5 problems

Let $h_{j,m}$, $j=1,\dots,m$, be the degree-one polynomials in two variables entering the decompositions of the heat kernel. For $m\geq 1$, let $C_+^{(m)}$ denote the corresponding…

Boundary-drift convergence conjecture. If the drift of the random walk lies in $\partial C$, then the sequence $\left(p_n^+\left(\,.,\,.,0\right)\right)_{n\geq 0}$ converges pointw…

Sharpness conjecture. For every $\varepsilon>0$ there exist $\delta=\delta(\varepsilon)\in(0,1)$ and $C=C(\varepsilon)>0$ such that, for all $n<\mathrm{e}^{d/(u+\varepsilon)}$,

Doubled-chamber conjecture. The set of oscillating lattice walks of length $n$ in $W_k$ starting at $\delta$ and ending at the boundary

Oscillating-walk conjecture. The set of $W_k$-oscillating lattice walks of length $n$ ending at the boundary