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 ψψ of a family G0G0 of probability measures on the one-dimensional Wiener space to the space Ξ0Ξ0 of reflectionless potentials. The mapping gives a probabilistic expression of reflectionless potential. In this paper, it will be shown that ψψ is bijective, and hence G0G0 and Ξ0Ξ0 can be identified. The space Ξ0Ξ0 was extended to the one ΞΞ 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 G0G0 and Ξ0Ξ0 via ψψ, taking advantage of the Brownian sheet, it will be seen that convergences of elements in G0G0 realizes the extension of Ξ0Ξ0 to ΞΞ.