1 The Haar functions and the Brownian motion

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1 1 The Haar fuctios ad the Browia motio 1.1 The Haar fuctios ad their completeess The Haar fuctios The basic Haar fuctio is 1 if x < 1/2, ψx) = 1 if 1/2 x < 1, otherwise. 1.1) It has mea zero 1 ψx)dx =, ad is ormalized so that 1 ψ 2 x)dx = 1. The rescaled ad shifted Haar fuctios are ψ jk x) = 2 j/2 ψ2 j x k), j, k Z. These fuctios form a orthoormal set i L 2 ) because if j = j ad k k the ψ jk x)ψ jk x)dx = 2 j ψ2 j x k)ψ2 j x k )dx = because ψy k)ψy k ) = for ay y ad k k. O the other had, if j j, say, j < j, the ψ jk x)ψ j k x)dx = 2j/2+j /2 = 2 j /2 j/2 = 2 j /2 j/2 1/2 ψy)ψ2 j j y + 2 j j k k )dy ψ2 j x k)ψ2 j x k )dx ψ2 j j y + 2 j j k k )dy 2 j /2 j/2 1 1/2 ψ2 j j y + 2 j j k k )dy. Both of the itegrals above equal to zero. Ideed, 2 j j 2, hece, for istace, because 1/2 j j 2 j j 1 ψ2 j j y + 2 j j k k )dy = 2 m ψy)dy =, ψy + 2 j j k k )dy =, for all m, Z, ad j > j. Fially, whe j = j, k = k we have ψ jk x) 2 = 2 j ψ2 j x k) 2 dx = ψx k) 2 dx = 1. 1

2 The Haar coefficiets of a fuctio f L 2 ) are defied as the ier products ad the Haar series of f is c jk = fx)ψ jk x)dx, 1.2) c jk ψ jk x). 1.3) j,k Z Orthoormality of the family {ψ jk } esures that c jk 2 f 2 L < +, 2 j,k ad the series 1.3) coverges i L 2 ). I order to show that it actually coverges to the fuctio f itself we eed to prove that the Haar fuctios form a basis for L 2 ). Completeess of the Haar fuctios To show that Haar fuctios form a basis i L 2 ) we cosider the dyadic projectios P defied as follows. Give f L 2 ), ad, k Z, cosider the itervals the P fx) = I k = k 1)/2, k/2 ], fdx = 2 I k I k fdx, for x I k. The fuctio P f is costat o each of the dyadic itervals I k. I particular, each Haar fuctio ψ jk satisfies P ψ jk x) = for j, while P ψ jk x) = ψ jk x) for j <. We claim that, actually, for ay f L 2 ) we have P +1 f P f = k Z c k ψ k x), 1.4) with the Haar coefficiets c k give by 1.2). Ideed, let x I k ad write I k = k 1), k ] 2k 1) 2k 1) ] 2k 1) 2k ] =,, = I ,2k 1 I+1,2k. The fuctio P f is costat o the whole iterval I k while P +1 f is costat o each of the sub-itervals I +1,2k 1 ad I +1,2k. I additio, This meas exactly that which is 1.4). I k P f)dx = I k P +1 f)dx. P +1 x) = P fx) + c k ψ k x) for x I k, 2

3 As a cosequece of 1.4) we deduce that P +1 fx) P m fx) = c jk ψ jk x), 1.5) j= m k Z for all m, Z with > m. It remais to show that for ay f L 2 ) we have lim P mfx) =, m + lim P fx) = fx), 1.6) + both i the L 2 -sese. The operators P f are uiformly bouded because for all, k Z we have 2 P f)x) 2 dx = fy)dy fy) 2 dy. I k I k I k Summig over k Z for a fixed we get P fx) 2 fx) 2, thus P f L 2 f L 2. Uiform boudedess of P implies that it is sufficiet to establish both limits i 1.6) for fuctios f C c ). However, for such f we have, o oe had, P m fx) 1 fx) dx as m +, 2 m ad, o the other, f is uiformly cotiuous o, so that P fx) fx) L as +, which, as both P f ad f are compactly supported, implies the secod limit i 1.6). Therefore, ψ jk form a orthoormal basis i L 2 ) ad every fuctio f L 2 ) has the reperesetatio fx) = j,k= 1.2 The Browia motio c jk ψ jk x), c jk = fy)ψ jk y)dy. 1.7) Browia motio is a radom process X t ω), t defied o a probability space Ω, F, P) which has the followig properties: i) The fuctio X t ω) is cotiuous i t for a.e. realizatio ω. ii) For all s < t < + the radom variable X t ω) X s ω) is Gaussia with mea zero ad variace t s: EXt) Xs)) =, EXt) Xs)) 2 = t s. iii) For ay subdivisio = t < t 1 <... < t N = t of the iterval [, t], the radom variables X t1 X t,..., X tn X tn 1 are idepedet. 3

4 Costructio of the Browia motio We will costruct the Browia motio o the iterval t 1 the restrictio to a fiite iterval is a simple coveiece but by o meas a ecessity. The Haar fuctios ψ jk x), with j, k 2 j 1, form a basis for the space L 2 [, 1]. Let us deote accordigly φ x) = ψ jk x) for = 2 j +k, k 2 j 1, ad φ x) = 1 so that {φ } form a orthoormal basis for L 2 [, 1]. Let Z ω),, be a collectio of idepedet Gaussia radom variables of mea zero ad variace oe, that is, We will show that the process P Z < y) = X t ω) = y e y2 Z ω) = t dy. 2π φ s)ds 1.8) is a Browia motio. First, we eed to verify that the series 1.8) coverges i L 2 Ω) for a fixed t [, 1]. Note that m 2 t m t 2 m E Z k ω) φ k s)ds) = φ k s)ds) = χ [,t], φ k 2. k= k= As φ k form a basis for L 2 [, 1], the series 1.8) satisfies the Cauchy criterio ad thus coverges i L 2 Ω). Moreover, for ay s < t 1 we have t k= E X t X s ) 2 = E Z k ω) t k= s = χ [s,t] 2 L 2 = t s, φ k u)du) 2 = k= s φ k u)du) 2 = χ [s,t], φ k 2 k= hece the icremets X t X s have the correct variace. Let us show that they are idepedet: for t < t 1 t 2 < t 3 1: t3 ) t1 E X t3 X t2 )X t1 X t )) = E φ k u)du φ k u )du t = k= t 2 χ [t2 t 3 ], φ k χ [t t 1 ], φ k = χ [t2 t 3 ], χ [t t 1 ] =. k= As the variables X t X s are joitly Gaussia, idepedece of the icremets follows. Cotiuity of the Browia motio I order to prove cotiuity of the process X t ω) defied by the series 1.8) we show that the series coverges uiformly i t almost surely i ω. To this ed let us show that Mω) = sup Z ω) log < + almost surely i ω. 1.9) 4

5 Note that, for each : P Z ω) 2 ) log e 2 log ) 2 /2 = 1, 2 thus = P Z ω) 2 ) log < +. The Borel-Catelli lemma implies that almost surely the evet { Z ω) 2 log } happes oly fiitely may times, so that Z ω) < 2 log for al ω) almost surely, ad 1.9) follows. Aother useful observatio is that for each fixed t ad j N there exists oly oe k so that ad for that k we have t φ 2 j +ks)ds, t φ 2 j +ks)ds 2j/2 2 j = 1 2. j/2 Hece, we may estimate the dyadic blocs, usig 1.9): 2 j 1 t Z 2 j +kω) φ 2 j +ks)ds Mω) j + 1) log 2 k= 2 j 1 k= t ψ jk s)ds jm1 ω). 2 j/2 Therefore, the dyadic blocs are bouded by a coverget series which does ot deped o t [, 1], hece the sum X t ω) of the series is a cotiuous fuctio for a.e. ω. Nowhere differetiability of the Browia motio Theorem 1.1 The Browia path X t ω) is owhere differetiable for almost every ω. Proof. Let us fix β >. The if Ẋ s exists at some s [, 1] ad Ẋs < β the there exists so that X t X s 2β t s if t s 2 1.1) for all >. Let A be the set of fuctios xt) C[, 1] for which 1.1) holds for some s [, 1]. The A A +1 ad the set A = =1 A icludes all fuctios xt) C[, 1] such that ẋs) β at some poit s [, 1]. The ext step is to replace 1.1) by a discrete set of coditios this is a stadard trick i such situatios. Assume that 1.1) holds for a fuctio xt) C[, 1] ad let k = sup{j : j/ s}, the ) ) ) ) ) )) k + 2 k + 1 y k = max x x, k + 1 k x x, k k 1 x x 8β. Therefore, if we deote by B the set of all fuctios xt) C[, 1] for which y k 8β/ for some k, the A B. Therefore, i order to show that PA) = it suffices to check that lim PB ) =. 1.11) 5

6 This, however, ca be estimated directly, usig traslatio ivariace of the Browia motio: 2 [ [ PB ) P max X k + 2 P k=1 = P [ X [ max[ X 1 ) Xk + 1 ), X k + 1 ) Xk ), X k ) Xk 1 ] ) 8β ] ) ) ) ) )] 3 2 X, 2 1 X X, 1 X 8β ] ) 8β ] 3 3 8β/ ) 3 = e dx) x2 /2 16β C 2π 2π, 8β/ which implies 1.11). It follows that PA) = as well, hece Browia motio is owhere differetiable with probability oe. Corollary 1.2 Browia motio does ot have bouded variatio with probability oe. 6