"Triple points: from non-Brownian filtrations to harmonic measures."
Geom. and Funct. Anal. 7:6, 1096-1142 (1997).
Available online from Springer-Verlag (not free, sorry):
www.link.springer.de/link/service/journals/00039/bibs/7007006/70071096.htm
or from my site: [download]

A long (47 pages) research paper. Bibl. 42 refs.

Two conjectures (geometric and probabilistic), seemingly independent, are proved:

• A conjecture by C. Bishop (1991) about harmonic measures for three arbitrary (not just regular) non-intersecting domains in Rⁿ. Roughly speaking, trilateral contact is always rare harmonically (though not topologically).
• A conjecture by M. Barlow, J. Pitman, and M. Yor (1989) about a special two-dimensional continuous martingale (known as Walsh's Brownian motion) that never leaves the union of tree rays (just the Brownian motion in a one-dimensional topological space with a branching point). It appears that the martingale is not a Brownian martingale.

1. Introduction.
1. From stochastic calculus to stochastic topology.
2. Joining two copies of a filtration.
3. Joining two copies of Walsh's Brownian motion.
4. Joining two copies of reflecting Brownian motion.
5. A generalization: change of measure, or drift.
6. Another generalization: asymmetric triple point.
7. Application to harmonic measures.
1. Appendix. A reformulation in terms of a measure in a linear space.