  In branching diffusions with immigration (BDI), particles travel on
independent diffusion paths in $\mathbb{R}^d$, branch at position-dependent
rates and leave offspring -- randomly scattered around the parent's death
position -- according to position-dependent laws. We specify a set of
conditions which grants ergodicity such that the invariant occupation measure
is of finite total mass and admits a continuous Lebesgue density. Under
discrete-time observation, BDI configurations being recorded at discrete times
$i\Delta$ only, $i\in\mathbb{N}_0$, we lose information about particle
identities between successive observation times. We present a reconstruction
algorithm which in a high-frequency setting (asymptotics $\Delta\downarrow 0$)
allows to reconstruct correctly a sufficiently large proportion of particle
identities, and thus allows to recover $\Delta$-increments of unobserved
diffusion paths on which particles are travelling. Picking some few well-chosen
observations we fill regression schemes which, on cubes $A$ where the invariant
occupation density is strictly positive, allow to estimate the diffusion
coefficient of the one-particle motion at nonparametric rates.
