Branching brownian motion with absorption

We consider a branching diffusion {Z_t}_t?0 in which particles move during their life time according to a Brownian motion with drift -μ and variance coefficient σ², and in which each particle which enters the negative half line is instantaneously removed from the population. If particles die with probability c dt+o(dt) in t,t+dt] and if the mean number of offspring per particle is m>1, then Z_t dies out w.p.l. if $\text{μ?μ}{}_0\text{≡{2σ}{}^2\text{c(m?1)}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. If μ<μ₀, then itZ_t grows exponentially with positive probability. Our main concern here is with the critical case where μ=μ₀. Even though $\text{E}\text{{}\text{Z}{}_{\mathrm{T}}\text{}∽}\text{const.}\text{T}{}^{\mathrm{<mtext>?</mtext><mtext>3</mtext><mtext>2</mtext>}}$ in this case, we find that $\text{P}\text{{}\text{Z}{}_{\mathrm{T}}\text{>0}}$ is only exp$\text{{–}\text{const.}\text{T}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{+0(}\text{log}\text{T)}{}^2\text{}}$, and conditionally on {Z_T>0} there are with high probability much fewer particles alive at time T than $\text{E}\text{{Z}{}_{\mathrm{T}}\text{|Z}{}_{\mathrm{T}}\text{0}}$.