$ \def\eps{\varepsilon}
\def\bone{{\bf 1}}
\def\R{{\bf R}}
\def\X{{\bf X}}
\def\n{{\bf n}}
\def\prt{\partial}
\def\ol{\overline}
$
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.
1. Probabilistic version
of McMillan's Theorem in higher dimensions
Let $X_t$ be the $d$-dimensional Brownian motion
starting from the origin, and let $D$ be an open
set in the $d$-dimensional space containing the
origin. Let $\tau= \inf\{t >0: X_t \notin D\}$
be the exit time from $D$. Consider
the set $A$ of ``asymptotic directions of approach,''
depending on the domain $D$ and the trajectory of $X_t$,
and defined as the set of all cluster points of
$${X_t - X_\tau \over |X_t - X_\tau|},$$
as $t \uparrow \tau$. It has been proved in Burdzy (1990a)
that for $d=2$, a.s., the set $A$ is equal to
either a circle or a semicircle. In some domains $D$,
the set $A$ is a circle with a non-trivial
probability, i.e., with a probability strictly between
$0$ and $1$.
Problem
Is it true that for every $d >2$
and every $d$-dimensional open set $D$, the set
$A$ of asymptotic directions of approach is either
a sphere or a hemisphere, a.s.?
2. Topology of planar Brownian trace
Let $X_t$ be the two-dimensional Brownian motion.
A Jordan arc is a set homeomorphic to a line segment.
Problem (i)
Is it true that for every pair
of points $x,y \notin X[0,1]$ one can find a Jordan arc $\Gamma$
with $x,y\in\Gamma$, and such that $\Gamma \cap X[0,1]$ contains
only a finite number of points, possibly depending on $x$ and
$y$?
Let $\{A_k\}_{k\geq 1}$ be the family of all
connected components of the complement
of $X[0,1]$, and let
$K = X[0,1] \setminus \bigcup_{k\geq 1} \partial A_k$.
We say that a set is totally disconnected
if it has no connected subsets containing more
than one point.
Problem (ii)
Is $K$ totally disconnected?
The negative answer to Problem (ii) and
a soft argument would yield the negative answer
to Problem (i). It is easy to see that for any fixed $t\in [0,1]$,
$X_t \in K$, a.s. Hence, the dimension of $K$ is equal to 2,
a.s. The problem is related to the existence
of ``cut points;'' see Burdzy (1989, 1995).
It is also related to the question of whether $X[0,1]$
is a ``universal planar curve'' or equivalently, whether
it contains a homeomorphic image of the Sierpi\'nski
carpet; see Mandelbrot (1982, Section VIII.25).
3. Percolation dimension of planar Brownian trace
Let $\hbox{dim} (A)$ denote the Hausdorff dimension
of a set $A$. The percolation dimension
of a set $B$ is the infimum of $\hbox{dim} A$,
where the infimum is taken over all Jordan arcs
$A\subset B$ which contain at least two distinct points.
Suppose that $X_t$ is a two-dimensional Brownian motion.
Problem
Is the percolation
dimension of $X[0,1]$ equal to 1?
Related paper: Burdzy (1990b).
4. Efficient couplings in acute triangles
Suppose that $D$ is a triangle with all angles
strictly smaller than $\pi/2$ and let $\mu_2>0$
be the second eiganvalue for the Laplacian in $D$
with Neumann boundary conditions. The first eigenvalue
is zero.
Problem
Can one construct
two reflected Brownian motions $X_t$ and $Y_t$ in $D$
starting from different points and
such that $\tau = \inf\{t\geq 0: X_t = Y_t \}< \infty$ a.s., and
for every fixed $\eps>0$,
$$P(\tau > t) \leq \exp(-(\mu_2 -\eps) t),$$
for large $t$?
See Burdzy and Kendall (2000) for the background of the problem.
5. Convergence of synchronous couplings
Suppose
$D\subset\R^2$ is an open connected set with smooth boundary, not
necessarily simply connected. Let $\n(x)$ denote the unit inward
normal vector at $x\in\prt D$ and suppose that $x_0,y_0 \in \ol
D$. Let $B$ be standard planar Brownian motion and consider
processes $X$ and $Y$ solving the following equations,
$$\eqalignno{
X_t &= x_0 + B_t + \int_0^t \n(X_s) dL^X_s
\qquad \hbox{for } t\geq 0, \cr
Y_t &= y_0 + B_t + \int_0^t \n(Y_s) dL^Y_s
\qquad \hbox{for } t\geq 0. }
$$
Here $L^X$ is the local time of $X$ on $\prt D$, i.e, it is a
non-decreasing continuous process which does not increase when $X$
is in $D$. In other words, $\int_0^\infty \bone_{D}(X_t) dL^X_t =
0$, a.s. The same remarks apply to $L^Y$. We call $(X, Y)$ a
``synchronous coupling.''
Problem
(i) Does there exist a bounded
planar domain such that with positive probability, $
\limsup_{t\to\infty} |X_t - Y_t| > 0$?
(ii) If $D$ is the complement of a non-degenerate closed disc, is
it true that with positive probability, $ \limsup_{t\to\infty}
|X_t - Y_t| > 0$?
If there exists a bounded domain $D$ satisfying the condition in
Problem 5 (i) then it must have at least two holes, by the results
in Burdzy, Chen and Jones (2006). See that paper and Burdzy and
Chen (2002) for the background of the problem.
6. Non-extinction of a Fleming-Viot particle model
Consider a branching particle system $\X_t =
(X^1_t, \dots, X^N_t)$ in which individual particles $X^j$ move
as $N$ independent Brownian motions and die when they hit the
complement of a fixed domain $D\subset \R^d$. To keep the
population size constant, whenever any particle $X^j$ dies,
another one is chosen uniformly from all particles inside $D$,
and the chosen particle branches into two particles.
Alternatively, the death/branching event can be viewed as a
jump of the $j$-th particle.
Let $\tau_k$ be the time of the $k$-th jump of $\X_t$. Since the
distribution of the hitting time of $\prt D$ by Brownian motion
has a continuous density, only one particle can hit $\prt D$ at
time $\tau_k$, for every $k$, a.s. The construction of the process
is elementary for all $t< \tau_\infty = \lim_{k\to \infty}
\tau_k$. However, there is no obvious way to continue the process
$\X_t$ after the time $\tau_\infty$ if $\tau_\infty < \infty$.
Hence, the question of the finiteness of $\tau_\infty$ is
interesting. Theorem 1.1 in Burdzy, Ho\l yst and March (2000) asserts that
$\tau_\infty = \infty$, a.s., for every domain $D$. Unfortunately,
the proof of that theorem contains an irreparable error. It has been shown in Bieniek, Burdzy and Finch (2009) that $\tau_\infty = \infty$,
a.s., if the domain $D \subset \R^d$ is Lipschitz with a
Lipschitz constant depending on $d$ and the number $N$ of
particles.
Problem
Is it true that $\tau_\infty = \infty$,
a.s., for any bounded open connected set $D \subset \R^d$?
7. Are shy couplings necessarily rigid? (Proposed by K. Burdzy and W. Kendall)
Suppose that $D\subset \R^d$, $d\geq 2$, is a bounded connected open set
and let $X_t$ and $Y_t$ be reflected Brownian motions in $D$ defined
on the same probability space.
Problem
Suppose that there exist reflected Brownian motions
$X_t$ and $Y_t$ in $D$ and $\eps>0$ such that
$\inf _{t\geq 0} |X_t - Y_t| \geq \eps$ with probability
greater than 0. Does this imply that there exist
reflected Brownian motions
$X'_t$ and $Y'_t$ in $D$, $\eps>0$ and
a deterministic function $f$ such that $f(X'_t) = Y'_t$
for all $t\geq 0$, a.s., and $\inf _{t\geq 0} |X'_t - Y'_t| \geq \eps$ with probability
greater than 0?
Example 3.9 of Benjamini, Burdzy and Chen (2007) shows that there exists a graph $\Gamma$
and Brownian motions $X_t$ and $Y_t$ on $\Gamma$ such that
$\inf _{t\geq 0} |X_t - Y_t| \geq \eps$ with probability
greater than 0 but $Y_t$ is not a deterministic function
of $X_t$. Moreover, all bijective isometries of $\Gamma$ have fixed points.
REFERENCES
I. Benjamini, K. Burdzy and Z. Chen (2007)
Shy couplings Probab. Theory Rel. Fields
137, 345--377.
M. Bieniek, K. Burdzy and S. Finch (2009) Non-extinction of a Fleming-Viot particle model (preprint)
K. Burdzy (1989) Cut points on Brownian paths.
Ann. Probab. 17, 1012--1036.
K. Burdzy (1990a) Minimal fine derivatives and Brownian
excursions. Nagoya Math. J. 119, 115--132.
K. Burdzy (1990b) Percolation dimension of fractals.
J. Math. Anal. Appl. 145, 282--288.
K. Burdzy (1995) Labyrinth dimension of Brownian trace.
Probability and Mathematical Statistics 15, 165--193.
K. Burdzy and Z. Chen (2002) Coalescence of synchronous
couplings Probab. Theory Rel. Fields 123, 553--578.
K. Burdzy, Z. Chen and P. Jones (2006) Synchronous
couplings of reflected Brownian motions in smooth domains
Illinois. J. Math., Doob Volume, 50, 189--268.
K. Burdzy, R. Holyst and P. March (2000)
A Fleming-Viot particle representation of Dirichlet Laplacian
Comm. Math. Phys. 214, 679--703.
K. Burdzy and W. Kendall (2000) Efficient Markovian
couplings: examples and counterexamples. Ann. Appl. Probab.
10, 362--409.
B.B. Mandelbrot (1982) The Fractal Geometry of
Nature. Freeman & Co., New York.
