"Triple points: from non-Brownian filtrations to harmonic
Geom. and Funct. Anal. 7:6, 1096-1142 (1997).
Available online from Springer-Verlag (not free, sorry):
or from my site:
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
- From stochastic calculus to stochastic topology.
- Joining two copies of a filtration.
- Joining two copies of Walsh's Brownian motion.
- Joining two copies of reflecting Brownian motion.
- A generalization: change of measure, or drift.
- Another generalization: asymmetric triple point.
- Application to harmonic measures.
- Appendix. A reformulation in terms of a measure in
a linear space.