I offer a collection of open problems,
mostly about Brownian motion.
I tried to solve every problem on the list
but I failed to do so.
Although I do not recall seeing these problems
published anywhere,
I do not insist on being their author-other
probabilists might have proposed them independently.
The rigorous statement of the problems
is available in
PDF.
1. Probabilistic version
of McMillan's Theorem in higher dimensions
This problem is concerned with the direction
of approach of the boundary for Brownian paths
in mutidimensional domains.
2. Topology of planar Brownian trace
Does the planar Brownian trace contain a
(non-trivial) connected subset which does not
interesect the boundary of any connected
component of the complement of the trace?
3. Percolation dimension of planar Brownian trace
What is the "shortest" (in the sense
of the Hausdorff dimension) path within
two-dimensional Brownian range?
4. Efficient couplings in acute triangles
Can one construct a pair of reflected Brownian motions
in an acute triangle whose coupling rate is the
same as the second Neumann eigenfunction?
5. Convergence of synchronous couplings
Does there exist a bounded planar domain and a pair
of reflected Brownian motions in this domain, driven
by the same Brownian motion, and such that their
distance does not converge to 0?
6. Non-extinction of a Fleming-Viot particle model
Can all Brownian particles in a branching system
hit the boundary of a domain at the same finite time?
7. Are shy couplings necessarily rigid? (Proposed by K. Burdzy and W. Kendall)
Suppose that two (dependent) reflected Brownian motions in a connected open
bounded Euclidean domain never approach each other closer
than some positive distance. Does it follow that
there exist two reflected Brownian motions in the same domain
such that one of them
is a deterministic function of the other one
and they never approach each other closer
than some positive distance?
