Gaussian Hilbert spaces

Transcription

1 Gaussian Hilbert spaces Jordan Bell Department of Mathematics, University of Toronto July 11, Gaussian measures Let γ be a Borel probability measure on. For a, if γ = δ a then we call γ a Gaussian measure with mean a and variance 0. f σ > 0 and γ has density p(t, a, σ ) = ( 1 exp πσ ) (t a) σ, t, with respect to Lebesgue measure λ 1 on, then we call γ a Gaussian measure with mean a and variance σ. A Borel probability measure γ on n is called Gaussian if for each ξ ( n ), the pushforward measure ξ γ is a Gaussian measure on. The characteristic function of a Borel probability measure µ on n is µ(y) = e i y,x dµ(x), y n. n We call a linear operator C L ( n ) positive when Cx, x 0 for all x n. 1 t can be proved that a Borel probability measure γ on n is Gaussian if and only if there is some a n and some positive self-adjoint C L ( n ) such that γ(y) = exp (i y, a 1 ) Cy, y, y n. We say that γ has mean a and covariance operator C. f C is invertible (which is equivalent to Cx, x > 0 for all nonzero x n ), then the density of γ with respect to Lebesgue measure λ n on n is ( 1 x (π)n det C exp 1 C 1 (x a), x a ), n. 1 We remark that n is a real Hilbert space, and differently than a complex Hilbert space it need not be true that a positive linear operator is self-adjoint. Theorem 5. 1

2 The standard Gaussian measure on n, denoted γ n, is the Gaussian measure on n with mean 0 and covariance operator : dγ n (x) = (π) n/ exp ( 1 ) x, x dλ n (x), where λ n is Lebesgue measure on n. Throughout the remainder of this note, when we speak of Gaussian measures we assume unless we say otherwise that they have mean 0. Let be the positive integers. For nonempty subsets J and K of with J K, let π K,J : K J be the projection map, and for i let π i = π,{i}. For a topological space X, let B X be the Borel σ-algebra of X. f B i B for each i and {i : B i } is finite, we call B i i a cylinder set. The σ-algebra generated by the collection of all cylinder sets is called the product σ-algebra, and is denoted by B. The product measure γ = i γ 1 is the unique probability measure 3 on the product σ-algebra B such that for each cylinder set i B i, ( ) γ B i = γ 1 (B i ). i i Because is countable and is a second-countable topological space, the Borel σ-algebra of, with the product topology, is equal to the product σ-algebra on : 4 B = B. Thus γ is a Borel probability measure on. On the one hand, is a real vector space. On the other hand, with the product topology it is a topological space. t can be proved that with the separating family of seminorms { π i : i } it is a Fréchet space, 5 whose topology (the product topology) is induced by the complete translation invariant metric d(x, y) = i x i y i 1 + x i y i, x, y. i We remark that because is a countable product of second-countable topological spaces, it is itself a second-countable topological space, and so is separable Theorem 7. 5 Charalambos D. Aliprantis and Kim C. Border, nfinite Dimensional Analysis: A Hitchhiker s Guide, third ed., p. 07, Example 5.78.

3 The dual space of, denoted ( ), is the collection of continuous linear maps. t turns out that the dual space ( ) of is equal to the collection of those x such that {i : π i (x) 0} is finite, 6 with the dual pair x, y = x i y i, x, y ( ). i 3 Gaussian Hilbert spaces Because is a second-countable topological space, its Borel σ-algebra B is countably generated, and because γ is a probability measure on B it is a fortiori σ-finite, so the real Hilbert space L (γ ) is separable. 7 Let H be a real separable Hilbert space, with inner product, and norm, let e i, i, be an orthonormal basis for H, and let V be the linear span of this basis. n particular, V is a dense linear subspace of H. For v V, define φ(v) = i v, e i π i. Using we calculate 8 x i dγ 1 (x i ) = 0, x i dγ 1 (x i ) = 1, φ(v) L (γ ) = φ(v)(x) dγ (x) = v, e i π i (x) dγ (x) i + v, e i v, e j π i (x)π j (x)dγ (x) i = i = v, v, e i showing that φ : V L (γ ) is a linear isometry. Because (i) V is a dense linear subspace of H, (ii) the operator φ : V L (γ ) is bounded, and (iii) L (γ ) is a Hilbert space, there is a unique bounded linear operator Φ : H L (γ ) whose restriction to V is equal to φ. 9 For v H there is a sequence v n in V that tends to v, and because Φ is continuous and : H is continuous, Φ(v) = lim n Φ(v n) = lim n φ(v n) = lim n v n = v, 6 Charalambos D. Aliprantis and Kim C. Border, nfinite Dimensional Analysis: A Hitchhiker s Guide, third ed., p. 58, Theorem Donald L. Cohn, Measure Theory, second ed., p. 10, Proposition Walter udin, Functional Analysis, second ed., p. 39, Exercise 19. 3

4 showing that Φ is a linear isometry. Define F : H C by F (v) = exp(iφ(v)(x))dγ (x). Because e is e it s t, and using the Cauchy-Schwarz inequality, for v, w H, F (v) F (w) exp(iφ(v)(x)) exp(iφ(w)(x)) dγ (x) Φ(v)(x) Φ(w)(x) dγ (x) = Φ(v w)(x) dγ (x) Φ(v w) L (γ ) = v w, which shows in particular that F is continuous. For v V, let v = {i : v, e i } 0, which is finite. We calculate using Fubini s theorem ( F (v) = exp i ) v, e i π i (x) dγ (x) i = exp(i v, e i π i (x))dγ (x) i = i v = exp(i v, e i t)dγ 1 (t) γ 1 ( v, e i ) i v = ( 1 ) v, e i i v exp = exp ( 1 v ). For v H (, there is a sequence v n V that tends to v, and because F and w exp 1 w ) are continuous, F (v) = lim F (v n) = lim ( exp 1 ) v n = exp ( 1 ) n n v. That is, for all v H, exp(iφ(v)(x))dγ (x) = exp 4 ( 1 v ).

5 For distinct v 1,..., v n H, write X = Φ(v 1 ) Φ(v n ), which is measurable n, and let µ be the pushforward measure of γ by X, namely the joint distribution of the random variables Φ(v 1 ),..., Φ(v n ). For y n we calculate using the change of variables theorem 10 and Fubini s theorem µ(y) = e i y,u dµ(u) n = e i y,x(x) dγ (x) = e i(y1φ(v1)(x)+ +ynφ(vn)(x)) dγ (x) = e iφ(y1v1+ +ynvn)(x) dγ (x) = exp ( 1 ) y 1v y n v n = exp 1 v i, v j y i y j, i,j which shows that µ is a Gaussian measure on n with covariance matrix C i,j = v i, v j. 11 Thus {Φ(v)} v H is a stochastic process with sample space (, B, γ ), index set H, and state space, which we call the Gaussian process with covariance,. Let T be a separable metric space and suppose that c : T T is continuous and that for any t 1,..., t n T, {c(t i, t j )} is a symmetric positive semidefinite matrix. For each t T let δ t be a formal symbol, and let V be the linear span of {δ t : t T }. For v, w V, there are distinct t 1,..., t n T and real numbers α 1,..., α n, β 1,..., β n such that v = n i=1 α iδ ti and w = n i=1 β iδ ti, and we define For a, [av, w] = [v, w] = (aα i )β j c(t i, t j ) = a α i β j c(t i, t j ). α i β j c(t i, t j ) = a[v, w]. For u, v, w V there are distinct t 1,..., t n T and real numbers α 1,..., α n, β 1,..., β n, γ 1,..., γ n such that v = n i=1 α iδ ti, w = n i=1 β iδ ti, u = n i=1 γ iδ ti, and [u + v, w] = (α i + γ i )β j c(t i, t j ) = [u, w] + [v, w]. 10 Charalambos D. Aliprantis and Kim C. Border, nfinite Dimensional Analysis: A Hitchhiker s Guide, third ed., p. 484, Theorem See Barry Simon, Functional ntegration and Quantum Physics, p. 16, Theorem.3A. 5

6 Because {c(t i, t j )} is symmetric, [v, w] = [w, v]. matrix {c(t i, t j )} is positive semidefinite, [v, v] = α i α j c(t i, t j ) 0. Finally, because the This establishes that [, ] is a positive semidefinite inner product on V. Then v = [v, v], v = α i α j c(t i, t j ), is a seminorm on V, and N = {v V : v = 0} is a closed linear subspace of V. Let V/N = {v + N : v V }, For v, w V and r, s N, because r = 0 and s = 0, by the Cauchy-Schwarz inequality (which is indeed true for a positive semidefinite inner product) 1 we have [v, s] v s = 0 and [r, w] r w = 0, hence [v + r, w + s] = [v, w] + [v, s] + [r, w] + [r, s] = [v, w]. Therefore it makes sense to define v + N, w + N = [v, w]. f v + N, w + N = 0 then [v, v] = 0, i.e. v = 0 and so v N, i.e. v +N = 0 V/N, and therefore, is an inner product on V/N. Then there is a Hilbert space (H,, ) and a linear isometry i : V/N H whose image is a dense subset of H, called a completion of V/H, and this completion is unique up to a unique isomorphism of Hilbert spaces. 13 T is separable so it has a countable dense subset S. For t T, either t S or t S. f t S, there is a sequence of distinct s k in S that tends to t, and because c is continuous, δ sk δ t = c(s k, s k ) c(s k, t) c(t, s k )+c(t, t) c(t, t) c(t, t) c(t, t)+c(t, t), so δ sk δ t in V, which shows that δ : T V is continuous. Let W be the Q-linear span of {δ s : s S}. W is countable, and it follows from the above that W is dense in V. Define π : V V/N by π(v) = v + N, which is an onto continuous linear map. Then the image of W under i π : V H is dense in H, thus (H,, ) is a separable Hilbert space. Now let {Φ(v)} v H be the Gaussian process with covariance,, and define q : T L (γ ) by q = Φ i π δ. 1 A. Ya. Helemskii, Lectures and Exercises on Functional Analysis, p. 68, Theorem A. Ya. Helemskii, Lectures and Exercises on Functional Analysis, p. 17, Proposition 3. 6

7 For distinct t 1,..., t n T, the vectors v j = (i π δ)(t j ) H, 1 j n, are distinct. Then (Φ(v 1 ) Φ(v n )) γ is the Gaussian measure on n with covariance matrix C i,j = v i, v j = (i π δ)(t i ), (i π δ)(t j ) = [δ ti, δ tj ] = c(t i, t j ). That is, the joint distribution of the random variables q t1,..., q tn is the Gaussian measure on n with covariance matrix C i,j = c(t i, t j ). 7