Notes 27 : Brownian motion: path properties

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1 Notes 27 : Browia motio: path properties Math : Theory of Probability Lecturer: Sebastie Roch Refereces:[Dur10, Sectio 8.1], [MP10, Sectio 1.1, 1.2, 1.3]. Recall: DEF 27.1 (Covariace) Let X = (X 1,..., X d ) be a radom vector with mea µ = E[X]. The covariace of X is the d d matrix Γ with etries Γ ij = Cov[X i, X j ] = E[(X i µ i )(X j µ j )]. DEF 27.2 (Gaussia distributio) A stadard Gaussia is a RV Z with CF φ Z (t) = exp ( t 2 /2 ), ad desity f Z (x) = 1 2π exp ( x 2 /2 ). I particular, Z has mea 0 ad variace 1. More geerally, X = σz + µ, is a Gaussia RV with mea µ R ad variace σ 2 > 0. DEF 27.3 (Multivariate Gaussia) A d-dimesioal stadard Gaussia is a radom vector X = (X 1,..., X d ) where the X i s are idepedet stadard Gaussias. I particular, X has mea 0 ad covariace matrix I. More geerally, a radom vector X = (X 1,..., X d ) is Gaussia if there is a vector b, a d r matrix A ad a r-dimesioal stadard Gaussia Y such that X = AY + b. The X has mea µ = b ad covariace matrix Γ = AA T. The CF of X is give by d φ X (t) = exp i t j µ j 1 d t j t k Γ jk. 2 1 j,k=1

2 Lecture 27: Browia motio: path properties 2 COR 27.4 (Idepedece) Let X = (X 1,..., X d ) be a multivariate Gaussia. The the X i s are idepedet if ad oly if Γ ij = 0 for all i j, that is, if they are ucorrelated. COR 27.5 (Liear combiatios) The radom vector (X 1,..., X d ) is multivariate Gaussia if ad oly if all liear combiatios of its compoets are Gaussia. DEF 27.6 (Gaussia process) A cotiuous-time stochastic process {X(t)} t 0 is a Gaussia process if for all 1 ad 0 t 1 < < t < + the radom vector (X(t 1 ),..., X(t )), is multivariate Gaussia. The mea ad covariace fuctios of X are E[X(t)] ad Cov[X(s), X(t)] respectively. DEF 27.7 (Browia motio: Defiitio I) The cotiuous-time stochastic process X = {X(t)} t 0 is a stadard Browia motio if X is a Gaussia process with almost surely cotiuous paths, that is, such that X(0) = 0, ad P[X(t) is cotiuous i t] = 1, E[X(t)] = 0, Cov[X(s), X(t)] = s t. More geerally, B = σx + x is a Browia motio started at x. DEF 27.8 (Statioary idepedet icremets) A SP {X(t)} t 0 has statioary icremets if the distributio of X(t) X(s) depeds oly o t s for all 0 s t. It has idepedet icremets if the RVs {X(t j+1 X(t j )), 1 j < } are idepedet wheever 0 t 1 < t 2 < < t ad 1. DEF 27.9 (Browia motio: Defiitio II) The cotiuous-time stochastic process X = {X(t)} t 0 is a stadard Browia motio if X has almost surely cotiuous paths ad statioary idepedet icremets such that X(s + t) X(s) is Gaussia with mea 0 ad variace t. THM (Existece) Stadard Browia motio B = {B(t)} t 0 exists.

3 Lecture 27: Browia motio: path properties 3 1 Ivariace We begi with some useful ivariace properties. The followig are immediate. THM (Time traslatio) Let s 0. If B(t) is a stadard Browia motio, the so is X(t) = B(t + s) B(s). THM (Scalig ivariace) Let a > 0. If B(t) is a stadard Browia motio, the so is X(t) = a 1 B(a 2 t). Proof: (Sketch) We compute the variace of the icremets: Var[X(t) X(s)] = Var[a 1 (B(a 2 t) B(a 2 s))] = a 2 (a 2 t a 2 s) = t s. THM (Time iversio) If B(t) is a stadard Browia motio, the so is { 0, t = 0, X(t) = tb(t 1 ), t > 0. Proof: (Sketch) We compute the covariace fuctio for s < t: Cov[X(s), X(t)] = Cov[sB(s 1 ), tb(t 1 )] = st ( s 1 t 1) = s. It remais to check cotiuity at 0. Note that { } lim B(t) = 0 = { B(t) 1/m, t Q (0, 1/)}, t 0 m 1 1 ad { } lim X(t) = 0 t 0 = m 1 { X(t) 1/m, t Q (0, 1/)}. 1 (We are usig cotiuity o t > 0.) The RHSs have the same probability because the distributios o all fiite-dimesioal sets (icludig 0) ad therefore o the ratioals are the same. The LHS of the first oe has probability 1. Typical applicatios of these are:

4 Lecture 27: Browia motio: path properties 4 COR For a < 0 < b, let T (a, b) = if {t 0 : B(t) {a, b}}. The E[T (a, b)] = a 2 E[T (1, b/a)]. I particular, E[T ( b, b)] is a costat multiple of b 2. Proof: Let X(t) = a 1 B(a 2 t). The, E[T (a, b)] = a 2 E[if{t 0, : X(t) {1, b/a}}] = a 2 E[T (1, b/a)]. COR Almost surely, t 1 B(t) 0. Proof: Let X(t) be the time iversio of B(t). The B(t) lim = lim X(1/t) = X(0) = 0. t t t 2 Modulus of cotiuity By costructio, B(t) is cotiuous a.s. I fact, we ca prove more. DEF (Hölder cotiuity) A fuctio f is said locally α-hölder cotiuous at x if there exists ε > 0 ad c > 0 such that f(x) f(y) c x y α, for all y with y x < ε. We refer to α as the Hölder expoet ad to c as the Hölder costat. THM (Holder cotiuity) If α < 1/2, the almost surely Browia motio is everywhere locally α-hölder cotiuous. Proof:

5 Lecture 27: Browia motio: path properties 5 LEM There exists a costat C > 0 such that, almost surely, for every sufficietly small h > 0 ad all 0 t 1 h, B(t + h) B(t) C h log(1/h). Proof: Recall our costructio of Browia motio o [0, 1]. Let ad D = {k2 : 0 k 2 }, D = =0D. Note that D is coutable ad cosider {Z t } t D a collectio of idepedet stadard Gaussias. Let Z 1, t = 1, F 0 (t) = 0, t = 0, liearly, i betwee. ad for 1 Fially 2 (+1)/2 Z t, t D \D 1, F (t) = 0, t D 1, liearly, i betwee. B(t) = F (t). =0 Each F is piecewise liear ad its derivative exists almost everywhere. By costructio, we have F F 2. Recall that there is N (radom) such that Z d < c for all d D with > N. I particular, for > N we have F < c 2 (+1)/2. Usig the mea-value theorem, assumig l > N, B(t + h) B(t) F (t + h) F (t) =0 l h F + =0 h =l+1 N F + ch =0 2 F, l 2 /2 + 2c =N =l+1 2 /2.

6 Lecture 27: Browia motio: path properties 6 (The idea above is that the sup orm ad the sup orm of the derivatives by themselves are ot good eough. But each is good i its ow domai: derivative for small because of the h, sup orm for large because the series is summable. You eed to combia them ad fid the right breakpoit, that is, whe both are essetially equal.) Take h small eough that the first term is smaller tha h log(1/h) ad l defied by 2 l < h 2 l+1 exceeds N. The approximatig the secod ad third terms by their largest elemet gives the result. We go back to the proof of the theorem. For each k, we ca fid a h(k) small eough so that the result applies to the stadard BMs ad {B(k + t) B(k) : t [0, 1]}, {B(k + 1 t) B(k + 1) : t [0, 1]}. (By the same kid of ivariace argumets we used before, time reversal preserves stadard BM. We eed the time reversal because the theorem is stated oly for icremets i oe directio.) Sice there are coutably may itervals [k, k + 1), such h(k) s exist almost surely o all itervals simultaeously. The ote that for ay α < 1/2, if t [k, k + 1) ad h < h(k) small eough, B(t + h) B(t) C h log(1/h) Ch α (= Ch 1/2 (1/h) (1/2 α) ). This cocludes the proof. I fact: THM (Lévy s modulus of cotiuity) Almost surely, lim sup h 0 sup 0 t 1 h B(t + h) B(t) 2h log(1/h) = 1. For the proof, see [MP10]. This result is tight. See [MP10, Remark 1.21]. 3 No-Mootoicity A first example of irregularity : THM Almost surely, for all 0 < a < b < +, stadard BM is ot mootoe o the iterval [a, b].

7 Lecture 27: Browia motio: path properties 7 Proof: It suffices to look at itervals with ratioal edpoits because ay geeral o-degeerate iterval of mootoicity must cotai oe of those. Sice there are coutably may ratioal itervals, it suffices to prove that ay particular oe has probability 0 of beig mootoe. Let [a, b] be such a iterval. Note that for ay fiite sub-divisio a = a 0 < a 1 < < a 1 < a = b, the probability that each icremet satisfies B(a i ) B(a i 1 ) 0, i = 1,...,, or the same with egative, is at most ( ) 1 2 0, 2 as by symmetry of Gaussias. More geerally, we ca prove the followig. THM Almost surely, BM satisfies: 1. The set of times at which local maxima occur is dese. 2. Every local maximum is strict. 3. The set of local maxima is coutable. Proof: Part (3). We use part (2). If t is a strict local maximum, it must be i the set + =1 { t : B(t, ω) > B(s, ω), s, s t < 1 }. But for each, the set must be coutable because two such t s must be separated by 1. So the uio is coutable. 4 No-differetiability So B(t) grows slower tha t. But the followig lemma shows that its limsup grows faster tha t. LEM Almost surely Ad similarly for lim if. lim sup + B() = +.

8 Lecture 27: Browia motio: path properties 8 Proof: By reverse Fatou, P[B() > c i.o.] lim sup + P[B() > c ] = lim sup P[B(1) > c] > 0, + by the scalig property. Thikig of B() as the sum of X = B() B( 1), the evet o the LHS is exchageable ad the Hewitt-Savage 0-1 law implies that it has probability 1 (where we used the positive lower boud). DEF (Upper ad lower derivatives) For a fuctio f, we defie the upper ad lower right derivatives as ad D f(t) = lim sup h 0 D f(t) = lim if h 0 We begi with a easy first result. f(t + h) f(t), h f(t + h) f(t). h THM Fix t 0. The almost surely Browia motio is ot differetiable at t. Moreover, D B(t) = + ad D B(t) =. Proof: Cosider the time iversio X. The D X(0) lim sup + X( 1 ) X(0) 1 = lim sup B() = +, + by the lemma above. This proves the result at 0. The ote that X(s) = B(t+s) B(s) is a stadard Browia motio ad differetiability of X at 0 is equivalet to differetiability of B at t. I fact, we ca prove somethig much stroger. THM Almost surely, BM is owhere differetiable. Furthermore, almost surely, for all t D B(t) = +, or or both. D B(t) =,

9 Lecture 27: Browia motio: path properties 9 Proof: Suppose there is t 0 such that the latter does ot hold. By boudedess of BM over [0, 1], we have B(t 0 + h) B(t 0 ) sup M, h [0,1] h for some M < +. Assume t 0 is i [(k 1)2, k2 ] for some k,. The for all 1 j 2 k, i particular, for j = 1, 2, 3, B((k + j)2 ) B((k + j 1)2 ) B((k + j)2 ) B(t 0 ) + B(t 0 ) B((k + j 1)2 ) M(2j + 1)2, by our assumptio. Defie the evets Ω,k = { B((k + j)2 ) B((k + j 1)2 ) M(2j + 1)2, j = 1, 2, 3}. It suffices to show that 2 3 k=1 Ω,k caot happe for ifiitely may. Ideed, [ ] B(t 0 + h) B(t 0 ) P t 0 [0, 1], sup M h [0,1] h [ 2 ] 3 P Ω,k for ifiitely may. (The the result follows by takig all [k, k + 1] itervals ad all M itegers.) But by the idepedece of icremets k=1 P[Ω,k ] = 3 P[ B((k + j)2 ) B((k + j 1)2 ) M(2j + 1)2 ] P [ B(2 ) 7M2 ] 3 [ ( 1 [ = P B ) 2 2 ] 2 [ = P B(1) 7M ] 3 2 ] 7M ( 7M 2 ) 3,

10 Lecture 27: Browia motio: path properties 10 because the desity of a stadard Gaussia is bouded by 1/2. (The choice of 3 comes from summability.) Hece P [ 2 3 k=1 Ω,k ] 2 ( 7M 2 ) 3 = (7M) 3 2 /2, which is summable. The result follows from BC. That is, the probability above is 0. 5 Quadratic variatio Recall: DEF (Bouded variatio) A fuctio f : [0, t] R is of bouded variatio if there is M < + such that k f(t j ) f(t j 1 ) M, for all k 1 ad all partitios 0 = t 0 < t 1 < < t k = t. Otherwise, we say that it is of ubouded variatio. Fuctios of bouded variatio are kow to be differetiable. Sice BM is owhere differetiable, it must have ubouded variatio. However, BM has a fiite quadratic variatio. THM (Quadratic variatio) Suppose the sequece of partitios 0 = t () 0 < t () 1 < < t () k() = t, is ested, that is, at each step oe or more partitio poits are added, ad the mesh () = coverges to 0. The, almost surely, k() lim + sup {t () j t () j 1 }, 1 j k() (B(t () j ) B(t () j 1 ))2 = t.

11 Lecture 27: Browia motio: path properties 11 Proof: By cosiderig subsequeces, it suffices to cosider the case where oe poit is added at each step. Let Let ad k() X = (B(t () j ) B(t () j 1 ))2. G = σ(x, X 1,...) G = G k. k=1 CLAIM We claim that {X } is a reversed MG. Proof: We wat to show that E[X +1 G ] = X. I particular, this will imply by iductio X = E[X 1 G ]. Assume that, at step, the ew poit s is added betwee the old poits t 1 < t 2. Write X +1 = (B(t 2 ) B(t 1 )) 2 + W, ad X = (B(s) B(t 1 )) 2 + (B(t 2 ) B(s)) 2 + W, where W is idepedet of the other terms. We claim that E[(B(t 2 ) B(t 1 )) 2 (B(s) B(t 1 )) 2 + (B(t 2 ) B(s)) 2 ] which follows from the followig lemma. = (B(s) B(t 1 )) 2 + (B(t 2 ) B(s)) 2, LEM Let X, Z L 2 be idepedet ad assume Z is symmetric. The E[(X + Z) 2 X 2 + Z 2 ] = X 2 + Z 2.

12 Lecture 27: Browia motio: path properties 12 Proof: By symmetry of Z, E[(X + Z) 2 X 2 + Z 2 ] = E[(X Z) 2 X 2 + ( Z) 2 ] = E[(X Z) 2 X 2 + Z 2 ]. Takig the differece we get E[XZ X 2 + Z 2 ] = 0. The fact that X is a reversed MG follows from the argumet above. (Exercise.) We retur to the proof of the theorem. By Lévy s Dowward Theorem, X E[X 1 G ], almost surely. Note that E[X 1 ] = E[X ] = t. Moreover, by (FATOU), the variace of the limit (the fourth cetral momet of the Gaussia i 3σ 4 ) So fially E[(E[X 1 G ] t) 2 ] lim if lim if = lim if 3t lim if = 0. E[(X t) 2 ] k() Var k() 3 (t () j () E[X 1 G ] = t. (B(t () j t () j 1 )2 ) B(t () j 1 ))2 Refereces [Dur10] Rick Durrett. Probability: theory ad examples. Cambridge Series i Statistical ad Probabilistic Mathematics. Cambridge Uiversity Press, Cambridge, [MP10] Peter Mörters ad Yuval Peres. Browia motio. Cambridge Series i Statistical ad Probabilistic Mathematics. Cambridge Uiversity Press, Cambridge, With a appedix by Oded Schramm ad Wedeli Werer.