Brownian sheet and reflectionless potentials.pdf

a r X i v : m a t h / 0 5 0 7 2 2 9 v 1 [ m a t h .P R ] 1 2 J u l 2 0 0 5 Brownian sheet and reflectionless potentials Setsuo Taniguchi ? Faculty of Mathematics, Kyushu University, Fukuoka 812-8581, Japan July 12, 2005 Abstract In this paper, the investigation into stochastic calculus related with the KdV equation, which was initiated by S. Kotani [4] and made in succession by N. Ikeda and the author [2, 11], is continued. Reflectionless potentials give important examples in the scattering theory and the study of the KdV equation; they are expressed concretely by their corresponding scattering data, and give a rise of solitons of the KdV equation. N. Ikeda and the au- thor [2] established a mapping ψ of a family G0 of probability measures on the 1-dimensional Wiener space to the space Ξ0 of reflectionless potentials. The mapping gives a probabilistic expression of reflectionless potential. In this paper, it will be shown that ψ is bijective, and hence G0 and Ξ0 can be identified. The space Ξ0 was extended to the one Ξ of general- ized reflectionless potentials, and was used by V. Marchenko to investigate the Cauchy problem for the KdV equation and by S. Kotani to construct KdV-flows. As an application of the identification of G0 and Ξ0 via ψ, taking advantage of the Brownian sheet, it will be seen that convergences of elements in G0 realizes the extension of Ξ0 to Ξ. Key words: Brownian sheet; reflectionless potential; Ornstein-Uhlenbeck process, AMS 2000 subject classifications: 60H30; 60B10; 34L25 1. Introduction Let W be the space of all R-valued continuous functions w on [0,∞) with w(0) = 0, and B be its Borel σ-field, W being equipped with the topology of uniform convergence on compacts. The coordinate mapping on W is denoted by X(x); X(x, w) = w(x), w ∈ W, x ∈ [0,∞). Let Σ0 be the set of measures on R of the form ∑n j=1 c 2 jδpj for some n ∈ N and pj ∈ R, cj 0, 1 ≤ j ≤ n with pi 6= pj if i 6= j, where δp is the Dirac measure concentrated at p. For σ