A converse to Pitman's theorem for a space-time Brownian motion in a type A_1^1 Weyl chamber

We prove an inverse Pitman’s theorem for a space-time Brownian motion conditioned in Doob’s sense to remain in an affine Weyl chamber. Our theorem provides a way to recover an unconditioned space-time Brownian motion from a conditioned one by applying a sequence of path transformations.