CHAPTER IV - BROWNIAN MOTION

CHAPTER IV - BROWNIAN MOTION JOSEPH G. CONLON 1. Construction of Brownian Motion There are two ways in which the idea of a Markov chain on a discrete state space can be generalized: (1) The discrete time variable of the chain i.e. n =, 1, 2,.., can be made continuous; (2) the state space can be made into a continuum. There are many such examples, but there is one which stands out-brownian motion- and we will study it here. Thus we have a probability space (Ω, F, P ) and a continuous set X t, t, of real variables which have the following properties: (a) X t is a Gaussian variable with mean and variance t. (b) For any sequence = t < t 1 < t m, the variables X tj X tj 1, j = 1,.., m, are independent Gaussian with mean and variance t j t j 1, j = 1,.., m. Evidently the properties (a), (b) determine the cdf of any finite set of variables X t1,..., X tm. Now let Q be the non-negative rational numbers. Then the Kolmogorov construction allows us to create a probability space (Ω, F, P ) and variables X t, t Q, which have the properties (a), (b). The main issue in constructing Brownian motion is to extend this set of variables to a continuous set of variables X t, t. The key to doing this is the continuity result: Proposition 1.1. Let (Ω, F, P ) be a probability space with variables X t, t Q, satisfying (a) and (b). Then for any a > the function t X t (ω), t Q [, a], is uniformly continuous with probability 1. Proof. We take a = 1 and use the notation X t = X(t). For n =, 1, 2,.., we associate to a dyadic interval I n,k = {x [, 1] : (k 1)/2 n < x k/2 n } a random variable Y n,k by (1.1) Y n,k = sup{ X(t) X((k 1)/2 n ) : t I n,k Q }. Observe that the variables Y n,k, k = 1,.., 2 n, are i.i.d., whence it follows that for any δ >, (1.2) P ( max 1 k 2 n Y n,k > δ ) 2 n k=1 P ( Y n,k > δ ) = 2 n P (Y n,1 > δ ). The key point now is to use the maximal function inequality (1.3) P (Y n,1 > δ ) 1 δ p E[ X(1/2n ) p ]. This follows from Proposition 2.1 of Chapter II since any sequence X(t j ), j = 1, 2,.., with t 1 < t 2 <, is a Martingale. One can also prove it using the reflection principle introduced in Chapter I. Since X(t) is Gaussian with mean and variance t we have that (1.4) E[ X(t) 4 ] = 3t 2. 1

2 JOSEPH G. CONLON Hence (1.2), (1.3) imply that (1.5) P ( max 1 k 2 n Y n,k > δ ) 3 δ 4 2 n. Evidently (1.5) implies that (1.6) P ( max n,k 1 k 2 n > δ ) <, n=1 Thus by the Borel-Cantelli lemma one has [ ] (1.7) lim sup max Y n,k δ with probability 1. n 1 k 2 n The inequality (1.7) implies uniform continuity of the function t X t (ω), t Q [, 1], with probability 1 Corollary 1.1. There exists a probability space (Ω, F, P ) and a continuous set X t, t, of real variables with the properties (a), (b). In addition the function t X t, t, is continuous on the interval [, ) with probability 1. Proof. We define the variable X t for non-rational t by X t (ω) = lim{x s (ω); s Q, s t}. From Proposition 1.1 this limit exists with probability 1 for all t and the resulting function t X t (ω) is continuous. It is easy to see that (a) and (b) hold now for all variables X t, t, since lim s t X s = X t with probability 1 implies that lim s t f(x s ) = f(x t ) for all bounded measurable functions f : R R by the dominated convergence theorem. We have constructed Brownian motion and in the process have seen that its paths t X t, t, are continuous with probability 1. The next result shows that they are however very irregular. Proposition 1.2 (Dvoretski, Erdös, Kakutani). Brownian paths t X t, t, are nowhere differentiable with probability 1. Proof. Similarly to Proposition 1.1 let us define random variable Y n,k for k = 1,.., 2 n 2, by (1.8) Y n,k = sup{ X(k/2 n ) X((k 1)/2 n ), X((k+1)/2 n ) X(k/2 n ), X((k+2)/2 n ) X((k+1)/2 n ) }. Then we have by independence of the 3 variables in the definition of Y n,k that ( ) 3 (1.9) P ( Y n,k a/2 n ) = [P ( X(1/2 n ) a/2 n )] 3 2a/2 n. 2π/2 n The inequality (1.9) implies that (1.1) P ( min Y n,k a/2 n ) k 2 n 2 k=1 Thus by the Borel-Cantelli lemma we have that P ( Y n,k a/2 n ) a3 2 2 2 n/2 π 3/2. (1.11) with probability 1 there exists N(ω) such that for n N(ω), Y n,k (ω) a/2 n for all k = 1,.., 2 n 2.

MATH 625-21 3 Fix β > and suppose that a function f : [, 1] R has a derivative at the point s [, 1] with f (s) < β. Then there exists δ > such that f(t) f(s) 2β t s if t s < δ. Assuming now that s I n,k+1 and δ > 2/2 n, we have that (1.12) f((k + 1)/2 n ) f(k/2 n ) 2β/2 n, f((k + 2)/2 n ) f((k + 1)/2 n ) 6β/2 n, f(k/2 n ) f((k 1)/2 n ) 6β/2 n. Choosing a > 6β in (1.11) we conclude that with probability 1 any point s of differentiability of the function t X t, t 1, has derivative which exceeds β in absolute value. The result follows by letting β. It is not difficult to see that Brownian motion X(t), t, satisfies the Markov property i.e. for any Borel set A R, (1.13) P (X(t) A X(t 1 ) = x 1,.., X(t m ) = x m ) = P (X(t) A X(t m ) = x m ), t 1 < t 2 < t m < t. For this reason Brownian motion is a Markov process. We can seek to establish the strong Markov property for Brownian motion. First for t let F t be the σ field generated by the variables X(s), s t. Note that by the continuity property of Brownian motion F t is actually generated by a countable set of variables X(s), s t. In fact any countable dense set in [, t] will suffice. A stopping time τ : Ω [, ) for Brownian motion is a measurable function such that {τ t} F t for all t. Lemma 1.1. Let X t (ω), t, ω Ω, be Brownian motion with probability space (Ω, F, P ) and consider the function from [, ) Ω R defined by X(t, ω) = X t (ω). The function X(, ) is measurable with respect to the product σ algebra B F, where B is the σ field generated by the open sets of [, ). Proof. Consider the function F m : [, 1] R m R defined by linear interpolation, so (1.14) k F m (t, x 1,..., x m ) = [mt k] x k+1 +[k + 1 mt] x k, m t k + 1, k =,.., m 1, m where we have set x =. Evidently F m is a continuous function and hence Borel measurable on [, 1] R m. For n = 1, 2,.. define X n : [, 1] Ω R by (1.15) X n (t, ω) = F m (t, X(1/2 n, ω), X(2/2 n, ω),..., X(1, ω)), where m = 2 n. The function X n : [, 1] Ω R is then measurable with respect to B 1 F, where B 1 is the Borel field generated by open sets of the interval [, 1]. The measurability of X : [, 1] Ω R then follows from the fact that (1.16) lim n X n(t, ω) = X(t, ω) for all t [, 1] with probability 1, ω Ω. Corollary 1.2. Let τ : Ω [, ) be a stopping time for Brownian motion. Then the function ω X(τ(ω), ω) from Ω to R is measurable. Proof. Since the mapping ω (τ(ω), ω) from Ω to [, ) Ω is obviously measurable, the result follows from Lemma 1.1.

4 JOSEPH G. CONLON Proposition 1.3 (Strong Markov Property). Suppose X(t), t, is Brownian motion and τ : Ω [, ) a stopping time. Then the process Y (t) = X(t + τ) X(τ), t, is also a copy of Brownian motion. Rather than proving Proposition 1.3 we shall concentrate on a particular case. The same method can be applied to prove the proposition. The case we have in mind is the Brownian motion version of the reflection principle, which we first encountered for the standard random walk on Z in Lemma 5.1 of Chapter I. Proposition 1.4 (Reflection Principle). Let X t, t, be Brownian motion and M t = sup s t X s be the maximal function. Then for a > there is the identity P (M t a) = 2P (X t a). Proof. We define a stopping time τ : Ω [, ) by (1.17) τ = inf{t : X(t) > a}. We first show that τ < with probability 1. We can see this by using a method to solve problem 6 on homework I. Thus observe that for n = 1, 2,.., (1.18) X(n) = ξ 1 + ξ 2 + + ξ n where the ξ j are i.i.d standard normal. Hence for any α >, ( ) X(n) (1.19) P lim sup n n α < 1 = or 1 since the LHS of (1.19) is a tail event. Assuming the probability is 1, it follows that if H( ) is the Heaviside function then (1.2) lim n E[ H(1 X(n)/nα ) ] = 1. Since X(n) is Gaussian with mean and variance n we have that (1.21) E[ H(1 X(n)/n α ) ] = 1 2π n α 1/2 e z2 /2 dz. Evidently if α < 1/2 then the RHS of (1.21) converges to 1/2 in contradiction to (1.2). We conclude that the probability in (1.19) is, whence τ < with probability 1. Next observe that (1.22) {ω Ω : τ(ω) t} = {ω Ω : M t (ω) a} F t, so τ is a stopping time and X(τ(ω), ω) = a, ω Ω. We have now from (1.17) that that (1.23) P (M t a) = P ( τ t, X(t) X(τ) )+P ( τ t, X(t) X(τ) < ). Since it is clear that (1.24) P ( τ t, X(t) X(τ) ) = P (X(t) a), we just need to establish that (1.25) P ( τ t, X(t) X(τ) < ) = P ( τ t, X(t) X(τ) ). To see this let us take t = 1 wlog and note from (1.2) that

(1.26) P ( τ 1, X(1) X(τ) < ) = 2 n P ( Y n,1 > δ ) + 2 n r=1 2 n P ( Y n,1 > δ ) + MATH 625-21 5 2 n r=1 P ( (r 1)/2 n < τ r/2 n, X(1) X(τ) < ) P ( (r 1)/2 n < τ r/2 n, X(1) X(τ) <, Y n,r δ ) 2 n r=1 where we have used the fact that (1.27) X(1) X(r/2 n ) X(1) X(τ)+ Since P ( (r 1)/2 n < τ r/2 n, X(1) X(r/2 n ) < 2δ ), X(τ) X((r 1)/2 n ) + X(r/2 n ) X((r 1)/2 n ) 2Y n,r. (1.28) {ω Ω : (r 1)/2 n < τ(ω) r/2 n } F r/2 n, we have from the standard reflection principle for fixed time that (1.29) P ( (r 1)/2 n < τ r/2 n, X(1) X(r/2 n ) < 2δ) We conclude then from (1.26)-(1.29) that = P ( (r 1)/2 n < τ r/2 n, X(1) X(r/2 n ) > 2δ). (1.3) P ( τ 1, X(1) X(τ) < ) 2 n P ( Y n,1 > δ )+ 2 n r=1 P ( (r 1)/2 n < τ r/2 n, X(1) X(r/2 n ) > 2δ ). We can do now an exactly parallel argument as in the previous paragraph to bound the RHS of (1.3) from above in terms of P ( τ 1, X(1) X(τ) > ) and a small error. Thus we have that (1.31) P ( (r 1)/2 n < τ r/2 n, X(1) X(r/2 n ) > 2δ ) P ( (r 1)/2 n < τ r/2 n, X(1) X(τ) > 4δ, Y n,r δ ) + P ( Y n,1 > δ ). Evidently (1.3), (1.31) imply that (1.32) P ( τ 1, X(1) X(τ) < ) P ( τ 1, X(1) X(τ) > 4δ )+2 n+1 P ( Y n,1 > δ ). Letting n in (1.32) and using (1.3) with p = 4 we conclude that (1.33) P ( τ 1, X(1) X(τ) < ) P ( τ 1, X(1) X(τ) > 4δ ) for every δ >. Hence by letting δ in (1.33) we see that the LHS of (1.25) does not exceed the RHS. A symmetry argument then implies the identity (1.25). Next we show how expectations for Brownian motion can be obtained by solving differential equations. First we see that the analogue of the backward Kolmogorov

6 JOSEPH G. CONLON equation for the countable state space Markov chain is the heat equation. Thus let T > and u(x, t), be defined by (1.34) u(x, t) = E[ f(x T ) X t = x] = 1 2π(T t) exp [ (x y)2 2(T t) ] f(y) dy. Then (1.34) is stating that the variable X T conditioned on X t = x is Gaussian with mean x and variance T t. One easily sees that u(x, t) is the solution to the terminal value problem, (1.35) u t + 1 2 u xx =, x R, t < T ; lim t T u(x, t) = f(x), x R. Expectations for functions of stopping times can sometimes be computed as solutions to time independent differential equations. We illustrate how this happens with an example: Proposition 1.5. Suppose a < b and λ. Let u λ : (a, b) R be the function (1.36) u λ (x) = E[ e λτ X() = x ], where τ is the first exit time for Brownian motion X(t), t, started at x (a, b) from the interval [a, b], whence X(τ) = a or X(τ) = b. Then u λ ( ) is the unique solution to the differential equation 1 (1.37) 2 with boundary conditions d 2 u λ (x) dx 2 = λu λ (x), a < x < b, (1.38) u λ (a) = u λ (b) = 1. Proof. Defining u λ (x) for a < x < b by (1.36), we first prove that (1.39) lim x a u λ (x) = 1, This follows from the inequality lim u λ (x) = 1. x b (1.4) 1 u λ (x) e λt P (M t > b x), a < x < b. From Proposition 1.4 we have that (1.41) P (M t > a) = 2P ( Z > a/ t ), a >, where Z is standard normal. Thus the limit of the RHS of (1.4) as x b is e λt. Now on letting t we obtain (1.39). The key to proving that the equation (1.37) holds is to observe that (1.42) E[ e λτ H(τ t) X() = x ] = e λt E[ u λ (X(t)) H(τ t) X() = x ], where H( ) is the Heaviside function. This follows since the function H(τ t) is F t measurable. Thus (1.43) E[ e λτ H(τ t) F t ] = e λt H(τ t)e[ e λ(τ t) X(t) ] = H(τ t)u λ (X(t)). Hence we have that (1.44) u λ (x) = e λt 2πt exp [ ] (x y)2 2t u λ (y) dy + Error(t).

MATH 625-21 7 In (1.44) we have extended the function u λ ( ) by 1 outside of the interval (a, b). The error term is bounded by (1.45) Error(t) 2P ( τ t X() = x ) 4P (Z > min{x a, b x}/ t ), where Z is standard normal. We obtain the equation (1.37) by dividing (1.44) by t and letting t. It is easy to see that lim t Error(t)/t =, whence we have that (1.46) lim t { uλ (x) e λt E[ u λ (X(t)) X( = x ] } /t =. The limit in (1.46) yields the equation (1.37) if we assume u λ (y) is C 2 for y in a neighborhood of x, by expanding u λ (X(t)) in a Taylor series about x to second order. Of course the definition (1.36) of u λ ( ) allows us to conclude only that it is a continuous function from the arguments we have been using. There is a gap here in our argument, which can really only be filled by going further into the theory of differential equations. We therefore omit it and leave the proof at this point. It is easy to see that the solution to the boundary value problem (1.37), (1.38) is given by (1.47) u λ (x) = cosh 2λ(x c) cosh 2λR, c = a + b 2, R = b a 2 so c is the center and R is the radius of the interval (a, b). Suppose now we consider the semi-infinite interval (, b) with b >. In that case it is easy to see that (1.48) u λ (x) = exp[ 2λ(b x) ], x < b. Assuming the exit time τ from the interval (, b) conditioned on X() = x, has a density ρ x (s), s >, with respect to Lebesgue measure, then (1.48) yields the formula (1.49) e λt ρ x (s) ds = exp[ 2λ(b x) ]. This is simply an explicit formula for the Laplace transform of the function ρ x (s), s >. We compare this with the formula given by the reflection principle Proposition 1.4. Thus (1.5) P ( τ t X() = ) = 2P ( X(t) > b X() = ) t 2 ρ (s) ds = e z2 /2 dz. 2π b/ t Differentiating the second equation in (1.5) with respect to t, we see that [ ] b (1.51) ρ (t) = exp b2. 2πt 3/2 2t On comparing (1.49) and (1.51) we conclude on setting b = 1/ 2 that [ 1 (1.52) Laplace transform of 2 exp 1 ] = exp[ λ]. πt3/2 4t One can of course verify (1.52) directly, but it is of some interest that the identity is a consequence of a symmetry property of Brownian motion i.e. the reflection principle.,

8 JOSEPH G. CONLON 2. Gaussian Processes We have so far in the study of Brownian motion noted that it is a Markov process. We shall see in this section that it is also a Gaussian Process. To see what this means we need to review some basic facts about Gaussian random variables. Let (Ω, F, P ) be a probability space and Y : Ω R n be a random variable with mean Y ( ) =. We shall think of Y (ω) R n as a column vector. Then Y is Gaussian if its pdf is determined by its covariant matrix Γ, where Γ is a positive definite symmetric n n matrix defined by (2.1) v Γv = [v Y ( )] 2, v R n. The characteristic function χ Y ( ) for Y is then given by the formula [ (2.2) χ Y (σ) = E[ e iσ Y ( ) ] = exp 1 ] 2 σ Γσ, σ R n. The probability measure associated with (2.2) is given by [ (2.3) exp 1 ] n 2 φ Γ 1 φ dφ(j) / normalization. The normalization in (2.3) can be computed explicitly as (2.4) normalization = (2π) n/2 (det Γ) 1/2, but we wish to de-emphasize this since the normalization in (2.3) is determined by the fact that the measure (2.3) is a probability measure. Since the Gaussian variable Y is determined by the symmetric positive definite covariance matrix Γ, its structure can be understood from the structure of the matrix Γ, in particular the fact that Γ has a basis of orthogonal eigenvectors, all with positive eigenvalues. Suppose the eigenvectors are an orthonormal set v 1,..., v n R n, i.e. orthogonal and with Euclidean norm v j = 1, j = 1,.., n. Then the n n matrix O = [v 1,.., v n ] is orthogonal so that (2.5) OO = O O = identity. If the eigenvalues of the v j are λ j, j = 1,.., n, then we can form the diagonal n n matrix Λ, λ 1 (2.6) Λ = λ 2 and one has the identity (2.7) Γ = OΛO. Observe now that since the v j, j = 1,., n, are an orthonormal basis for R n that (2.8) Y (ω) = n (Y (ω), v j )v j = n λj ξ j (ω)v j, ω Ω, where (, ) denotes the Euclidean inner product on R n. Thus (2.8) defines the set of random variables ξ j, j = 1,.., n. It is easy to see that the ξ j, j = 1,.., n, are i.i.d.

MATH 625-21 9 standard normal. This follows from (2.2). Thus let ξ(ω) = (ξ 1 (ω),.., ξ n (ω)) R n and observe from (2.8) that for any σ R n, (2.9) σ ξ(ω) = where n (2.1) w = Noting now that (2.11) Γw = σ j λj (Y (ω), v j ) = (Y (ω), w), n σ j λj v j. n σ j λj v j, we conclude that [ (2.12) χ ξ (σ) = E[ e iσ ξ( ) ] = exp 1 ] 2 w Γw [ = exp 12 ] σ 2, σ R n, whence the ξ j, j = 1,.., n, are i.i.d. standard normal. We wish now to generalize the above considerations to Gaussian processes, by which we mean a continuous set of random variables Y (t), t R, such that any finite set of them (Y (t 1 ),.., Y (t n )) have joint distribution which is Gaussian. Its covariance matrix is now a function Γ(t, s) defined by (2.13) Γ(t, s) = Y (t)y (s) s, t R. We have already seen that Brownian motion is a Gaussian process, and its covariance is given by (2.14) Γ(t, s) = X(t)X(s) = min[s, t]. We would like to obtain an infinite dimensional version of (2.8) and write Brownian motion as a sum of i.i.d. standard normal variables. Thus we wish to write for some suitable functions a j ( ), j =, 1,.., (2.15) X(t) = a j (t)ξ j j= where the ξ j, j =, 1, 2,.., are i.i.d. standard normal. Let us try to find such a representation at least for t in a finite interval, say the interval [, π]. If we were to follow the method above for Gaussian variables Y R n, we would look to find the eigenfunctions of the self-adjoint operator Γ defined by (2.16) Γf(t) = π Γ(t, s)f(s) ds, t π. It is known from the theory of integral equations that the operator (2.16) is compact on L 2 ([, π]), and this implies that there is an orthonormal basis for L 2 ([, π]) consisting of eigenfunctions of Γ. Denoting the orthonormal basis of eigenfunctions

1 JOSEPH G. CONLON by v j (t), t π, j =, 1, 2,..., with corresponding non-negative eigenvalues λ j, j =, 1, 2,.. we expect following (2.8) the identity (2.17) X(t, ω) = λj ξ j (ω)v j (t), ω Ω, t π. j= We cannot however find the eigenfunctions in (2.17) explicitly. To get an explicit representation we need to use the fact that Brownian motion is the derivative of the white noise Gaussian process W (t), t R. This process is defined as the distributional derivative of Brownian motion, so (2.18) f(s)w (s) ds = f (s)x(s) ds for all C functions f : R R with compact support. Formally W (t) = dx(t)/dt or in other words white noise paths are the infinitesimal time increments for Brownian motion. We have already seen that Brownian paths are differentiable nowhere with probability 1, which is why we must define white noise as a distribution. The covariance Γ of (2.13) for white noise is Γ(t, s) = δ(t s), where δ( ) is the Dirac delta function. Thus the corresponding integral operator (2.16) is simply the identity. To see this observe from (2.13) that if X( ) denotes Brownian motion then (2.19) g(t)γ(t, s)f(s) dt ds = [X(s 2 ) X(s 1 )][X(t 2 ) X(t 1 )] = g(t)f(t) dt if g( ) is the characteristic function of the interval [s 1, s 2 ] and f( ) is the characteristic function of the interval [t 1, t 2 ]. Now any orthonormal basis of L 2 ([, π]) is an orthonormal basis of eigenfunctions for the identity operator Γ on L 2 ([, π]) with corresponding eigenvalues 1. Choosing the basis consisting of the functions (2.2) v (t) = 1 π, v j (t) = we conclude from (2.17) that (2.21) W (t) = ξ π + 2 cos jt, π j = 1, 2,..., t π, 2 π ξ j cos jt, t π, where the ξ j, j =, 1,.., are i.i.d. standard normal. We have not rigorously proven (2.21) of course, and in fact we have been rather vague about what we mean about the white noise process W ( ). Let us continue however and formally integrate equation (2.21) from to t, observing now that Brownian motion is the integral of white noise. Thus we obtain the representation (2.22) X(t) = tξ 2 sin jt + ξ j, t π, π π j for Brownian motion. The formula (2.22) is known as the Polya-Wiener representation for Brownian motion. One might ask what is the advantage of (2.22) over

MATH 625-21 11 the representation (2.23) X(t) = t [ξ + ξ 1 + + ξ m ], t = (m + 1) t, m =, 1,..., where the ξ j, j =, 1,.., are i.i.d. standard normal. One advantage is that the partial sums in (2.22) converge in L 2 (Ω) uniformly in t for t 1 since 2 (2.24) sin jt ( ) 2 ξ j sin jt j = C j N, j=n+1 j=n+1 for some constant C. Hence representations like (2.22) can be efficient ways of generating Brownian motion starting with i.i.d. standard normal variables. We have so far defined two Gaussian processes-brownian motion and its derivative the white noise process. We shall define other Gaussian processes by considering the infinite dimensional version of the formula (2.3) for the probability density. Consider Brownian motion φ(t), t, for which we know that if < t 1 < t 2 < t n, the joint pdf for the variables φ(t j ), j = 1,.., n, is given by n {φ(t j ) φ(t j 1 )} (2.25) exp 2 n dφ(t j ) / normalization, 2(t j t j 1 ) where t =. We can rewrite the sum in (2.25) as n {φ(t j ) φ(t j 1 )} 2 tn [ ] 2 dφ(t) (2.26) = dt 2(t j t j 1 ) dt where [t, φ(t)], t t n, is the graph obtained by linear interpolation of the points [, ] and [t j, φ(t j )], j = 1,.., n. Thus we might be tempted to write the Brownian motion measure as [ (2.27) exp 1 [ ] ] 2 dφ(t) dt dφ(t) / normalization. 2 dt Comparing (2.27) with (2.3) we expect that if Γ is the covariance operator for Brownian motion given by (2.14), then [ ] 2 dφ(t) (2.28) [φ( ), Γ 1 φ( )] = dt, dt where (2.29) [f( ), g( )] = t> f(t)g(t) dt, for functions f, g L 2 ([, )). We conclude that for Brownian motion (2.3) Γ 1 g(t) = d2 g(t), t <. dt2 We can verify this intuition directly by observing from (2.14) that (2.31) Γf(t) = t Differentiating (2.31) twice we see that sf(s) ds + t t f(s) ds. (2.32) d2 Γf(t) = f(t), t <, dt2

12 JOSEPH G. CONLON and hence Γ 1 is given by (2.3). It is clear now how we can naturally generalize Brownian motion by considering Gaussian processes for which Γ 1 is a second order positive definite differential operator. Differential operators come together with boundary conditions so we need to be a little careful here. Thus Γ 1 for Brownian motion is actually defined on functions f(t), t, with the Dirichlet boundary condition f() =. Thus (2.33) [f( ), Γ 1 g( )] = f(t)g (t) dt = f (t)g (t) dt, where we have used the boundary condition f() = to integrate by parts. It follows from the second integral formula on the RHS of (2.33) that Γ 1 is positive definite. Consider now Γ 1 defined on functions f : [, 1] R with Dirichlet boundary conditions f() = f(1) = by (2.34) [f( ), Γ 1 g( )] = 1 f(t)g (t) dt = 1 f (t)g (t) dt, whence Γ 1 is positive definite. The Gaussian process associated with (2.34) is called the Brownian bridge. One can easily see from (2.34) that (2.35) Γ(t, s) = s(1 t) if s < t, Γ(t, s) = t(1 s) if s > t. A realization of the Brownian bridge process α(t), t 1, can be given in terms of Brownian motion X(t), t, by the formula α(t) = X(t) tx(1). To see this all we need to do is to verify from (2.14) that the covariance of α(t), t 1, is given by (2.35). Note that α(t), t 1, is not Markovian. Next consider Γ 1 defined on functions f : (, ) R by (2.36) [f( ), Γ 1 g( )] = f(t){ g (t) + g(t)} dt = {f (t)g (t) + f(t)g(t)} dt, whence Γ 1 is positive definite. It seems that we are not imposing any boundary conditions on the functions f( ), g( ), but in the integration by parts we are implicitly assuming Dirichlet boundary condition at t = ± i.e. lim t ± f(t) =. The Gaussian process Y (t), t R, associated with (2.36) is called the Ornstein- Uhlenbeck process. One sees from (2.36) that (2.37) Γ(t, s) = 1 2 e t s. The process Y (t), t R, is Markovian and can be represented by Brownian motion X(s), s, as (2.38) Y (t) = Y ()e t/2 + 1 X(t) 1 2 2 e (s t)/2 X(s) ds, t >. 2 In (2.38) the variable Y () is independent of the Brownian motion. Taking Y () to be Gaussian with mean and variance 1/2, we see from (2.14) that Y (t)y (s) is given by the RHS of (2.37) for s, t. In particular Y (t) is Gaussian with mean and variance 1/2 for all t. Hence the Markov process Y (t) has an invariant measure which is the Gaussian variable with mean zero and variance 1/2. University of Michigan, Department of Mathematics, Ann Arbor, MI 4819-119 E-mail address: conlon@umich.edu t