Consider this simple random particle system on $\mathbb{R}$:
Every particle on the system has a independent exponential clock of parameter $1$ . If a clock rings the corresponding particle $p$ have an offspring in the position $x_p + D$ where $x_p$ is the position of the the particle $p$ at the time of the clock ring and $D$ is a random variable with fix distribution $F$.
Let $v\in \mathbb{R}, h >0$ and consider the event $A_{v,h}$ where there exists a infinite descending path on the genealogical tree of the system such that the positions of the corresponding particles stay above the line of slope $v$ at times of the form $ih$ $i\in \mathbb{N}$. Similarly consider $A_v$ the event where there exists a infinite descending path on the genealogical tree of the system such that the positions of the corresponding particles stay above the line of slope $v$ at every time $t\geq 0$.
Let $p(v)=\mathbb{P}(A_v)$ and $p_h(v)=\mathbb{P}(A_{v,h})$
Clearly for every $h>0$ $A_{v,h} \supseteq A_v$ and we can consider the bound $$p(v) \leq p_h(v)\leq p(v) + \mathbb{P}(A_{v,h}\cap \Omega \setminus A_v)$$ Naturally I would like to prove that $ \mathbb{P}(A_{v,h}\cap \Omega \setminus A_v)$ go to $0$ when $h \to 0$ but i not even sure this is true.
Any help will be appreciated