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

TL;DR

This paper establishes an inverse Pitman's theorem for space-time Brownian motion in an affine Weyl chamber, enabling the recovery of unconditioned processes from conditioned ones through path transformations.

Contribution

It introduces a novel inverse Pitman's theorem for space-time Brownian motion in affine Weyl chambers, expanding understanding of conditioned stochastic processes.

Findings

Proves an inverse Pitman's theorem for affine Weyl chambers.

Provides a method to recover unconditioned processes from conditioned ones.

Uses path transformations to achieve the inverse relationship.

Abstract

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.