212a1416 Wiener measure.

Transcription

1 212a1416 Wiener measure. November 11, 2014

2 1 Wiener measure. A refined version of the Riesz representation theorem for measures The Big Path Space. The heat equation. Paths are continuous with probability one. Embedding in S. 2 Stochastic processes and generalized stochastic processes. 3 Gaussian measures. Generalities about expectation and variance. Gaussian measures and their variances. The variance of a Gaussian with density. The variance of Brownian motion. 4 The derivative of Brownian motion is white noise.

3 We begin by constructing Wiener measure following a paper by Nelson, Journal of Mathematical Physics 5 (1964) The construction depends on the refined version of the Riesz representation theorem from last week. I review the statement of the theorem here.

4 The refined version of the Riesz representation theorem for measures Regular measures. Let X be a locally compact Hausdorff space. Let F be a σ-field which contains B(X ). A (non-negative valued) measure µ on F is called regular if 1 µ(k) < for any compact subset K X. 2 For any A F 3 If U X is open then µ(a) = inf{µ(u) : A U, U open} µ(u) = sup{µ(k) : K U, K compact }. The second condition is called outer regularity and the third condition is called inner regularity.

5 The refined version of the Riesz representation theorem for measures The improved version of the Riesz representation theorem. Theorem Let X be a locally compact Hausdorff space, L the space of continuous functions of compact support on X, and I a non-negative linear functional on L. Then there exists a σ-field F containing B(X ) and a non-negative regular measure µ on F such that I (f ) = fdµ (1) for all f L. Furthermore, the restriction of µ to B(X ) is unique.

6 The Big Path Space. The big path space Let Ṙ n denote the one point compactification of R n. Let Ω := Ṙ n (2) 0 t< be the product of copies of Ṙ n, one for each non-negative t. This is an uncountable product, and so a huge space, but by Tychonoff s theorem, it is compact and Hausdorff. We can think of a point ω of Ω as being a function from R + to Ṙ n, i.e. as a curve with no restrictions whatsoever.

7 The Big Path Space. Finite functions Let F be a continuous function on the m-fold product: m F : Ṙ n R, i=1 and let t 1 t 2 t m be fixed times. Define by φ = φ F ;t1,...,t m : Ω R φ(ω) := F (ω(t 1 ),..., ω(t m )). We can call such a function a finite function since its value at ω depends only on the values of ω at finitely many points. The set of such functions satisfies our abstract axioms for a space on which we can define integration.

8 The Big Path Space. Furthermore, the set of such functions is an algebra containing 1 and which separates points, so is dense in C(Ω) by the Stone-Weierstrass theorem. Let us call the space of such functions C fin (Ω). If we define an integral I on C fin (Ω) then, by the Stone-Weierstrass theorem it extends to C(Ω) and therefore, by the Riesz representation theorem, gives us a regular Borel measure on Ω.

9 The Big Path Space. The integrals I x For each x R n we are going to define such an integral, I x by I x (φ) = F (x 1, x 2,..., x m )p(x, x 1 ; t 1 )p(x 1, x 2 ; t 2 t 1 ) p(x m 1, x m, t m t m 1 )dx 1... dx m when φ = φ F,t1,...,t m where p(x, y; t) = 1 (2πt) n/2 e (x y)2 /2t (3) (with p(x, ) = 0) and all integrations are over Ṙ n.

10 The Big Path Space. Checking that this is well defined In order to check that this is well defined, we have to verify that if F does not depend on a given x i then we get the same answer if we define φ in terms of the corresponding function of the remaining m 1 variables. This amounts to the computation p(x, y; s)p(y, z, t)dy = p(x, z; s + t).

11 The Big Path Space. If n = 1 this is the computation 1 e (x y)2 /2s e (y z)22t dy = 2πt R 1 2π(s + t) e(x z)2 /2(s+t). If we make the change of variables u = x y this becomes n t n s = n t+s where n r (x) := 1 r e x2 /2r.

12 The Big Path Space. In terms of our scaling operator S a given by S a f (x) = f (ax) we can write n r = r 1 2 Sr 1 2 n where n is the unit Gaussian n(x) = e x2 /2. Now the Fourier transform takes convolution into multiplication, satisfies (S a f )ˆ= (1/a)S 1/aˆf, and takes the unit Gaussian into the unit Gaussian. Thus, upon taking the Fourier transform, the equation n t n s = n t+s becomes the obvious fact that e sξ2 /2 e tξ2 /2 = e (s+t)ξ2 /2. The same proof (or an iterated version of the one dimensional result) applies in n-dimensions.

13 The Big Path Space. The measure pr x So, for each x R n we have defined a measure on Ω. We denote the measure corresponding to I x by pr x. It is a probability measure in the sense that pr x (Ω) = 1.

14 The Big Path Space. The intuitive idea behind the definition of pr x The intuitive idea behind the definition of pr x is that it assigns probability pr x (E) := p(x, x 1 ; t 1 )p(x 1, x 2 ; t 2 t 1 ) p(x m 1, x m, t m t m 1 )dx 1... dx m E 1 E m to the set of all paths ω which start at x and pass through the set E 1 at time t 1, the set E 2 at time t 2 etc. and we have denoted this set of paths by E.

15 The heat equation. The heat semi-group We pause to reflect upon the computation we did in the preceding section. Define the operator T t on the space S (or on S ) by (T t f )(x) = p(x, y, t)f (y)dy. (4) R n In other words, T t is the operation of convolution with We have verified that t n/2 e x2 /2t. T t T s = T t+s. (5) Also, we have verified that when we take Fourier transforms, If we let t 0 in this equation we get (T t f )ˆ(ξ) = e tξ2 /2ˆf (ξ). (6)

16 The heat equation. lim T t = Identity. (7) t 0 So the T t form a continuous semi-group of operators. If we differentiate (6) with respect to t, and let u(t, x) := (T t f )(x) we see that u is a solution of the heat equation u t = 2 u ( x 1 ) with the initial conditions u(0, x) = f (x). 2 u ( x n ) 2

17 The heat equation. In terms of the operator ( ) 2 := ( x 1 ) ( x n ) 2 we see that T t = e t.

18 Paths are continuous with probability one. The purpose of the next few slides is to prove that if we use the measure pr x, then the set of discontinuous paths has measure zero.

19 Paths are continuous with probability one. We begin with some technical issues. We recall that the statement that a measure µ is regular means that for any Borel set A and for any open set U µ(a) = inf{µ(g) : A G, G open } µ(u) = sup{µ(k) : K U, K compact}. This second condition has the following consequence: Suppose that Γ is any collection of open sets which is closed under finite union. If O = G then µ(o) = sup µ(g) G Γ G Γ since any compact subset of O is covered by finitely many sets belonging to Γ.

20 Paths are continuous with probability one. The importance of this stems from the fact that we can allow Γ to consist of uncountably many open sets, and we will need to impose uncountably many conditions in singling out the space of continuous paths, for example. Indeed, our first task will be to show that the measure pr x is concentrated on the space of continuous paths in R n which do not go to infinity too fast.

21 Paths are continuous with probability one. We begin with the following computation in one dimension: ( ) 1 1/2 pr 0 ({ ω(t) > r}) = 2 e x2 /2t dx 2πt r ( ) 2 1/2 πt r ( ) 2 1/2 t x /2t πt r r t e x2 dx = x r e x2 /2t dx = ( ) 2t 1/2 e r 2 /2t. π r For fixed r > 0 this tends to zero (very fast) as t 0.

22 Paths are continuous with probability one. In n dimensions, y > ɛ in the Euclidean norm implies that at least one of its coordinates y i satisfies y i > ɛ/ n so we find that pr x ({ ω(t) x > ɛ}) c ɛ e ɛ2 /2nt for a suitable constant c that depends only on n. In particular, if we let ρ(ɛ, δ) denote the supremum of the above probability over all 0 t δ then ρ(ɛ, δ) = o(δ). (8)

23 Paths are continuous with probability one. Lemma Let 0 t 1 t m with t m t 1 δ. Let A := {ω ω(t j ) ω(t 1 ) > ɛ for some j = 1,... m}. Then pr x (A) 2ρ( 1 ɛ, δ) (9) 2 independently of the number m of steps.

24 Paths are continuous with probability one. Proof. Let let and let B := {ω ω(t 1 ) ω(t m ) > 1 2 ɛ} C i := {ω ω(t i ) ω(t m ) > 1 2 ɛ} D i := {ω ω(t 1 ) ω(t i ) > ɛ and ω(t 1 ) ω(t k ) ɛ k = 1,... i 1}. If ω A, then ω D i for some i by the definition of A, by taking i to be the first j that works in the definition of A. If ω B and ω D i then ω C i since it has to move a distance of at least 1 2 ɛ to get back from outside the ball of radius ɛ to inside the ball of radius 1 2ɛ. So we have

25 Paths are continuous with probability one. and hence m A B (C i D i ) i=1 pr x (A) pr x (B) + m pr x (C i D i ). (10) Now we can estimate pr x (C i D i ) as follows. For ω to belong to this intersection, we must have ω D i and then the path moves a distance at least ɛ 2 in time t m t i and these two events are independent, so pr x (C i D i ) ρ( ɛ 2, δ) pr x(d i ). Here is this argument in more detail: i=1

26 Paths are continuous with probability one. Let so that F = 1 {(y,z) y z > 1 2 ɛ} 1 Ci = φ F,ti,t n. Similarly, let G be the indicator function of the subset of Ṙ n Ṙ n Ṙ n (i copies) consisting of all points with x k x 1 ɛ, k = 1,..., i 1, x 1 x i > ɛ so that 1 Di = φ G,t1,...,t i. Then

27 Paths are continuous with probability one.... pr x (C i D i ) = p(x, x 1 ; t 1 ) p(x i 1, x i ; t i t i 1 )F (x 1,..., x i )G(x i, x n ) p(x i, x n ; t m t i )dx 1 dx n. The last integral (with respect to x m ) is ρ( 1 2ɛ, δ). Thus pr x (C i D i ) ρ( ɛ 2, δ) pr x(d i ). The D i are disjoint by definition, so prx (D i ) pr x ( D i ) 1. So pr x (A) pr x (B) + ρ( 1 2 ɛ, δ) 2ρ(1 ɛ, δ). 2 QED

28 Paths are continuous with probability one. Let E : {ω ω(t j ) ω(t k ) > 2ɛ for some 1 j < k m}. Then E A since if ω(t j ) ω(t k ) > 2ɛ then either ω(t 1 ) ω(t j ) > ɛ or ω(t 1 ) ω(t k ) > ɛ (or both). So Lemma Let 0 a < b with b a δ. Let pr x (E) 2ρ( 1 ɛ, δ). (11) 2 E(a, b, ɛ) := {ω ω(s) ω(t) > 2ɛ for some s, t [a, b]}. Then pr x (E(a, b, ɛ)) 2ρ( 1 ɛ, δ). 2

29 Paths are continuous with probability one. Proof. Here is where we are going to use the regularity of the measure. Let S denote a finite subset of [a, b] and and let E(a, b, ɛ, S) := {ω ω(s) ω(t) > 2ɛ for some s, t S}. Then E(a, b, ɛ, S) is an open set and pr x (E(a, b, ɛ, S)) < 2ρ( 1 2ɛ, δ) for any S. The union over all S of the E(a, b, ɛ, S) is E(a, b, ɛ). The regularity of the measure now implies the lemma. QED

30 Paths are continuous with probability one. Let k and n be integers, and set Let δ := 1 n. F (k, ɛ, δ) := {ω ω(t) ω(s) > 4ɛ for some t, s [0, k], with t s < δ} Then we claim that pr x (F (k, ɛ, δ)) < 2k ρ( 1 2ɛ, δ). (12) δ Indeed, [0, k] is the union of the nk = k/δ subintervals [0, δ], [δ, 2δ],..., [k δ, δ]. If ω F (k, ɛ, δ) then ω(s) ω(t) > 4ɛ for some s and t which lie in either the same or in adjacent subintervals. So ω must lie in E(a, b, ɛ) for one of these subintervals, and there are kn of them. QED

31 Paths are continuous with probability one. Let ω Ω be a continuous path in R n. Restricted to any interval [0, k] it is uniformly continuous. This means that for any ɛ > 0 it belongs to the complement of the set F (k, ɛ, δ) for some δ. We can let ɛ = 1/p for some integer p. Let C denote the set of continuous paths from [0, ) to R n. Then C = k ɛ δ F (k, ɛ, δ)c so the complement C c of the set of continuous paths is F (k, ɛ, δ), a countable union of sets of measure zero since ) pr x ( δ k ɛ F (k, ɛ, δ) δ lim 2kρ( 1 ɛ, δ)/δ = 0. δ 0 2 We have thus proved a famous theorem of Wiener:

32 Paths are continuous with probability one. Theorem [Wiener.] The measure pr x is concentrated on the space of continuous paths, i.e. pr x (C) = 1. In particular, there is a probability measure on the space of continuous paths starting at the origin which assigns probability pr 0 (E) = p(0, x 1 ; t 1 )p(x 1, x 2 ; t 2 t 1 ) p(x m 1, x m, t m t m 1 )dx 1... dx m E 1 E m to the set of all paths ω which start at 0 and pass through the set E 1 at time t 1, the set E 2 at time t 2 etc. and we have denoted this set of paths by E.

33 Embedding in S. For convenience in notation let me now specialize to the case n = 1. Let W C consist of those paths ω with ω(0) = 0 and 0 (1 + t) 2 w(t) dt <. Proposition [Stroock] The Wiener measure pr 0 is concentrated on W.

34 Embedding in S. Indeed, we let E( ω(t) ) denote the expectation of the function ω(t) of ω with respect to Wiener measure, so E( ω(t) ) = 1 2πt Thus, by Fubini, ( ) E (1 + t) 2 w(t) dt = 0 R x e x2 /2t dx = 1 2πt t 0 0 x t e x2 /t dx = Ct 1/2. (1 + t) 2 E( w(t) ) <. Hence the set of ω with 0 (1 + t) 2 w(t) dt = must have measure zero. QED

35 Embedding in S. Now each element of W defines a tempered distribution, i.e. an element of S according to the rule ω, φ = 0 ω(t)φ(t)dt. (13)

36 Embedding in S. We claim that this map from W to S is measurable and hence the Wiener measure pushes forward to give a measure on S.

37 Embedding in S. To see this, let us first put a different topology (of uniform convergence) on W. In other words, for each ω W let U ɛ (ω) consist of all ω 1 such that sup ω 1 (t) ω(t) < ɛ, t 0 and take these to form a basis for a topology on W. Since we put the weak topology on S it is clear that the map (13) is continuous relative to this new topology. So it will be sufficient to show that each set U ɛ (ω) is of the form A W where A is in B(Ω), the Borel field associated to the (product) topology on Ω.

38 Embedding in S. So first consider the subsets V n,ɛ (ω) of W consisting of all ω 1 W such that sup ω 1 (t) ω(t) ɛ 1 t 0 n. Clearly U ɛ (ω) = n V n,ɛ (ω), a countable union, so it is enough to show that each V n,ɛ (ω) is of the form A n W where A n B(X ). Now by the definition of the topology on Ω, if r is any real number, the set A n,r := {ω 1 ω 1 (r) ω(r) ɛ 1 n is closed.

39 Embedding in S. So if we let r range over the non-negative rational numbers Q +, then A n = r Q + A n,r belongs to B(Ω). But if ω 1 is continuous, then if ω 1 A n then sup t R+ ω 1 (t) ω(t) ɛ 1 n, and so as was to be proved. A n W = V n,ɛ (ω)

40 In the standard probability literature a stochastic process is defined as follows: one is given an index set T and for each t T one has a random variable X (t). More precisely, one has some probability triple (Ω, F, P) and for each t T a real valued measurable function on (Ω, F). So a stochastic process X is just a collection X = {X (t), t T } of random variables. Usually T = Z or Z + in which case we call X a discrete time random process or T = R or R + in which case we call X a continuous time random process. Thus the word process means that we are thinking of T as representing the set of all times.

41 A realization of X, that is the set X (t)(ω) for some ω Ω is called a sample path. If T is one of the above choices, then X is said to have independent increments if for all t 0 < t 1 < t 2 < < t n the random variables X (t 1 ) X (t 0 ), X (t 1 ) X (t 1 ),..., X (t n ) X (t n 1 ) are independent.

42 For example, consider Wiener measure, and let X (t)(ω) = ω(t) (say for n = 1). This is a continuous time stochastic process with independent increments which is known as (one dimensional) Brownian motion. The idea (due to Einstein) is that one has a small (visible) particle which, in any interval of time is subject to many random bombardments in either direction by small invisible particles (say molecules) so that the central limit theorem applies to tell us that the change in the position of the particle is Gaussian with mean zero and variance equal to the length of the interval.

43 Suppose that X is a continuous time random variable with the property that for almost all ω, the sample path X (t)(ω) is continuous. Let φ be a continuous function of compact support. Then the Riemann approximating sums to the integral X (t)(ω)φ(t)dt T will converge for almost all ω and hence we get a random variable where X, φ (ω) = X, φ T X (t)(ω)φ(t)dt, the right hand side being defined (almost everywhere) as the limit of the Riemann approximating sums.

44 The same will be true if φ vanishes rapidly at infinity and the sample paths satisfy (a.e.) a slow growth condition such as given by Stroock s proposition above in addition to being continuous a.e. The notation X, φ is justified since X, φ clearly depends linearly on φ.

45 But now we can make the following definition due to Gelfand. We may restrict φ further by requiring that φ belong to D or S. We then consider a rule Z which assigns to each such φ a random variable which we might denote by Z(φ) or Z, φ and which depends linearly on φ and satisfies appropriate continuity conditions. Such an object is called a generalized random process. The idea is that (just as in the case of generalized functions) we may not be able to evaluated Z(t) at a given time t, but may be able to evaluate a smeared out version Z(φ).

46 The purpose of the next few sections is to do the following computation: We wish to show that for the case Brownian motion, X, φ is a Gaussian random variable with mean zero and with variance 0 0 min(s, t)φ(s)φ(t)dsdt. First we need some results about Gaussian random variables.

47 Generalities about expectation and variance. Let V be a vector space (say over the reals and finite dimensional). Let X be a V -valued random variable. That is, we have some measure space (M, F, µ) (which will be fixed and hidden in this section) where µ is a probability measure on M, and X : M V is a measurable function. If X is integrable, then E(X ) := Xdµ is called the expectation of X and is an element of V. M

48 Generalities about expectation and variance. The function X X is a V V valued function, and if it is integrable, then Var(X ) = E(X X ) E(X ) E(X ) = E (X E(X )) ((X E(X )) is called the variance of X and is an element of V V. It is by its definition a symmetric tensor, and so can be thought of as a quadratic form on V.

49 Generalities about expectation and variance. If A : V W is a linear map, then AX is a W valued random variable, and E(AX ) = AE(X ), Var(AX ) = (A A) Var(X ) (14) assuming that E(X ) and Var(X ) exist. We can also write this last equation as Var(AX )(η) = Var(X )(A η), η W (15) if we think of the variance as quadratic function on the dual space.

50 Generalities about expectation and variance. The function on V given by ξ E(e iξ X ) is called the characteristic function associated to X and is denoted by φ X. Here we have used the notation ξ v to denote the value of ξ V on v V. It is a version of the Fourier transform (with the conventions used by the probabilists). More precisely, let X µ denote the push forward of the measure µ by the map X, so that X µ is a probability measure on V.

51 Generalities about expectation and variance. Then φ X is the Fourier transform of this measure except that there are no powers of 2π in front of the integral and a plus rather than a minus sign is before the i in the exponent. These are the conventions of the probabilists. What is important for us is the fact that the Fourier transform determines the measure, i.e. φ X determines X µ. The probabilists would say that the law of the random variable (meaning X µ) is determined by its characteristic function.

52 Generalities about expectation and variance. To get a feeling for (15) consider the case where A = ξ is a linear map from V to R. Then Var(X )(ξ) = Var(ξ X ) is the usual variance of the scalar valued random variable ξ X. Thus we see that Var(X )(ξ) 0, so Var(X ) is non-negative definite symmetric bilinear form on V. The variance of a scalar valued random variable vanishes if and only if it is a constant. Thus Var(X ) is positive definite unless X is concentrated on hyperplane.

53 Generalities about expectation and variance. Suppose that A : V W is an isomorphism, and that X µ is absolutely continuous with respect to Lebesgue measure, so X µ = ρdv where ρ is some function on V (called the probability density of X ). Then (AX ) µ is absolutely continuous with respect to Lebesgue measure on W and its density σ is given by σ(w) = ρ(a 1 w) det A 1 (16) as follows from the change of variables formula for multiple integrals.

54 Gaussian measures and their variances. Let d be a positive integer. We say that N is a unit (d-dimensional) Gaussian random variable if N is a random variable with values in R d with density (2π) d/2 e (x2 1 +x2 d )/2. It is clear that E(N) = 0 and, since (2π) d/2 x i x j e (x2 1 +x2 d )/2 dx = δ ij, that Var(N) = i δ i δ i (17) where δ 1,..., δ d is the standard basis of R d. We will sometimes denote this tensor by I d.

55 Gaussian measures and their variances. In general we have the identification V V with Hom(V, V ), so we can think of the Var(X ) as an element of Hom(V, V ) if X is a V -valued random variable. If we identify R d with its dual space using the standard basis, then I d can be thought of as the identity matrix.

56 Gaussian measures and their variances. We can compute the characteristic function of N by reducing the computation to a product of one dimensional integrals yielding φ N (t 1,..., t d ) = e (t t2 d )/2. (18) A V -valued random variable X is called Gaussian if (it is equal in law to a random variable of the form) AN + a where A : R d V is a linear map, where a V, and where N is a unit Gaussian random variable.

57 Gaussian measures and their variances. Clearly E(X ) = a, Var(X ) = (A A)(I d ) or, put another way, Var(X )(ξ) = I d (A ξ) and hence φ X (ξ) = φ N (A ξ)e iξ a = e 1 2 I d (A ξ) e iξ a or φ X (ξ) = e Var(X )(ξ)/2+iξ E(X ). (19)

58 Gaussian measures and their variances. It is a bit of a nuisance to carry along the E(X ) in all the computations, so we shall restrict ourselves to centered Gaussian random variables meaning that E(X ) = 0. Thus for a centered Gaussian random variable we have φ X (ξ) = e Var(X )(ξ)/2. (20)

59 Gaussian measures and their variances. Conversely, suppose that X is a V valued random variable whose characteristic function is of the form φ X (ξ) = e Q(ξ)/2, where Q is a quadratic form. Since φ X (ξ) 1 we see that Q must be non-negative definite. Suppose that we have chosen a basis of V so that V is identified with R q where q = dim V. By the principal axis theorem we can always find an orthogonal transformation (c ij ) which brings Q to diagonal form. In other words,

60 Gaussian measures and their variances. if we set η j := i c ij ξ i then Q(ξ) = j λ j η 2 j. The λ j are all non-negative since Q is non-negative definite. So if we set a ij := λ 1 2 j c ij, and A = (a ij ) we find that Q(ξ) = I q (A ξ). Hence X has the same characteristic function as a Gaussian random variable hence must be Gaussian.

61 Gaussian measures and their variances. As a corollary to this argument we see that A random variable X is centered Gaussian if and only if ξ X is a real valued Gaussian random variable with mean zero for each ξ V.

62 The variance of a Gaussian with density. In our definition of a centered Gaussian random variable we were careful not to demand that the map A be an isomorphism. For example, if A were the zero map then we would end up with the δ function (at the origin for centered Gaussians) which (for reasons of passing to the limit) we want to consider as a Gaussian random variable.

63 The variance of a Gaussian with density. But suppose that A is an isomorphism. Then by (16), X will have a density which is proportional to e S(v)/2 where S is the quadratic form on V given by S(v) = J d (A 1 v) and J d is the unit quadratic form on R d : J d (x) = x x 2 d or, in terms of the basis {δ i } of the dual space to Rd, J d = i δ i δ i.

64 The variance of a Gaussian with density. Here J d (R d ) (R d ) = Hom(R d, (R d ) ). It is the inverse of the map I d. We can regard S as belonging to Hom(V, V ) while we also regard Var(X ) = (A A) I d as an element of Hom(V, V ). I claim that Var(X ) and S are inverses to one another. Indeed, dropping the subscript d which is fixed in this computation, Var(X )(ξ, η) = I (A ξ, A η) = η (A I A )ξ when thought of as a bilinear form on V V, and hence Var(X ) = A I A when thought of as an element of Hom(V, V ).

65 The variance of a Gaussian with density. Similarly thinking of S as a bilinear form on V we have S(v, w) = J(A 1 v, A 1 w) = J(A 1 v) A 1 w so S = A 1 J A 1 when S is thought of as an element of Hom(V, V ). Since I and J are inverses of one another, the two above displayed expressions for S and Var(X ) show that these are inverses on one another. This has the following very important computational consequence:

66 The variance of a Gaussian with density. Suppose we are given a random variable X with (whose law has) a density proportional to e S(v)/2 where S is a quadratic form which is given as a matrix S = (S ij ) in terms of a basis of V. Then Var(X ) is given by S 1 in terms of the dual basis of V.

67 The variance of Brownian motion. For example, consider the two dimensional vector space with coordinates (x 1, x 2 ) and probability density proportional to exp 1 ( x s + (x 2 x 1 ) 2 ) t s where 0 < s < t. This corresponds to the matrix ( ) t s(t s) 1 t s = 1 t s 1 t s 1 t s whose inverse is ( s ) s s t which thus gives the variance. ( t s ) 1 1 1

68 The variance of Brownian motion. So, if we let we can write the above variance as ( ) B(s, s) B(s, t). B(t, s) B(t, t) B(s, t) := min(s, t) (21) Now suppose that we have picked some finite set of times 0 < s 1 < < s n and we consider the corresponding Gaussian measure given by our formula for Brownian motion on a one-dimensional space for a path starting at the origin and passing successively through the points x 1 at time s 1, x 2 at time s 2 etc. We can compute the variance of this Gaussian to be (B(s i, s j )) since the projection onto any coordinate plane (i.e. restricting to two values s i and s j ) must have the variance given above.

69 The variance of Brownian motion. Let φ S. We can think of φ as a (continuous) linear function on S. For convenience let us consider the real spaces S and S, so φ is a real valued linear function on S. Applied to Stroock s version of Brownian motion which is a probability measure living on S we see that φ gives a real valued random variable. Recall that this was given by integrating φ ω where ω is a continuous path of slow growth, and then integrating over Wiener measure on paths.

70 The variance of Brownian motion. This is the limit of the Gaussian random variables given by the Riemann approximating sums 1 n (φ(s 1)x φ(s n 2)x n 2) where s k = k/n, k = 1,..., n 2, and (x 1,..., x n 2) is an n 2 dimensional centered Gaussian random variable whose variance is (min(s i, s j )). Hence this Riemann approximating sum is a one dimensional centered Gaussian random variable whose variance is 1 n 2 min(s i, s j )φ(s i )φ(s j ). i,j

71 The variance of Brownian motion. Passing to the limit we see that integrating φ ω defines a real valued centered Gaussian random variable whose variance is min(s, t)φ(s)φ(t)dsdt = 2 sφ(s)φ(t)dsdt, 0 0 as claimed. 0 0 s t (22)

72 The variance of Brownian motion. Let us say that a probability measure µ on S is a centered generalized Gaussian process if every φ S, thought of as a function on the probability space (S, µ) is a real valued centered Gaussian random variable; in other words φ (µ) is a centered Gaussian probability measure on the real line. If we denote this process by Z, then we may write Z(φ) for the random variable given by φ. We clearly have Z(aφ + bψ) = az(φ) + bz(ψ) in the sense of addition of random variables, and so we may think of Z as a rule which assigns, in a linear fashion, random variables to elements of S. With some slight modification (we, following Stroock, are using S instead of D as our space of test functions) this notion was introduced by Gelfand some sixty five years ago. (See Gelfand and Vilenkin, Generalized Functions volume IV.)

73 The variance of Brownian motion. If we have generalized random process Z as above, we can consider its derivative in the sense of generalized functions, i.e. Ż(φ) := Z( φ).

74 To see how this derivative works, let us consider what happens for Brownian motion. Let ω be a continuous path of slow growth, and set ω h (t) := 1 (ω(t + h) ω(t)). h The paths ω are not differentiable (with probability one) so this limit does not exist as a function. But the limit does exist as a generalized function, assigning the value 0 φ(t)ω(t)dt to φ. Now if s < t the random variables ω(t + h) ω(t) and ω(s + h) ω(s) are independent of one another when h < t s since Brownian motion has independent increments.

75 Hence we expect that this limiting process be independent at all points in some generalized sense. (No actual, as opposed to generalized, process can have this property. We will see more of this point in a moment when we compute the variance of Ż.)

76 In any event, Ż(φ) is a centered Gaussian random variable whose variance is given (according to (22)) by ( t ) 2 s φ(s)ds φ(t)dt. 0 We can integrate the inner integral by parts to obtain t 0 0 s φ(s)ds = tφ(t) Integration by parts now yields and 0 0 tφ(t) φ(t)dt = 1 2 ( t 0 t 0 0 ) φ(s)ds φ(t)dt = φ(s)ds. φ(t) 2 dt 0 φ(t) 2 dt.

77 We conclude that the variance of Ż(φ) is given by which we can write as φ(t) 2 dt δ(s t)φ(s)φ(t)dsdt. Notice that now the covariance function is the generalized function δ(s t). The generalized process (extended to the whole line) with this covariance is called white noise because it is a Gaussian process which is stationary under translations in time and its covariance function is δ(s t), signifying independent variation at all times, and the Fourier transform of the delta function is a constant, i.e. assigns equal weight to all frequencies.