arXiv subjects
Publications and source records attributed to Cole Graham.
We study the non-uniformity of probability measures on the interval and the circle. On the interval, we identify the Wasserstein-$p$ distance with the classical $L^p$-discrepancy. We thereby derive sharp estimates in Wasserstein distances for the irregularity of distribution of sequences on the interval and the circle. Furthermore, we prove an $L^p$-adapted Erdős$-$Turán inequality.
We consider the one-dimensional Fisher-KPP equation with step-like initial data. Nolen, Roquejoffre, and Ryzhik showed that the solution $u$ converges at long time to a traveling wave $ϕ$ at a position $\tilde σ(t) = 2t - (3/2)\log t + α_0- 3\sqrtπ/\sqrt{t}$, with error $O(t^{γ-1})$ for any $γ>0$. With their methods, we find a refined shift $σ(t) = \tilde σ(t) + μ_* (\log t)/t + α_1/t$ such that in the frame moving with $σ$, the solution $u$ satisfies $u(t,x) = ϕ(x) + ψ(x)/t + O(t^{γ-3/2})$ for a certain profile $ψ$ independent of initial data. The coefficient $α_1$ depends on initial data, but $μ_* = 9(5-6\log 2)/8$ is universal, and agrees with a finding of Berestycki, Brunet, and Derrida in a closely-related problem. Furthermore, we predict the asymptotic forms of $σ$ and $u$ to arbitrarily high order.
We study transition fronts for one-dimensional reaction-diffusion equations with compactly perturbed ignition-monostable reactions. We establish an almost sharp condition on reactions which characterizes the existence and non-existence of fronts. In particular, we prove that a strong inhomogeneity in the reaction prevents formation of transition fronts, while a weak inhomogeneity gives rise to a front. Our work extends results and methods introduced by J. Nolen, J.M. Roquejoffre, L. Ryzhik, and A. Zlatoš.