Abstract

A conditioned hypoelliptic process on a compact manifold, satisfying the strong Hormander’s condition, is a hypoelliptic bridge. If the Markov generator satisfies the two step strong Hormander condition, the drift of the conditioned hypoelliptic bridge is integrable on and the hypoelliptic bridge is a continuous semi-martingale.

Highlights

  • We are motivated by the path integration formula and by the L2 analysis on the space of pinned continuous curves where the Brownian bridge plays an important role

  • For the L2 analysis, it is standard to equip the space with the probability measure determined by the Brownian bridge, which fuelled the study of the logarithm of the heat kernel and their derivatives

  • For example on a Lie group, a basic object is a diffusion operator built from a family of left invariant vector fields generated by elements of the Lie algebra

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Summary

Introduction

We are motivated by the path integration formula and by the L2 analysis on the space of pinned continuous curves where the Brownian bridge plays an important role. Given a hypoelliptic L, the probability distribution of the L diffusion process conditioned to reach the terminal value y at time 1 is absolutely continuous on [0, t], for any t < 1, with respect to that of the L-diffusion. It is not so clear how it approaches the terminal value at the terminal time. It is tempting to argue that the integral bound obtained here, for diffusions satisfying two-step Hörmander condition, fails when l(x) is sufficiently large. Since the L1 bound and the semi-martingale property depend on properties of the heat kernel for small time, and since the sub-Riemannian geodesic is horizontal in whose direction the singularity in t should be exactly t−. We tend to believe these conclusions hold much more generally

Preliminaries
L1 integrability and the semi-martingale Property
Full Text
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