Invariance of white noise for KdV on the line
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Invariance of white noise for KdV on the line Rowan Killip1 · Jason Murphy2 · Monica Visan1
Received: 25 May 2019 / Accepted: 11 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We consider the Korteweg–de Vries equation with white noise initial data, posed on the whole real line, and prove the almost sure existence of solutions. Moreover, we show that the solutions obey the group property and follow a white noise law at all times, past or future. As an offshoot of our methods, we also obtain a new proof of the existence of solutions and the invariance of white noise measure in the torus setting. 1 Introduction The Korteweg–de Vries equation d dt q
= −q + 6qq
(1.1)
takes its name from the paper [26] where it is derived as a model for long waves of small amplitude in shallow water. Since that time, it has grown to be
B Rowan Killip
[email protected] Jason Murphy [email protected] Monica Visan [email protected]
1
Department of Mathematics, University of California, Los Angeles, USA
2
Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, USA
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R. Killip et al.
one of the central models in mathematical physics because it sits at the nexus of numerous strands of research, both pure and applied. It is one of the simplest models synthesizing nonlinear and dispersive effects, it is Hamiltonian, it supports solitons, and it is completely integrable. In seeking to consider statistical ensembles of initial data for a mechanical system, one is naturally led to Gibbs measures, which model such a system in thermal equilibrium. Such measures are constructed directly from the Hamiltonian structure and the temperature. (More commonly, the temperature is 1 where T is the temperaexpressed through the inverse temperature β = kT ture and k is the Boltzmann constant.) The KdV equation admits multiple Hamiltonian descriptions; this is a common symptom of being completely integrable. The best known of these descriptions is the following (cf. [20]): The Hamiltonian G 2 3 1 (1.2) HKdV (q) := 2 q (x) + q(x) d x, generates the dynamics (1.1) via the Poisson bracket ∂ δG δF {F, G}0 = (x) (x) d x. δq ∂ x δq
(1.3)
(Regarding our notation for functional derivatives, see (2.5).) The second description is based on the Magri–Lenard bracket (cf. [30]): {F, G}1 =
∂3 ∂ δG δF ∂ (x) − 3 + 2 q(x) + 2q(x) (x) d x, δq ∂x ∂x ∂ x δq (1.4)
under which (1.1) is generated by M HKdV (q)
:=
2 1 2 q(x) d x.
(1.5)
Evidently, (1.2) and (1.5) both define conserved quantities for the KdV flow. We should also mention the simplest of all the conserved quantities: M(q) = q(x) d x, (1.6) which in the water-wave setting represents any surplus/deficit of water relative to equilibrium and thereby conservation of matter. M , and H G , are merely the first members These three quantities, M, HKdV KdV of an infinite hierarchy of such conservation laws; see [36]. In fact, the full list
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Invariance of white noise
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