The spreading speed of solutions of the non-local Fisher-KPP equation

We consider the Fisher-KPP equation with a non-local interaction term. Hamel and Ryzhik showed that in solutions of this equation, the front location at a large time $t$ is $\sqrt 2 t +o(t)$. We study the asymptotics of the second order term in the front location. If the interaction kernel $\phi(x)$ decays sufficiently fast as $x\rightarrow \infty$ then this term is given by $-\frac{3}{2\sqrt 2 }\log t +o(\log t)$, which is the same correction as found by Bramson for the local Fisher-KPP equation. However, if $\phi$ has a heavier tail then the second order term is $-t^{\beta +o(1)}$, where $\beta \in (0,1)$ depends on the tail of $\phi$. The proofs are probabilistic, using a Feynman-Kac formula. Since solutions of the non-local Fisher-KPP equation do not obey the maximum principle, the proofs differ from those in Bramson's work, although some of the ideas used are similar.