A ratio limit theorem for erased branching Brownian motion (Q1198595)

It is considered the erasure transform \(C^ \rho\), \(\rho\geq 0\), of the branching Brownian motion \(\hat X=\{\hat X_ t=(\hat X_ t^{(1)},\dots,\hat X_ t^{(\hat N_ t)})\), \(t\geq 0\}\), \(\hat X_ 0=x\in R^ d\), inside of a bounded domain \({\mathcal O}\) in \(\mathbb{R}^ d\) with killing at the boundary \(\partial{\mathcal O}\). Under assumption that the off-spring distribution \(\{p_ n\), \(n=0,1,\dots\}\) is state independent, \(p_ 0=p_ 1=0\), \(\sum_{n=2}^ \infty p_ n n\ln n<\infty\), it is proved that for every bounded, positive, continuously two times differentiable function \(f\) on \({\mathcal O}\) \[ \lim_{t\to\infty} \sum_{i=1}^{\hat N_ t} f(\hat X_ t^{(i)}\circ c^ \rho) / \sum_{i=1}^{\hat N_ t}f(\hat X_ t^{(i)}) = \int_{\mathcal O} u^ \rho (x)\varphi_ 0(x)f(x)dx / \int_{\mathcal O}\varphi_ 0(x)f(x)dx, \] \(\hat P\)-a.s. on the set \(\{\hat N_ t\to\infty\}\), where \(u^ \rho(x)=\hat P_ x(T\geq\rho)\), \(T\) is the extinction time of \(\hat X\) and \(\varphi_ 0\) is the principal eigenfunction with the normalization \(\int_{\mathcal O}\varphi_ 0(x)dx=1\) of the following eigenvalue problem: \[ \begin{cases} ({1\over 2}\Delta+\lambda)\varphi(x)=0 &\text{ for } x\in{\mathcal O}\\ \varphi(x)=0 &\text{ for } x\in \partial{\mathcal O}\end{cases}. \]