Abstract Wiener Spaces in a Few Slides

Abstract Wiener Spaces in a few slides

Ambar N. Sengupta

17 March 2012

1 / 21 Introduction

Standard Gaussian in (in)ﬁnite dimensions

Measurable norm

Constructing the Gaussian

AWS summary I

Continuity of the injection

Details

Running AWS in reverse

Some references

2 / 21 Introduction

Abstract Wiener spaces (AWS) form just the right setting for inﬁnite-dimensional integration. The framework was built by L. Gross [1], who also proved all the fundamental results concerning abstract Wiener spaces.

3 / 21 Standard Gaussian measure on RN

Standard Gaussian measure on RN is the product of N copies of the one-dimensional standard Gaussian whose density is

1 2 √ e−x /2 2π

Thus standard Gaussian measure on RN is the distribution of the RN -valued random variable

Z = Z1u1 + ··· + ZN uN ,

N where u1,..., uN form an orthonormal basis of R and Z1,..., ZN are iid standard Gaussian random variables.

4 / 21 The problem is: with probability 1 the series (1) is not P P 2 convergent in H, because [ n Zn < ∞] is 0. The key idea then is to introduce a new norm | · | such that (1) is convergent with probability 1 as a series in the completion B of H relative to | · |.

Standard Gaussian measure for a Hilbert space

Let H be a separable inﬁnite dimensional real Hilbert space. The standard Gaussian measure for H should be the distribution measure of the H-valued random variable

Z = Z1u1 + Z2u2 + ··· , (1)

where u1, u2,... form an orthonormal basis of H, and Z1, Z2,... are iid standard Gaussians, all deﬁned on some (Ω, F, P).

5 / 21 The key idea then is to introduce a new norm | · | such that (1) is convergent with probability 1 as a series in the completion B of H relative to | · |.

Standard Gaussian measure for a Hilbert space

Let H be a separable inﬁnite dimensional real Hilbert space. The standard Gaussian measure for H should be the distribution measure of the H-valued random variable

Z = Z1u1 + Z2u2 + ··· , (1)

5 / 21 Standard Gaussian measure for a Hilbert space

Let H be a separable inﬁnite dimensional real Hilbert space. The standard Gaussian measure for H should be the distribution measure of the H-valued random variable

Z = Z1u1 + Z2u2 + ··· , (1)

where u1, u2,... form an orthonormal basis of H, and Z1, Z2,... are iid standard Gaussians, all deﬁned on some (Ω, F, P). The problem is: with probability 1 the series (1) is not P P 2 convergent in H, because [ n Zn < ∞] is 0. The key idea then is to introduce a new norm | · | such that (1) is convergent with probability 1 as a series in the completion B of H relative to | · |.

5 / 21 Then, assuming H is separable, by choosing a sequence of such subspaces, and orthonormal bases within each, we can construct an orthonormal basis u1, u2,... in H such that P [|Z1u1 + ··· Zk1 uk1 | > 1/2] < 1/2 P[| + ··· | > / ] < / Zk1+1uk1+1 Zk2 uk2 1 4 1 4 (2) . . . < .

for some 1 ≤ k1 < k2 < . . .. (For details see Frame 16 below.)

Measurable norm

A norm | · | on H is a measurable norm if for any > 0 there is a ﬁnite dimensional subspace F of H such that for any ﬁnite dimensional subspace F1 of H orthogonal to F

P[|Z | > ] <

for any F1-valued standard Gaussian variable Z .

6 / 21 Measurable norm

A norm | · | on H is a measurable norm if for any > 0 there is a ﬁnite dimensional subspace F of H such that for any ﬁnite dimensional subspace F1 of H orthogonal to F

P[|Z | > ] <

for any F1-valued standard Gaussian variable Z . Then, assuming H is separable, by choosing a sequence of such subspaces, and orthonormal bases within each, we can construct an orthonormal basis u1, u2,... in H such that P [|Z1u1 + ··· Zk1 uk1 | > 1/2] < 1/2 P[| + ··· | > / ] < / Zk1+1uk1+1 Zk2 uk2 1 4 1 4 (2) . . . < .

for some 1 ≤ k1 < k2 < . . .. (For details see Frame 16 below.)

6 / 21 If f ∈ B∗ then

k XN hf , Z i = lim hf , uniZn P-a.s. (3) N→∞ n=1 This is Gaussian, as we see below.

Constructing a Gaussian variable

By Borel-Cantelli it follows that Z = Z1u1 + ··· + Zk1 uk1 + Zk1+1u1 + ··· Zk2 uk2 + ···

is convergent almost surely in the completion

B = H

of H relative to | · |.

7 / 21 Constructing a Gaussian variable

By Borel-Cantelli it follows that Z = Z1u1 + ··· + Zk1 uk1 + Zk1+1u1 + ··· Zk2 uk2 + ···

is convergent almost surely in the completion

B = H

of H relative to | · |. If f ∈ B∗ then

k XN hf , Z i = lim hf , uniZn P-a.s. (3) N→∞ n=1 This is Gaussian, as we see below.

7 / 21 The inclusion i : H → B being continuous (Frame 14 below), for any f ∈ B∗ the restriction

f |H = f ◦ i

is continuous linear on H and hence in H∗. Hence (5) is ﬁnite.

Variance bound

 2 kN h 2i X E hf , Z i ≤ lim inf E  hf , uniZn  (4) N→∞   n=1

using Fatou. Next since the Zn’s are independent Gaussians, E 2 each with [Zn ] = 1, we have

∞ h 2i X 2 E hf , Z i ≤ hf , uni . (5) n=1

8 / 21 Variance bound

 2 kN h 2i X E hf , Z i ≤ lim inf E  hf , uniZn  (4) N→∞   n=1

using Fatou. Next since the Zn’s are independent Gaussians, E 2 each with [Zn ] = 1, we have

∞ h 2i X 2 E hf , Z i ≤ hf , uni . (5) n=1 The inclusion i : H → B being continuous (Frame 14 below), for any f ∈ B∗ the restriction

f |H = f ◦ i

is continuous linear on H and hence in H∗. Hence (5) is ﬁnite.

8 / 21 Variance

Then the convergence in (3) is in L2(P), and hence also in L1(P). Then

k XN E [hf , Z i] = lim hf , uniE[Zn] = 0 N→∞ n=1 and h i kN E 2 X 2 2 hf , Z i = lim hf , uni = ||f ||H∗ N→∞ n=1 (Technically the element on the right is f ◦ i, not just f .)

9 / 21 Gaussian nature

Then for any t ∈ R we have

k ithf ,Zi it P N hf ,u iZ e = lim e n=1 n n N→∞

in L1(P) [using |eia − eib| ≤ |a − b|] and so applying E we obtain the characteristic function of Z :

2 h ithf ,Z ii it·0− t ||f ||2 E e = e 2 H∗ (6)

2 Thus hf , Zi is Gaussian with mean 0 and variance ||f ||H∗ .

10 / 21 Deﬁnition: The distribution measure µ of Z is Gaussian measure on B.

AWS summary

Thus, starting with a measurable norm | · | on a real separable Hilbert space we have constructed a process Z with values in B = H such that

2 hf , Zi is Gaussian with mean 0 and variance ||f ||H∗ for all f ∈ B∗ ⊂ H∗.

11 / 21 AWS summary

Thus, starting with a measurable norm | · | on a real separable Hilbert space we have constructed a process Z with values in B = H such that

2 hf , Zi is Gaussian with mean 0 and variance ||f ||H∗ for all f ∈ B∗ ⊂ H∗. Deﬁnition: The distribution measure µ of Z is Gaussian measure on B.

11 / 21 Dense duals

Consider continuous linear

i : H → B,

where H is Hilbert and B is Banach. Then

i∗ : B∗ → H∗ : f 7→ i∗(f ) = f ◦ i

is continuous linear. If Ran(i∗) is not dense in H∗ then all element of i∗(B∗) vanish on some nonzero v ∈ H, and this means f i(v) = 0 for all f ∈ B∗ and hence i(v) = 0. So, if i is injective then i∗ has dense range. Moreover, if i(H) is dense in B then i∗ is injective.

12 / 21 By identifying H ' H∗ we can view I as being deﬁned on H. We can verify then that

Cov (I(v), I(w)) = hv, wiH

for all v, w ∈ H.

AWS summary II

Since i∗(B∗) is dense in H∗ and every f ∈ B∗ produces a 2 ∗ Gaussian variable hf , ·i on B with mean 0 and L -norm ||i (f )||H∗ we can extend isometrically to a linear map

H∗ → L2(B, µ): h 7→ I(h)

2 where each I(h) is Gaussian of mean 0 and variance ||h||H∗ .

13 / 21 We can verify then that

Cov (I(v), I(w)) = hv, wiH

for all v, w ∈ H.

AWS summary II

Since i∗(B∗) is dense in H∗ and every f ∈ B∗ produces a 2 ∗ Gaussian variable hf , ·i on B with mean 0 and L -norm ||i (f )||H∗ we can extend isometrically to a linear map

H∗ → L2(B, µ): h 7→ I(h)

2 where each I(h) is Gaussian of mean 0 and variance ||h||H∗ . By identifying H ' H∗ we can view I as being deﬁned on H.

13 / 21 AWS summary II

Since i∗(B∗) is dense in H∗ and every f ∈ B∗ produces a 2 ∗ Gaussian variable hf , ·i on B with mean 0 and L -norm ||i (f )||H∗ we can extend isometrically to a linear map

H∗ → L2(B, µ): h 7→ I(h)

2 where each I(h) is Gaussian of mean 0 and variance ||h||H∗ . By identifying H ' H∗ we can view I as being deﬁned on H. We can verify then that

Cov (I(v), I(w)) = hv, wiH

for all v, w ∈ H.

13 / 21 Continuity of i : H → B

Suppose i is not continuous. Then for any ﬁnite dimensional subspace F of H the restriction of i to F ⊥ is discontinuous (because i|F is continuous). Consider any > 0, and ﬁx any t > 0. By discontinuity of i|F ⊥ there are vectors in F ⊥ of unit H-norm but of arbitrarily large | · |B-norm; hence there is a ⊥ v ∈ F with ||v||H = 1 and |v|B > /t. With Z the standard Gaussian on R we have Z ∞ −1/2 −x2/2 P[|Zv|B > ] ≥ P[|Z| > t] = 2 (2π) e dx, t with the right side a positive value having nothing to do with . This contradicts the measurable norm property of | · |B

14 / 21 In analysis we often work with multiple norms that are continuous, in one way or another, with respect to each other. The notion of measurable norm is distinct and unique in that it is a relationship between norms that involves the measure of balls.

Measurable norm II

The condition for | · | to be a measurable norm on H is thus: for any > 0 there is a ﬁnite-dimensional subspace F ⊂ H such that

GaussF1 [v ∈ F1 : |v| > ] < ⊥ for any ﬁnite dimensional subspace F1 ⊂ F .

15 / 21 Measurable norm II

The condition for | · | to be a measurable norm on H is thus: for any > 0 there is a ﬁnite-dimensional subspace F ⊂ H such that

GaussF1 [v ∈ F1 : |v| > ] < ⊥ for any ﬁnite dimensional subspace F1 ⊂ F . In analysis we often work with multiple norms that are continuous, in one way or another, with respect to each other. The notion of measurable norm is distinct and unique in that it is a relationship between norms that involves the measure of balls.

15 / 21 Orthonormal basis detail Let i : H → B be an AWS, where H is separable. Let x1, x2,... be dense in H. Choose a ﬁnite dimensional subspace S1 ⊂ H such that GaussT [|v|B > 1/2] < 1/2 ⊥ where T is any ﬁnite dimensional subspace of S1 . Let

F1 = span of S1 and x1.

⊥ Next choose ﬁnite dimensional subspace S2 ⊂ F1 with

GaussT [|v|B > 1/2] < 1/2

⊥ where T is any ﬁnite dimensional subspace of S2 . Let

F2 = span of S2 and x2.

Deﬁne Fn for every positive integer n thus. Then ∪nFn is dense, containing each xi . Choose an orthonormal basis of F1, extend to an orthonormal basis of F2, and so on. 16 / 21 On B∗ there is the covariance form

hf , giµ = Cov(f , g),

assumed nondenerate. Assume also that each f has mean 0. Thus B∗ ⊂ L2(B, µ). The Cameron-Martin space H for (B, µ) is the Hilbert space:

H = H∗ = B∗ ⊂ L2(µ)

Gaussian measure on Banach spaces

A Borel measure µ on a Banach space B is Gaussian if every f ∈ B∗ is Gaussian as a random variable on (B, µ).

17 / 21 Thus B∗ ⊂ L2(B, µ). The Cameron-Martin space H for (B, µ) is the Hilbert space:

H = H∗ = B∗ ⊂ L2(µ)

Gaussian measure on Banach spaces

A Borel measure µ on a Banach space B is Gaussian if every f ∈ B∗ is Gaussian as a random variable on (B, µ). On B∗ there is the covariance form

hf , giµ = Cov(f , g),

assumed nondenerate. Assume also that each f has mean 0.

17 / 21 Gaussian measure on Banach spaces

A Borel measure µ on a Banach space B is Gaussian if every f ∈ B∗ is Gaussian as a random variable on (B, µ). On B∗ there is the covariance form

hf , giµ = Cov(f , g),

assumed nondenerate. Assume also that each f has mean 0. Thus B∗ ⊂ L2(B, µ). The Cameron-Martin space H for (B, µ) is the Hilbert space:

H = H∗ = B∗ ⊂ L2(µ)

17 / 21 Gaussian measure on Banach spaces For any φ ∈ B∗ and v ∈ H we have the L2(µ) pairing Z hφ, viµ = φ(x)v(x) dµ(x), (7) B This is the evaluation of φ ∈ B∗ on the B-valued integral Z xv(x) dµ(x) ∈ B (8) B (Badly disappointing, to have to use the Bochner integral, but see Driver’s notes [4] for a detailed development of the Bochner integral.) From B∗ ⊂ H∗ we cannot just conclude that H lies inside B. But we can imbed H into B using: Z i : H → B : v 7→ xv(x) dµ(x), (9) B

18 / 21 I Wiener measure on the sup-normed Banach space d C0([0, 1]; R ), with initial value 0, is Gaussian measure whose Cameron-Martin space works out to be the Hilbert space of all absolutely continuous functions d 0 f ∈ C0([0, 1]; R ) for which the a.e.-deﬁned derivative f is in L2([0, 1]; Rd ).

I If i : H → B is an AWS in which the norm on B is actually a Hilbert norm then i is Hilbert-Schmidt. Conversely any HS injection gives an AWS.

Other results

Among the vast array of other results we note:

I If µ is centered nondegenerate Gaussian on a separable Banach space B then i : H → B is an AWS, where H is the Cameron-Martin space.

19 / 21 I If i : H → B is an AWS in which the norm on B is actually a Hilbert norm then i is Hilbert-Schmidt. Conversely any HS injection gives an AWS.

Other results

Among the vast array of other results we note:

I If µ is centered nondegenerate Gaussian on a separable Banach space B then i : H → B is an AWS, where H is the Cameron-Martin space.

I Wiener measure on the sup-normed Banach space d C0([0, 1]; R ), with initial value 0, is Gaussian measure whose Cameron-Martin space works out to be the Hilbert space of all absolutely continuous functions d 0 f ∈ C0([0, 1]; R ) for which the a.e.-deﬁned derivative f is in L2([0, 1]; Rd ).

19 / 21 Other results

Among the vast array of other results we note:

I If µ is centered nondegenerate Gaussian on a separable Banach space B then i : H → B is an AWS, where H is the Cameron-Martin space.

I If i : H → B is an AWS in which the norm on B is actually a Hilbert norm then i is Hilbert-Schmidt. Conversely any HS injection gives an AWS.

19 / 21 BibliographyI There is a huge literature, but the following is a list I consulted (6= read) for these notes in decreasing order of depth of consultation: L. Gross, Abstract Wiener Spaces, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 1. Berkeley, Calif.: Univ. California Press. pp. 31-42. (1967) H.-H. Kuo, Gaussian measures on Banach spaces, Lecture Notes in Math., no. 463, Springer-Verlag (1975). Nate Eldredge, online notes at Cornell: http: //www.math.cornell.edu/~neldredge/7770/ Bruce Driver’s online notes on probability theory: follow link in [3]

20 / 21 BibliographyII

Daniel W. Stroock, Abstract Wiener Spaces, Revisited, Communications on Stochastic Analysis Vol. 2, No. 1 (2008) 145-151.

21 / 21