Brownian sheet and reflectionless potentials

In this paper, the investigation into stochastic calculus related with the KdV equation, which was initiated by S. Kotani [Construction of KdV-flow on generalized reflectionless potentials, preprint, November 2003] and made in succession by N. Ikeda and the author [Quadratic Wiener functionals, Kalman–Bucy filters, and the KdV equation, Advanced Studies in Pure Mathematics, vol. 41, pp. 167–187] and S. Taniguchi [On Wiener functionals of order 2 associated with soliton solutions of the KdV equation, J. Funct. Anal. 216 (2004) 212–229] 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. Ikeda and the author established a mapping ψ of a family G 0 of probability measures on the one-dimensional Wiener space to the space Ξ 0 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 G 0 and Ξ 0 can be identified. The space Ξ 0 was extended to the one Ξ of generalized 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 G 0 and Ξ 0 via ψ , taking advantage of the Brownian sheet, it will be seen that convergences of elements in G 0 realizes the extension of Ξ 0 to Ξ .