152 problems

Let $\bm W=(\bm W(t):t\geq 0)$ be planar Brownian motion and let … be the convex hull of its path up to time $t$. Lévy's smoothness conjecture. The boundary of $\mathcal{H}(t)$ sho…

Let $W$ be a Wiener process, and consider adaptive algorithms that construct a trajectory $h(t)$ using only the observed values $W(s)$ for $s\leq t$. The energy of a trajectory is…

Let $b(t)$, $t\in[0,1]$, be a two-dimensional Brownian motion with $b(0)=0$. Its frontier of Brownian motion (FBM) is the boundary of the unbounded connected component of the compl…

Let $\mathbf{X}(t)$ be a deterministic function such that $\int_0^t ||\mathbf{X}(u)||^2du$ is strictly increasing and finite for $t\leq T<\infty$, and let $Z_{\mathbf{X}}(t)$ denot…

Let a single Brownian prey start at $x_0(0)=1$ and let $n$ independent Brownian predators start at $x_1(0)=\dots=x_n(0)=0$, with all motion restricted to a line. Bramson–Griffeath'…

Let $U$ and $R$ be domains in the setting of Theorem, let $z$ be the specified starting point, and write $\tau_U$, $\tau_R$, $\tau_{U^+}$, and $\tau_{R^+}$ for the corresponding fi…

Let $W=(W_t,0\leq t\leq 1)$ be a two-dimensional Brownian motion, let its path be radially projected onto the unit circle, and let $\mu_\Theta$ be the occupation measure of the pro…

Let $W=(W_t,0\leq t\leq 1)$ be a two-dimensional Brownian motion started at the center of the unit disk, and let its radial projection onto the unit circle have occupation measure…

Let the c3-finite bead process be conditioned to survive up to distance $r$, equivalently up to time $r^2$ or imaginary part $r$, and let $r\rightarrow\infty$. Scaling limit conjec…

Let the Brownian frontier denote the outer boundary of a planar Brownian curve, and let the scaling limit of long self-avoiding curves be the conjectural continuum limit of self-av…

Brownian favourite-site upper-tail conjecture. There exists a constant $\nu>1$ such that

Let $K$ be a scaling limit of loop-erased random walk on $\mathbb{Z}^3$, and let the loop-erased property be the property stated in the referenced conditional uniqueness theorem. S…

Single-sign-switch conjecture. The function $\iota$ switches sign once on the positive part of $I$.

Consider the Brownian Pool model obtained by replacing the continuous-time and continuous-space random walks in the Pool model by Brownian motion, and let $\lambda=1$ be the critic…

Let $(M,g_\alpha)$ be a two-dimensional smooth manifold with a family of gas giant metrics $g_\alpha$. Assume that the function $\beta^{(1)}(\theta)$ from the boundary value proble…

Joint-tail conjecture. As $u\to\infty$,

Let $\mathfrak{h}$ be an upper-semicontinuous function that is bounded above by a linear function, and let $\mathfrak{h}^{\mathrm{cutoff}}_\mathrm{p}$ be its periodized cutoff. Let…

Brownian motion and cutoff conjecture. For every $t\geq 0$, the state $\varphi_t$ is not absolutely continuous with respect to the Haar state. Moreover,

For each $N\geq 1$, let $T_{2N}$ and $T_{2N+1}$ be the absorption times in the discrete Whittaker process. Let $M_N$ be the maximum height of $N$ non-intersecting reflected Brownia…

Consider the slow-bond random walk described above, with parameters $\alpha>0$ and $\beta\geq 0$, whose rescaled process converges to standard Brownian motion for $\beta\in[0,1)$,…

Let $K\subset\mathbb{R}$ be a nonrandom compact set, let $\mathfrak S(\theta)$ be the random set of points of slow growth at parameter $\theta$, and let $\lambda(\theta)$ denote th…

Let $\theta_c$ be the critical parameter and let $mathfrak S(thetac)$ denote the corresponding random set of points of slow growth. Write $operatorname{dim{{rm H}}}$ for Hausdorf…

Surjectivity conjecture. The functional $\rho_{\mid\mathscr{B}_a}$ is onto $(0,1]$. Since the preceding theorem establishes only $\rho(\mathscr{B}_a)\subseteq(0,1]$, the conjecture…

Let $U$ be a bounded simply connected domain containing the origin. Let $Z_t$ be a standard planar Brownian motion run inside $U$, let $\tau_U$ be its exit time, and let $\mu_U$ be…

Let $W$ be the Brownian motion and let $G$ be the Green kernel appearing in the definition of the renormalised Amperean area. Write $I_W$ for the recentred self-intersection local…