KdV PRESERVES WHITE NOISE

It is shown that white noise is an invariant measure for the Korteweg-deVries equation on T. This is a consequence of recent results of Kappeler and Topalov establishing the well-posedness of the equation on appropriate negative Sobolev spaces, together with a result of Cambronero and McKean that white noise is the image under the Miura transform (Ricatti map) of the (weighted) Gibbs measure for the modified KdV equation, proven to be invariant for that equation by Bourgain. 1. KdV on H−1(T) and White Noise The Korteweg-deVries equation (KdV) on T = R/Z, ut − 6uux + uxxx = 0, u(0) = f (1.1) defines nonlinear evolution operators Stf = u(t) (1.2) −∞ < t <∞ on smooth functions f : T → R. Theorem 1.1. (Kappeler and Topalov [KT1]) St extends to a continuous group of nonlinear evolution operators S̄t : H−1(T)→ H−1(T). (1.3) In concrete terms, take f ∈ H−1(T) and let fN be smooth functions on T with ‖fN−f‖H−1(T) → 0 as N →∞. Let uN (t) be the (smooth) solutions of (1.1) with initial data fN . Then there is a unique u(t) ∈ H−1(T) which we call u(t) = S̄tf with ‖uN (t)− u(t)‖H−1(T) → 0. White noise on T is the unique probability measure Q on the space D(T) of distributions on T satisfying ∫ ei〈λ,u〉dQ(u) = e− 1 2‖λ‖ 2 2 (1.4) for any smooth function λ on T where ‖ · ‖2 = 〈·, ·〉 are the L(T, dx) norm and inner product (see [H]). Let {en}n=0,1,2,... be an orthonormal basis of smooth functions in L(T) with e0 = 1. White noise is represented as u = ∑∞ n=0 xnen where and xn are independent Gaussian random variables, each with mean 0 and variance 1. Hence Q is supported in H−α(T) for any α > 1/2. Mean zero white noise Q0 on T is the probability measure on distributions u with ∫ T u = 0 satisfying ∫ edQ0(u) = e− 1 2‖λ‖ 2 2 (1.5) for any mean zero smooth function λ on T. It is represented as u = ∑∞ n=1 xnen. Recall that if f : X1 → X2 is a measurable map between metric spaces and Q is a probability measure on (X1,B(X1)), then the pushforward f∗Q is the measure on X2 given by f∗Q(A) = Q({x : f(x) ∈ A}) for any Borel set A ∈ B(X2). Date: November 6, 2006.