located at $x=0$, then the initial condition is $Q(t=0,x)=\theta(x)$.

With such an initial condition, the solution of the FKPP equation tends to a so-called “traveling wave”. The position of a FKPP traveling wave, which is related to the average position of the rightmost particle in the BRW, is known in the large-time limit: $$\left[\text{FKPP front position}\right]=2t-\frac{3}{2}\ln t
+\text{const}
+\frac{C_X}{\sqrt{t}}+\cdots\ \ 
\text{with}\ \ 
C_X=-3\sqrt{\pi},
\label{eq:FKPPpos}$$ where the last term was found by Ebert and Van Saarloos [@EVS:2000]. (The additive constant depends on the way one defines the position of the front. It is uninteresting for our purpose.) Note that the Ebert-Van Saarloos term is a decreasing but [*positive*]{} contribution to the front velocity. Equation (\[eq:FKPPpos\]) may easily be extended to different branching diffusion models by appropriately replacing some numerical constants (see below, Sec. \[sec:numerics\]).

More generally, if $N(t)$ is the number of particles at time $t$, and $\{x_i(t)\}$ is the set of their positions in a given realization, then G\_t(x)\_[i=1]{}\^[N(t)]{} f(x-x\_i(t))\[eq:G\] for any given function $f$ satisfies the same FKPP equation as $Q$, the initial condition being the function $f(x)$ itself in the case of a BRW starting with one single particle at the origin. If $f$ is a monotonous function 