Brownian Motion, the Gaussian Lévy Process

Transcription

1 Brownian Motion, the Gaussian Lévy Process Deconstructing Brownian Motion: My construction of Brownian motion is based on an idea of Lévy s; and in order to exlain Lévy s idea, I will begin with the following line of reasoning. Assume that {Bt : t 0} is a Brownian motion in R N. That is, {Bt : t 0} starts at 0, has indeendent increments, any increment Bs + t Bs has distribution γ 0,tI, and the aths t Bt are continuous. Next, given n N, let t B n t be the olygonal ath obtained from t Bt by linear interolation during each time interval [m n, m + n ]. Thus, B n t = Bm n + n t m n B m + n Bm n for m n t m + n. The distribution of {B 0 t : t 0} is very easy to understand. Namely, if X m,0 = Bm Bm for m, then the X m,0 s are indeendent, standard normal R N -valued random variables, B 0 m = m n X m,0, and B 0 t = m tb 0 m + t m + B 0 m for m t m. To understand the relationshi between successive B n s, observe that B n+ m n = B n m n for all m N and that X m,n+ n + B n+ m n B n m n = n + B m n B m n + B m n = n [B m n B m n B m n B m n ], and therefore {X m,n+ : m } is again a sequence of indeendent standard normal random variables. What is less obvious is that {X m,n : m, n Z + N} is also a family of indeendent random variables. In fact, checking this requires us to make essential use of the fact that we are dealing with Gaussian random variables. In rearation for roving the receding indeendence assertion, say that G L P; R is a Gaussian family if G is a linear subsace and each element of G is a centered i.e., mean value 0, R-valued Gaussian random variable. { My interest in Gaussian families at this oint is that the linear san GB of ξ, Bt : t 0 and ξ R } N is one. To see this, simly note that, for any R N 0 = t 0 < t < t n and ξ,..., ξ n R N, n ξm, Bt m n n = ξm, Bt R N l Bt l R N l= m=l which, as a linear combination of indeendent centered Gaussians, is itself a centered Gaussian. The crucial fact about Gaussian families is the content of the next lemma. R N,

2 Lemma. Suose that G L P; R is a Gaussian family. Then the closure of G in L P; R is again a Gaussian family. Moreover, for any S G, S is indeendent of S G, where S is the orthogonal comlement of S in L P; R. Proof: The first assertion is easy since Gaussian random variables are closed under convergence in robability. Turning to the second art, what I must show is that if X,..., X n S and X,..., X n S G, then [ n ] [ P e n ξ m X m e n ] [ ξ m X m = P e n ] ξ m X m P e ξ m X m for any choice of {ξ m : m n} {ξ m : m n} R. But the exectation value on the left is equal to ex n P ξm X m + ξ mx m = ex n P [ n ] [ = P e n ξ m X m P ξ m X m n P since P [X m X m ] = 0 for all m, m n. e ξ m X m ], ξ mx m Armed with Lemma, we can now check that {X m,n : m, n Z + N} is indeendent. Indeed, since, for all m, n Z + N and ξ R N, ξ, X m,n R N a member of the Gaussian family GB, all that we have to do is check that, for each m, n Z + N, l N, and ξ, η R N, P[ ξ, X m,n+ R N η, Bl n R N ] = 0. But, since, for s t, Bs is indeendent of Bt Bs, P[ ξ, Bs R N η, Bt R N ] = P [ ξ, Bs R N η, Bs and therefore n P[ ξ, X m,n+ R N η, Bl n R N ] = P[ ξ, B m n R N η, Bl n RN ] R N ] = s ξ, η R N P[ ξ, B m n + B m n η, Bl n RN ] R ] N = n ξ, η R N [ m l m l + m l = 0.

3 Lévy s Construction of Brownian Motion: Lévy s idea was to invert the reasoning just given. That is, start with a family {X m,n : m, n Z + N} of indeendent N0, I-random variables. Next, define {B n t : t 0} inductively so that t B n t is linear on each interval [m n, m n ], B 0 m = l m X l,0, m N, B n+ m n = B n m n for m N, and B n+ m n = B n m n + n X m,n+ for m Z +. If Brownian motion exists, then the distribution of {B n t : t 0} is the distribution of the rocess obtained by olygonalizing it on each of the intervals [m n, m n ], and so the limit lim n B n t should exist uniformly on comacts and should be Brownian motion. To see that this rocedure works, one must first verify that the receding definition of {B n t : t 0} gives a rocess with the correct distribution. That is, we need to show that { B n m+ n B n m n : m N } is a sequence of indeendent N0, n I-random variables. But, since this sequence is contained in the Gaussian family sanned by {X m,n : m, n Z + N}, Lemma says that we need only show that 3 P[ ξ, B n m + n B n m n R N ξ, B n m + n B n m n ] = n ξ, ξ δ R N m,m RN for ξ, ξ R N and m, m N. When n = 0, this is obvious. Now assume that it is true for n, and observe that B n+ m n B n+ m n = B nm n B n m n n X m,n+ and B n+ m n B n+ m n = B nm n B n m n + n X m,n+. Using these exressions and the induction hyothesis, it is easy to check the required equation. Second, and more challenging, we must show that, P-almost surely, these rocesses are converging uniformly on comact time intervals. For this urose, consider the difference t B n+ t B n t. Since this ath is linear on each interval [m n, m + n ], max Bn+ t B n t = max Bn+ m n B n m n t [0, L ] m L+n+ L+n 4 = n max X m,n+ n X m,n+ 4. m L+n

4 4 Thus, by Jensen s Inequality, P[ ] L+n B n+ B n [0,L ] n P[ X m,n+ 4] 4 = n L 4 4 C N, where C N P[ X,0 4] 4 <. Starting from the receding, it is an easy matter to show that there is a measurable B : [0, Ω R N such that B0 = 0, B, ω C [0, ; R N for each ω Ω, and B n B [0,t] 0 both P-almost surely and in L P; R for every t [0,. Furthermore, since Bm n = B n m n P-almost surely for all m, n N, it is clear that { B m + n Bm n : m 0 } is a sequence of indeendent N0, n I-random variables for all n N. Hence, by continuity, it follows that {Bt : t 0} is a Brownian motion. We have now comleted the Lévy s construction, but, before moving on, it is only roer to recognize that, clever as his method is, Lévy was not the first to construct a Brownian motion. Instead, it was N. Wiener who was the first. In fact, his famous 93 article Differential Sace in J. Math. Phys. # contains three different aroaches. Lévy s Construction in Context: There are elements of Lévy s construction that admit interesting generalizations, erhas the most imortant of which is Kolmogorov s Continuity Criterion. Theorem. Suose that {Xx : x [0, R] N } is a family of random variables taking values in a Banach sace B, and assume that, for some [,, C <, and r 0, ], [ Xy Xx B ] C y x N +r for all x, y [0, R] N. Then there exists a family { Xx : x [0, R] N } of random variables such that Xx = Xx P-almost surely for each x [0, R] N and x [0, R] N Xx, ω B is continuous for all ω Ω. In fact, for each α [0, r, there is a K <, deending only on N,, r, and α, such that su x,y [0,R] N x y Xy Xx B y x α KR N +r α. Wiener s article is remarkable, but I must admit that I have never been convinced that it is comlete. Undoubtedly, my sketicism are more a consequence of my own inetitude than of his.

5 Proof: First note that, by sm easy rescaling argument, it suffices to treat the case when R =. Given n 0, set and observe that [ Mn ] M n = max Xm n X k n k,m N N [0, n ] N B, m k = k,m N N [0, n ] N m k = k,m N N [0, n ] N m k = Xm n X k n B [ Xm n X k n ] B K nr, where K = CN. Let n be given. Because X n x X n x is a multilinear function on each cude m n + [0, n ] N, { su X n y X n x B = max Xn m n X n m n } B. x,y [0,] N m N N [0, n ] N Since X n m n = Xm n and either X n m n = Xm n or X n m n = k N N [0, n ] k m = θ m.k Xk n, where the θ m,k s are non-negative and sum to, it follows that and therefore that [ su x,y [0,] N X n y X n x B M n su x,y [0,] N X n y X n x B Given x R N, x = max j N x j. ] K nr. 5

6 6 Hence [ su su X n y X n x B n >n x,y [0,] N ] K r nr, and so there exists a measurable ma X : [0, ] N Ω B such that x Xx, ω is continuous for each ω Ω and [ su x [0,] N Xx X n x B ] K r nr. Furthermore, Xx = Xx a.s. if x = m n for some n 0 and m N N [0, n ] N. Hence, since x Xx is continuous and [ X[ n x] n Xx B ] C n N +r, it follows that Xx = Xx a.s. for all x [0, ] N. Finally, to rove the final estimate, suose that n < y x n. Then X n y X n x B N n x y M n, and so Xy Xx B su Xξ X n ξ B + N n x y M n. ξ [0,] N Hence, by the receding, su x,y [0,] N n < y x n Xy Xx B y x α K nr α, where K = 4K + N r K, and therefore su x,y [0,] N y x Xy Xx B y x α K r α.

7 Corollary 3. Assume that there is a [,, β > N, and C < such that [ Xy Xx ] B C y x β for all x, y [0, N. Then, for each for each γ > β, Xx lim x x γ = 0 a.s.. Proof: Take α = 0 in Theorem. Then and so [ [ su x [ n, n ] N [ γn su x [ m, N Xx X0 B x γ ] su x [0, n n] N Xx X0 B Xx X0 B x γ ] ] γ K β γn, γ K β γ β γm. 7 Wiener Measure and Brownian Motion: Given a robability sace Ω, F, P, a non-decreasing family {F t : t 0} of sub-σ-algebras, and a family {Bt : t 0} of R N -valued random variables, one says that Bt, F t, P is an R N -valued Brownian motion if i P-almost surely, B0 = 0 and t Bt is continuous. ii For each s 0, Bs is F s -measurable, and, for t > s, Bt Bs is indeendent of F s and has distribution γ 0,t si. ndow C [0, ; R N with the toology of uniform convergence on comacts. quivalently, if the metrice ρ on C [0, ; R N is defined by ρw, w = m w w [0,m] + w w [0,m] for w, w C [0, ; R N, then we are giving C [0, ; R N the toology determined by ρ. It is an elementary exercise to show that this metric sace is searable and comlete. Furtheremore, the associated Borel field B is the smallest σ-algebra σ {wt : t 0} for which all the mas w wt are measurable. Indeed, since w wt is continuous, it is Borel measurable, and

8 8 therefore σ {wt : t 0} B. To rove the oosite inclusion, begin by observing that every oen subset of C [0, ; R N can be written as the countable union of sets of the form B T w = {w : w w [0,T ] r}. Since B T w = t Q [0,T ] {w : w t wt r} σ {wt : t 0}, where Q denotes the set of rational numbers, there is nothing more to do. In view of the receding, we know that two Borel, robability measures µ and µ on C [0, ; R N are equal if, for all n and 0 t < < t n, the distribution of w wt,..., wt n is the same under µ and µ. In articular, the measure induced on C [0, ; R N by one Brownian motion is the same as that induced by any other Brownian motion. Namely, if µ is such a measure, then µ {w : w0 = 0} = and, for each 0 s < t, wt ws is indeendent of B s σ {wτ : τ [0, s]} and has distribution γ 0,t si. Hence, for any n and 0 = t 0 < t < < t n, wt wt 0,..., wt n wt n are mutually indeendent and the mth one has distribution γ 0,tm t m I. This unique measure is called Wiener measure, and I will use W is denote it.