6 problems

Precise lower-deviation asymptotics. There exists a constant $C>0$, depending on the law of the branching random walk and $\hat{c}_1$, such that

Extremal-particle separation conjecture. The property that the majority of the roughly $(\log x)x^{\frac{d-1}{2}}$ particles beyond $x$ at time $t_x$ are separated in either the fi…

Let $B_x$ be the shifted unit ball at Euclidean distance $x$ from the origin, let $\tau_x$ denote the first passage time of the branching random walk to $B_x$, and let…

Ma's conjecture. As $\lambda\to\infty$, where $\lambda$ is the mortality parameter, the conditional mean first passage times satisfy

Consider the stochastic periodically forced leaky integrate-and-fire (LIF) model neuron described above. Let $\mathbf{K}$ be the operator mapping the density $f_k$ of the phase of…

Let $\tau_0$ be the first passage time to $0$, let $\lambda_x(t)$ be its rate function, and let $\mu^n_1$ be the principal eigenvalue of the Dirichlet problem on $(0,n)$ associated…