  For the two-dimensional delta-Bose gas, we prove the first Feynman-Kac type
diffusion representation of the Schr\"odinger semigroup generated by the
relative motion of two particles. The representation builds on a unique
skew-product diffusion, in the sense of Erickson [22], such that the Markov
process ${\rm BES}(0,\beta{\downarrow})$ from Donati-Martin and Yor [20]
defines the radial part. Moreover, the law of this skew-product diffusion is
singular to the law of planar Brownian motion. As an application, we determine
some exact forms of non-Gaussianity for the stochastic heat equation. We give
two different proofs of the representation: one applies the original
characterization of ${\rm BES}(0,\beta{\downarrow})$ by excursion theory, and
the other extends [20] by approximations of SDEs and resolvents. In the latter,
we establish a new characterization of ${\rm BES}(0,\beta{\downarrow})$ as a
strongly well-posed SDE. The approximations of resolvents construct the
two-dimensional delta potentials from local times of lower-dimensional Bessel
processes, which may be of independent interest for being an alternative to the
long-standing approach of using mollifiers.
