Robert O. Bauer

A probabilistic approach to the Yang-Mills heat equation

We construct a parallel transport U in a vector bundle E, along the paths of a Brownian motion in the underlying manifold, with respect to a time dependent covariant derivative ∇ on E, and consider the covariant derivative ∇0U of the parallel transport with respect to perturbations of the Brownian motion. We show that the vertical part U −1 ∇0U of this covariant derivative has quadratic variation twice the Yang–Mills energy density (i.e., the square norm of the curvature 2-form) integrated along the Brownian motion, and that the drift of such processes vanishes if and only if ∇ solves the Yang–Mills heat equation. A monotonicity property for the quadratic variation of U −1 ∇0U is given, both in terms of change of time and in terms of scaling of U −1 ∇0U . This allows us to find a priori energy bounds for solutions to the Yang–Mills heat equation, as well as criteria for non-explosion given in terms of this quadratic variation.  2002 Editions scientifiques et medicales Elsevier SAS.

The correlator toolbox, metrics and moduli

Abstract We discuss the possible set of operators from various boundary conformal field theories to build meaningful correlators that lead via a Lowner type procedure to generalisations of SLE ( κ , ρ ) . We also highlight the necessity of moduli for a consistent kinematic description of these more general stochastic processes. As an illustration we give a geometric derivation of SLE ( κ , ρ ) in terms of conformally invariant random growing compact subsets of polygons. Further, we also mention a related class of polyhedral SLE ( κ , ρ , ρ ) processes. In the case of polygons, the parameters ρ j are related to the exterior angles. We also show that SLE ( κ , ρ ) can be generated by a Brownian motion in a gravitational background, where the metric and the Brownian motion are coupled. The metric is obtained as the pull-back of the Euclidean metric of a fluctuating polygon.

Khintchine–Pollaczek formula for random walks whose steps have one geometric tail

We derive a Khinchine–Pollaczek formula for random walks whose steps have a geometric left tail. The construction rests on the memory-less property of the geometric distribution. An example from a tandem queue modeling dynamic isnstability for microtubules is given.