I offer a collection of open problems. 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.

** 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.?
*

** 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).

** Problem **
*
Is the percolation
dimension of $X[0,1]$ equal to 1?
*

Related paper: Burdzy (1990b).

** 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.

** 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.

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 (2012) and Grigorescu and Kang (2012) 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 $D \subset \R^d$?
*

** 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.

For each $k\in \Z $, let $\{B^k_t, t\geq 0\}$ be a Brownian motion and let $T_k$ be a stopping time. Assume that $(T_k, \{B^k_t, t\in[0,T_k]\})$, $k\in \Z $, are independent and, a.s., $0 \leq T_k < \infty$, for $k\in \Z $, $\sum_{k=1}^\infty T_k = \infty$ and $\sum_{k=-\infty}^{-1} T_k = \infty$. Let $S_0=0$, and note that the conditions $S_{k+1} - S_{k} = T_k$, $k\in\Z$, define uniquely $S_k$ for all $k\in \Z$. Let $X$ be the unique continuous process such that $X_0 =0$ and $X_{S_k +t} - X_{S_k} = B^{k}_{t}$ for $t\in [0, T_{k})$, $k \in\Z$.

** Problem **
*
Suppose that $T_k < c$, a.s., for all $k\in\Z$, where $c<\infty$ is non-random. Is
$\{X_t,\, t \in \R\}$ a two-sided Brownian motion?
*

Theorem 5.3 of Burdzy and Scheutzow (2014) shows that it is not sufficient to assume that $\sup_k E (T_k^\alpha) < \infty$ for some $\alpha < \infty$.

** Problem **
*
Condition the random labeling of an $n$ by $n$ discrete
square on having exactly two peaks. Let $\rho_n$ be the (random) graph
distance between the two peaks. Does $\rho_n/n$ converge
in distribution to 0 when $n\to \infty$?
*

See Billey, Burdzy and Sagan (2013) for related one-dimensional results or Burdzy and Pal (2020).

** Problem **
*
If $(X,Y)$ is coherent, and $X$ and $Y$ are independent, is it true that
$$
P( | X-Y| \ge 1 - \delta ) \le 2\delta(1-\delta) \qquad\text{ for } \delta \in [0, 1/2) ?
$$
*

The inequality cannot be improved; see Section 5 of Burdzy and Pitman (2020).

** Solution **
See the
article by Cichomski and Petrov for the proof
of the inequality (uploaded to Arxiv on April 14, 2022).

I. Benjamini, K. Burdzy and Z. Chen (2007)
Shy couplings * Probab. Theory Rel. Fields ***
137**, 345-377.

M. Bieniek, K. Burdzy and S. Finch (2012) Non-extinction of a Fleming-Viot particle model * Probab. Theory Rel. Fields *** 153** 293-332.

S. Billey, K. Burdzy and B. Sagan (2013)
Permutations with given peak set * Journal of Integer Sequences* ** 16**
Article 13.6.1.

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.

K. Burdzy and S. Pal (2020) Twin peaks
* Random Structures Algorithms * ** 56** 432-460.

K. Burdzy and S. Pal (2021)
Can coherent predictions be contradictory?
* Adv. in Appl. Probab. * ** 53** 133-161.

K. Burdzy and J. Pitman (2020)
Bounds on the probability of radically different opinions
* Electr. Comm. Probab. * ** ** paper no. 14.

K. Burdzy and M. Scheutzow (2014)
Forward Brownian motion * Probab. Theory Rel. Fields* ** 160** 95-126.

I. Grigorescu and M. Kang (2012)
Immortal particle for a catalytic branching process.
* Probab. Theory Related Fields*** 153** 333-361.

B.B. Mandelbrot (1982) * The Fractal Geometry of
Nature*. Freeman & Co., New York.