This question can be seen as a continuation of Probability of a particle surviving forever.

Consider the particle system: for $1\le i\le N$,

$$Y^i_t= y + t + W_t + C\min\big(1,(Y^i_t+1)^+\big)\log\left(\frac{1}{N}\sum_{j=1}^N {\bf 1}_{\{\tau_j>t\}}\right),\quad \mbox{for all } 0\le t<\tau_*$$

where $y>0$, $0<C<1$, $(W_t)_{t\ge 0}$ is a standard Brownian motion

$$\tau_*:=\max_{1\le j\le N}\tau_j \quad \mbox{and} \quad \tau_i:=\inf\{t\ge 0: Y^i_t\le 0\}\quad \mbox{for all } 1\le i\le N.$$

Here we say particle $i$ is absorbed at $t$ once it hits zero, i.e. $\tau_i\le t$. My question is whether one may find some lower bound $c>0$, uniformly in $N$ (large enough), s.t.

$$\frac{1}{N}\mathbb E \left[\sum_{j=1}^N {\bf 1}_{\{\tau_j>t\}}\right]=\frac{1}{N}\sum_{j=1}^N \mathbb P(\tau_j>t)\ge c,\quad \mbox{for all } t\ge 0?$$

If so, we may deduce in particular

$$\frac{1}{N}\sum_{j=1}^N \mathbb P(\tau_j=\infty)\ge c$$

which means the number of particles surviving forever is proportional to the total number of particles. A similar question (by letting $N\to\infty$) is considered in Probability of a particle surviving forever.

Any answers, remarks or references are highly appreciated!


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