KdV PRESERVES WHITE NOISE

Transcription

1 KdV PESEVES WHIE NOISE JEEMY QUASEL AND BENEDEK VALKÓ Abstract. It is shown that white noise is an invariant measre for the Korteweg-deVries eqation on. his is a conseqence of recent reslts of Kappeler and opalov establishing the well-posedness of the eqation on appropriate negative Sobolev spaces, together with a reslt of Cambronero and McKean that white noise is the image nder the Mira transform (icatti map) of the (weighted) Gibbs measre for the modified KdV eqation, proven to be invariant for that eqation by Borgain. 1. KdV on H 1 () and White Noise he Korteweg-deVries eqation (KdV) on = /Z, t 6 x + xxx =, () = f (1.1) defines nonlinear evoltion operators S t f = (t) (1.) < t < on smooth fnctions f :. heorem 1.1. (Kappeler and opalov [K1]) S t extends to a continos grop of nonlinear evoltion operators S t : H 1 () H 1 (). (1.3) In concrete terms, take f H 1 () and let f N be smooth fnctions on with f N f H 1() as N. Let N (t) be the (smooth) soltions of (1.1) with initial data f N. hen there is a niqe (t) H 1 () which we call (t) = S t f with N (t) (t) H 1 (). White noise on is the niqe probability measre Q on the space D() of distribtions on satisfying e i λ, dq() = e 1 λ (1.4) for any smooth fnction λ on where =, are the L (, dx) norm and inner prodct (see [H]). Let {e n } n=,1,,... be an orthonormal basis of smooth fnctions in L () with e = 1. White noise is represented as = n= x ne n where and x n are independent Gassian random variables, each with mean and variance 1. Hence Q is spported in H α () for any α > 1/. Mean zero white noise Q on is the probability measre on distribtions with = satisfying e i λ, dq () = e 1 λ (1.5) for any mean zero smooth fnction λ on. It is represented as = n=1 x ne n. ecall that if f : X 1 X is a measrable map between metric spaces and Q is a probability measre on (X 1, B(X 1 )), then the pshforward f Q is the measre on X given by f Q(A) = Q({x : f(x) A}) for any Borel set A B(X ). Date: November 6, 6. 1

2 JEEMY QUASEL AND BENEDEK VALKÓ Or main reslt is: heorem 1.. White noise Q is invariant nder KdV; for any t, S t Q = Q. (1.6) emarks. 1. In terms of classical soltions of KdV, the meaning of heorem 1. is as follows. Let f N, N = 1,,... be a seqence of smooth mean zero random initial data approximating mean zero white noise. For example, one cold take f N (ω) = N n=1 x n(ω)e n where x n and e n are as above. Solve the KdV eqation for each ω p to a fixed time t to obtain S t f N. he limit in N exists [K1] in H 1 (), for almost every vale of ω, and is again a white noise.. It follows immediately that Ŝ t : L (Q ) L (Q ), (ŜtΦ)(f) = Φ( S t f) (1.7) are a grop of nitary transformations of L (Q ), defining a continos Markov process (t), t (, ) on H 1 () with Gassian white noise one dimensional marginals, invariant nder time+space inversions. he correlation fnctions S(x, t) = f() S t f(x)dq may have an interesting strctre. 3. A similar reslt holds withot the mean zero condition, bt now the mean m = is distribted not as an independent Gassian, bt as one conditioned to have m λ () where λ () is the principal eigenvale of d dx +. Since the addition of constants prodces a trivial rotation in the KdV eqation it seems more natral to consider the mean zero case. 4. Q is certainly not the only invariant measre for KdV. he Gibbs measre formally written as Z 1 1 ( K ) e H where H () = 3 1 x (1.8) is known to be invariant [Bo]. Note that (after sbtraction of the mean) this Gibbs measre is spported on a set of Q -measre. Q is also a Gibbs measre, corresponding to the Hamiltonian H 1 () =. (1.9) he existence of two Gibbs measres corresponds to the bihamiltonian strctre of KdV: It can be written δh i = J i, i = 1, (1.1) δ with symplectic forms J 1 = x x + x and J = x. Becase of all the conservation laws of KdV, there are many other invariant measres as well. 5. We were led to heorem 1. after noticing that the discretization of KdV sed by Krskal and Zabsky in the nmerical investigation of solitons, i = ( i+1 + i + i 1 )( i+1 i 1 ) ( i+ i+1 + i 1 i ), (1.11) preserves discrete white noise (independent Gassians mean and variance σ > ). he invariance follows from two simple properties of the special discretization (1.11). First of all i = b i preserves Lebesge measre whenever b = i ib i =, and (1.11) is of this form. Frthermore, it is easy to check (thogh something of a miracle) that i i is invariant nder (1.11). Hence Z 1 e 1 σ Pi i d i is also invariant. Note that the discretization (1.11) is not completely integrable, and we are not aware of a completely integrable discretization which does conserve discrete white noise. For example, consider the

3 KdV PESEVES WHIE NOISE 3 following family of completely integrable discretizations of KdV, depending on a real parameter α [AL]; i = (1 α i ){ α i 1 ( i i ) α( i 1 + i + i+1 )( i 1 i+1 ) α i+1 ( i i+ ) + i i 1 + i+1 i+ }. (1.1) hey conserve Lebesge measre by the Lioville theorem. We want α ; otherwise the qadratic term of KdV is not represented. In that case the conserved qantity analagos to is Q = i i + i i+1. (1.13) Bt Q is non-definite, and hence the corresponding measre e Q i d i cannot be normalized to make a probability measre. 6. At a completely formal level the proof proceeds as follows. Note first of all that the flow generated by t = xxx is easily solved and seen to preserve white noise. So consider the Brgers flow t = x. t f((t))e = δf δ, t e = δf δ, ( ) x e = f δ δ ( ) x e (1.14) and δ δ ( ) x e = ( x ( ) x ) e. (1.15) he last term vanishes becase ( ) x = 3 (3 ) x and becase of periodic bondary conditions any exact derivative integrates to zero: f x = 1 f =. Sch an argment is known in physics [S]. Note that the problem is sbtle, and reqires an appropriate interpretation. In fact the reslt is not correct for the standard mathematical interpretation of the Brgers flow as the limit as ɛ of ɛ t = ɛ ɛ x + ɛ ɛ xx, as can be checked with the Lax-Oleinik formla. On the other hand, the argment is rigoros for (1.11).. Invariant measres for mkdv on Let P denote Wiener measre on φ C() conditioned to have φ =. It can be derived from the standard circlar Brownian motion P on C() defined as follows: Condition a standard Brownian motion β(t), t [, 1] starting at β() = x to have β(1) = x as well, and now distribte x on the real line according to Lebesge measre. P is obtained from P by conditioning on φ =. Define P (4) to be the measre absoltely continos to P given by P (4) (B) = Z 1 J(φ)e 1 φ4 dp, (.1) B for Borel sets B C() where Z is the normalizing factor to make P (4) a probability measre and where and J(φ) = (π) 1/ K(φ)K( φ)e 1 ( φ ) (.) K(φ) = Φ(x) = 1 x e Φ(x) dx (.3) φ(y)dy. (.4) For smooth g and < t <, let φ(t) = M t g denote the (smooth) soltion of the modified KdV (mkdv) eqation, φ t 6φ φ x + φ xxx =, φ() = g. (.5)

4 4 JEEMY QUASEL AND BENEDEK VALKÓ heorem.1. (Kappeler and opalov [K3]) M t extends to a continos grop of nonlinear evoltion operators M t : L () L (). (.6) Let H(φ) = 1 mkdv can be written in Hamiltonian form, φ 4 + φ x. (.7) φ t = x δh δφ. (.8) P (4) gives rigoros meaning to the weighted Gibbs measre J(φ)e H(φ) on φ =. heorem.. (Borgain [Bo]) P (4) is invariant for mkdv, M t P (4) = P (4). (.9) Proof. In fact what is proven in [Bo] is that Z 1 e 1 φ4 dp is invariant for mkdv. he main obstacle at the time was a lack of well-posedness for mkdv on the spport H 1/ of the measre. his statement follows with less work once one has the reslts of Kappeler and opalov proving well-posedness on a larger set (heorem.1). We have in addition to show that J(φ) is a conserved qantity for mkdv. It is well known that φ is preserved. So the problem is redced to showing that K(φ) and K( φ) are conserved. Let φ(t) be a smooth soltion of mkdv. Note that Hence t K = Bt integrating by parts we have, since φ is periodic and Φ x = φ, 1 φ xx e Φ(x) dx = 1 t Φ = φ 3 φ xx. (.1) 1 φ x φe Φ(x) dx = (φ 3 φ xx )e Φ(x) dx. (.11) 1 (φ ) x e Φ(x) dx = 1 φ 3 e Φ(x) dx. (.1) herefore t K(φ(t)) =. One can easily check with the analogos integration by parts that t K( φ(t)) =. Now sppose M t φ = φ(t) with φ L (). From heorem.1 we have smooth φ n with φ n φ and φ n (t) φ(t) in L (). K(φ(t)) K(φ) = [K(φ(t)) K(φ n (t))] [K(φ n ) K(φ)] (.13) so if K is a continos fnctions on L () then K(φ) and K( φ) are conserved by that K is continos simply note that K(φ) K(ψ) = 1 M t. o prove e x φ [e x ψ φ 1]dx e φ L () [e ψ φ L () 1]. (.14)

5 KdV PESEVES WHIE NOISE 5 3. he Mira ransform on L () he Mira transform φ φ x + φ maps smooth soltions of mkdv to smooth soltions of KdV. It is basically two to one, and not onto. Bt this is mostly a matter of the mean φ. Since the mean is conserved in both mkdv and KdV, it is more natral to consider the map corrected by sbtracting the mean. he corrected Mira transform is defined for smooth φ by, µ(φ) = φ x + φ φ. (3.1) Let L () and H 1 () denote the sbspaces of L () and H 1 () with φ =. heorem 3.1. (Kappeler and opalov [K]) he corrected Mira transform µ extends to a continos map µ : L H 1 (3.) which is one to one and onto. µ takes soltions φ of mkdv (.5) on L (), to soltions = µ(φ) of KdV (1.1) on H 1 (); S t µ = µ M t. (3.3) emark. he icatti map is given by r(φ, λ) = φ x + φ + λ. (3.4) Note that Kappeler and opalov se the term icatti map for µ = r(φ, φ ). 4. he Mira ransform on Wiener Space heorem 4.1. (Cambronero and McKean [CM]) he corrected Mira transform µ maps P (4) into mean zero white noise Q ; µ P (4) = Q. (4.1) Proof. Let ˆP (4) be given by ˆP (4) (B) = 1 π (φ,λ) B K(φ)K( φ)e 1 (φ +λ) dp dλ (4.) where B is a Borel sbset of C(). Let ˆr = (r, φ). Let ˆQ on C() be given by ˆQ = Q Lebesge measre. What is actally proved in [CM] is that (4.1) is obtained by conditioning on λ = φ and φ =. ˆr ˆP (4) = ˆQ. (4.3) emark. here is a simple heristic argment explaining (4.1). Formally dp (4) = Z1 1 K(φ)K( φ)e 1 1 (φ +φ 1 φ ) df (φ), dq = Z 1 e 1 1 df () (4.4) where F is the (mythical) flat measre on 1 (4) φ =. Note that in the exponent of dp we have assmed that integration by parts gives 1 φ φ =. Since the corrected Mira transform = φ + φ 1 φ the only mystery is the form of the Jacobian CK(φ)K( φ). Let D be the map Df = f and φ stand for the map of mltiplication by φ with a sbtraction to make the reslt mean zero, φf = φ f 1 φ f. he Jacobian is then, f(φ) = det(1 + φd 1 ) = exp{r log(1 + φd 1 )} (4.5)

6 6 JEEMY QUASEL AND BENEDEK VALKÓ For fixed x, y let xy = (φ(y) φ(x)), i.e. the Gâteax derivative in the direction δ y δ x, xy F (φ) = lim ɛ ɛ 1 (F (φ + ɛ(δ y δ x )) F (φ). We have xy log f(φ) = xy r log(1 + ϕd 1 ) = r[{ xy (1 + φd 1 )}{1 + φd 1 ) 1 }]. (4.6) If we let G(x, y) denote the Green fnction of D + φ this gives xy log f(φ) = [G(y, y) G(x, x)].. It is not hard to compte the Green fnction with the reslt that xy log f(φ) = y x e Φ y x 1 e Φ 1. (4.7) e Φ e Φ he argment is completed by a straightforward verification that this is satisfied by f(φ) = K(φ)K( φ). he heristic argment can be made rigoros by taking finite dimensional approximations where this set of eqations actally identifies the determinant. Since the comptations become exactly those of [CM], we do not repeat them here. 5. Proof of heorem 1. S t hm 4.1 Q = S t µ P (4) hm 3.1 = µ M t P (4) hm. = µ P (4) hm 4.1 = Q (5.1) Acknowledgements. conversations. hanks to K. Khanin, M. Goldstein and J. Colliander for enlightening eferences [AL] Ablowitz, M. J.; Ladik, J. F. On the soltion of a class of nonlinear partial difference eqations. Stdies in Appl. Math. 57 (1976/77), no. 1, 1 1. [Bi] Billingsley, P., Convergence of probability measres. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons, Inc., New York, [Bo] Borgain, J., Periodic nonlinear Schrdinger eqation and invariant measres. Comm. Math. Phys. 166 (1994), no. 1, 1 6. [CKS] Colliander, J.; Keel, M.; Staffilani, G.; akaoka, H.; ao,., Sharp global well-posedness for KdV and modified KdV on and. J. Amer. Math. Soc. 16 (3), no. 3, [CM] Cambronero, S.; McKean, H. P. he grond state eigenvale of Hill s eqation with white noise potential. Comm. Pre Appl. Math. 5 (1999), no. 1, [H] Hida,., Brownian motion. Applications of Mathematics, 11. Springer-Verlag, New York-Berlin, 198. [KPV] Kenig, C., Ponce, G., Vega, L., A bilinear estimate with applications to the KdV eqation. J. Amer. Math. Soc., 9:573 63, [K1] Kappeler,.; opalov, P., Well-posedness of KdV on H 1 (). Mathematisches Institt, Georg-Agst- Universität Göttingen: Seminars 3/4, , Universitätsdrcke Göttingen, Göttingen, 4. [KM] Kappeler,.; Möhr, C.; opalov, P., Birkhoff coordinates for KdV on phase spaces of distribtions. Selecta Math. (N.S.) 11 (5), no. 1, [K] Kappeler,.; opalov, P., iccati map on L () and its applications. J. Math. Anal. Appl. 39 (5), no., [K3] Kappeler,.; opalov, P., Global well-posedness of mkdv in L (, ). Comm. Partial Differential Eqations 3 (5), no. 1-3, [S] Spohn, H., Large scale dynamics of interacting particles, exts and Monographs in Physics, Springer-Verlag (1991) p. 67. [] akaoka, H.; stsmi, Y., Well-posedness of the Cachy problem for the modified KdV eqation with periodic bondary condition. Int. Math. es. Not. 4, no. 56, Departments of Mathematics and Statistics, University of oronto qastel@math.toronto.ed, valko@math.toronto.ed