. Proposition 2.2.1. Let X be a separable Hilbert space. If µ : B(X) [0, + ] is a ! 1 σ-additive set function such that: (i) µ(x + B)=µ(B) for every x X, B B(X), 2 2 (ii) µ(B(0,r)) > 0 for every r>0, then µ(A)=+ for every open set A. 1 Proof. Assume that µ satisfies (i) and (ii), and let e be an orthonormal basis in X. { n} For any n N consider the balls Bn with centre 2ren and radius r>0; they are pairwise 2 disjoint and by assumption they have the same measure, say µ(Bn)=m>0 for all n N. 2 Then, 1 1 1 B B(0, 3r)= µ(B(0, 3r)) µ(B )= m =+ , n ⇢ ) ≥ n 1 n=1 n=1 n=1 [ X X hence µ(A)=+ for every open set A. 1 Definition 2.2.2 (Gaussian measures on X). Let X be a Banach space. A probability 1 measure γ on (X, B(X)) is said to be Gaussian if γ f − is a Gaussian measure in R ◦ for every f X .Themeasureγ is called centred (or symmetric) if all the measures 2 ⇤ γ f 1 are centred and it is called nondegenerate if for any f =0the measure γ f 1 is ◦ − 6 ◦ − nondegenerate. Our first task is to give characterisations of Gaussian measures in infinite dimensions in terms of characteristic functions analogous to those seen in Rd, Proposition 1.2.4. Notice that if f X then f Lp(X, γ) for every p 1: indeed, the integral 2 ⇤ 2 ≥ p p 1 f(x) γ(dx)= t (γ f − )(dt) | | | | ◦ ZX ZR 1 is finite because γ f − is Gaussian in R. Therefore, we can give the following definition. ◦ 16 Definition 2.2.3. We define the mean aγ and the covariance Bγ of γ by a (f):= f(x) γ(dx),fX⇤, (2.2.1) γ 2 ZX B (f,g):= [f(x) a (f)] [g(x) a (g)] γ(dx),f,gX⇤. (2.2.2) γ − γ − γ 2 ZX Observe that f a (f) is linear and (f,g) B (f,g) is bilinear in X . Moreover, 7! γ 7! γ ⇤ B (f,f)= f a (f) 2 0 for every f X . γ k − γ kL2(X,γ) ≥ 2 ⇤ Theorem 2.2.4. A Borel probability measure γ on X is Gaussian if and only if its char- acteristic function is given by 1 γˆ(f)=exp ia(f) B(f,f) ,fX⇤, (2.2.3) − 2 2 n o where a is a linear functional on X⇤ and B is a nonnegative symmetric bilinear form on X⇤. Proof. Assume that γ is Gaussian. Let us show thatγ ˆ is given by (2.2.3) with a = aγ and B = Bγ.Indeed,wehave: 1 1 2 γ(f)= exp i⇠ (γ f − )(d⇠)=exp im σ , R { } ◦ − 2 Z n o where m and σ2 areb the mean and the covariance of γ f 1, given by ◦ − 1 m = ⇠(γ f − )(d⇠)= f(x)γ(dx)=a (f), ◦ γ ZR ZX and 2 2 1 2 σ = (⇠ m) (γ f − )(d⇠)= (f(x) a (f)) γ(dx)=B (f,f). − ◦ − γ γ ZR ZX Conversely, let γ be a Borel probability measure on X and assume that (2.2.3) holds. Since a is linear and B is bilinear, we can compute the Fourier transform of γ f 1, for ◦ − f X , as follows: 2 ⇤ 1 1 γ\f (⌧)= exp i⌧t (γ f − )(dt)= exp i⌧f(x) γ(dx) ◦ − { } ◦ { } ZR ZX 1 =exp i⌧a(f) ⌧ 2B(f,f) . − 2 n o According to Remark 1.2.2, γ f 1 = N (a(f),B(f,f)) is Gaussian and we are done. ◦ − Remark 2.2.5.