14 problems

Let $\mathbf{U}$ and $\mathbf{V}_a$ be random elements with laws $mu$ and $pi_a$, respectively, coupled on a common probability space. Write $U_i$ and $V_{a,i}$ for their $i$th coo…

Let $Z_t$ be reflected Brownian motion in the generalized parabolic domain ${\mathcal D}$, let $X_t$ denote its horizontal coordinate, and define … where $b$ describes the domain's…

Non-negative compensation conjecture. There exists a unique choice of $C$ and $C'$ up to a global scaling such that

Near-optimality conjecture. As the upper-bound parameter tends to infinity, the solution obtained from the approximating drift-control problem is nearly optimal for the original si…

Let a reflected Brownian motion in a convex polyhedral cone have a stationary distribution, and consider the joint process used to represent sensitivities of its stationary distrib…

Let $\operatorname{GRBM}(R,\mu,\Gamma,U)$ be the generalized reflected Brownian motion defined by the strong solution of the stochastic differential equation in the paper, with tra…

Let $u^R(t,x,y):=\mathbb{P}_y(M^R_t\leq x)$, where $\mathbb{P}_y$ is the law of branching reflected Brownian motion starting at $y\geq 0$. Let $q^R_{\delta}(y,t)$ be defined as in…

Three-segment optimal-path conjecture. The optimal path from the origin to $z$ is composed of three segments: the first lies on the first coordinate axis, the second runs from the…

General large deviations principle. For every measurable $A\subset\mathbb{R}^d_+$,

Lagrangian measure convergence conjecture. The measures ${\bf Q}_{r,t}$ converge to a random measure ${\bf Q}_r$ on $D_1\setminus{\mathcal B}(0,r)$ as $t\to\infty$, in the sense of…

Many-hole domain conjecture. If $k$ is sufficiently large and

Let $D$ be a bounded convex domain, and consider reflected Brownian motion in $D$. A coupling of two reflected Brownian motions is shy if the coupled processes remain separated by…

Let $D\subset\mathbb{R}^n$ be a bounded connected domain, let $D_K$ be its connected square-lattice approximation with mesh $1/K$, and let $P_K$ be the distribution of the resultin…

Convergence conjecture. As $N\to\infty$, $P_N$ should converge to the distribution of reflected Brownian motion in $[0,1]$ with inert drift.