Brownian sheet and reflectionless potentials

The bijectivity of the mapping, which is represented as expectation, from a family of Gaussian measures parametrized by linear combinations of Dirac measures to the space of classical reflectionless potentials is shown. It is also shown that the bijectivity extends to the space of generalized reflectionless potentials, which was used by V. Marchenko to study the Cauchy problem for the KdV equation. In the extension, the stochastic calculus based on the Brownian sheet plays a key role.