## My favorite open problems

$\def\eps{\varepsilon} \def\bone{{\bf 1}} \def\R{\mathbb{R}} \def\Z{\mathbb{Z}} \def\X{{\bf X}} \def\n{{\bf n}} \def\prt{\partial} \def\ol{\overline}$

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.

#### 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 (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 $D \subset \R^d$?

#### 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.

#### 8. Do concatenated bounded Brownian pieces form a two-sided Brownian motion?

We call $\{X_t,\, t \in \R\}$ two-sided Brownian motion if there exists a random time $S$ such that $\{X_{S+t}-X_S,\,t \ge 0\}$ and $\{X_{S-t}-X_S,\,t \ge 0\}$ are independent standard Brownian motions.

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 $\alpha < \infty$.

#### 9. Do peaks of random labelings repell each other?

A labeling of a graph with $n$ vertices is a one-to-one function mapping the vertices to $1,2,\dots, n$. A vertex is a peak if all adjacent vertices have lower labels. The random labeling is a labeling chosen uniformly from all labelings.

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).

#### 10. Divergence of coherent opinions

Let $A \in \mathcal{F}$ be an event in some probability space $(\Omega, \mathcal{F}, P )$, and let $$X = P(A \mid \mathcal{G}) \qquad \mbox{and} \qquad Y = P(A \mid \mathcal{H})$$ for two sub-$\sigma$-fields $\mathcal{G}, \mathcal{H} \subseteq \mathcal{F}$. We call $(X,Y)$ coherent. It has been proved in Burdzy and Pal (2021) that $$\sup P( | X-Y| \ge 1 - \delta ) = 2\delta/(1-\delta) ,$$ where the supremum is taken over all probability spaces, $\sigma$-fields and events $A$.

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).

#### 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 (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.