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

Manon Defosseux, Charlie Herent

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.

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

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.
Paper Structure (13 sections, 24 theorems, 140 equations, 1 figure)

This paper contains 13 sections, 24 theorems, 140 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of the theorem
  3. The affine Lie algebra $A_1^{1}$ and its representations
  4. Highest weight representations.
  5. Verma modules
  6. Pitman transforms and Littelmann modules
  7. String coordinates
  8. Demazure character
  9. Random walks and Littelmann paths
  10. The continuous counterpart
  11. An inverse Pitman's theorem
  12. Proof of the convergence corresponding to the first arrow of the diagram
  13. Proof of the convergence corresponding to the third arrow of the diagram

Key Result

Theorem 2.1

The sequence of processes converges, in the sense of finite dimensional distributions, towards the space-time Brownian motion $\{B(t),t\ge 0\}$.

Figures (1)

  • Figure 1: A commutative diagram of finite dimensional distributions convergences

Theorems & Definitions (41)

  • Theorem 2.1
  • Definition 4.1
  • Proposition 5.1
  • Proposition 5.2
  • Lemma 5.3
  • proof
  • Lemma 5.4
  • proof
  • Proposition 5.5
  • proof
  • ...and 31 more