# Ergodic branching diffusions with immigration: properties of invariant occupation measure, identification of particles under high-frequency observation, and estimation of the diffusion coefficient at nonparametric rates

## TL;DR

This paper studies ergodic branching diffusions with immigration, providing conditions for invariant measure properties, a high-frequency particle identification algorithm, and nonparametric estimation of the diffusion coefficient.

## Contribution

It introduces a set of conditions for ergodicity, a high-frequency reconstruction algorithm for particle identities, and a nonparametric method to estimate the diffusion coefficient.

## Key findings

- Invariant occupation measure has finite mass and a continuous density.
- High-frequency observations enable accurate particle identity reconstruction.
- The diffusion coefficient can be estimated at nonparametric rates.

## Abstract

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.