Simulations for the (augmented) multiplicative coalescent

$$ B_t(s) = W_t(s) - \inf_{0 \leq u \leq s} W_t(u),$$ where $ W_t(s) = W(s) - \frac{1}{2}s^2 + t s $.

  • $X_i(t)$: $i$th largest excursion of $B_t(\cdot)$ above zero,

  • $\big( \pmb{X}(t), t \ge 0 \big) $ is a realization of an eternal multiplicative coalescent.

$W_t(s) = W(s) - \frac{s^2}{2} + t s$, for $t = 0.5$
$W_t(s) = W(s) - \frac{s^2}{2} + t s$, for $t = 0.5$

$B_t(s) = W_t(s) - \inf\limits_{0 \le u \le s} W_t(u)$, for $t = 0.5$
$B_t(s) = W_t(s) - \inf\limits_{0 \le u \le s} W_t(u)$, for $t = 0.5$

$W_t(s) = W(s) - \frac{s^2}{2} + t s$, for $t = 0.5$
$W_t(s) = W(s) - \frac{s^2}{2} + t s$, for $t = 0.5$

  • $\Lambda$: homogeneous Poisson point process on $\mathbb{R}_+^2$,
  • $X_i(t)$: $i$th largest excursion above zero,
  • $Y_i(t)$: nb. of marks below the curve during the $i$th largest excursion.

$\big( (\pmb{X}(t), \pmb{Y}(t)), t \ge 0 \big) $ is a realization of an eternal augmented multiplicative coalescent.

$B_t(s) = W_t(s) - \inf\limits_{0 \le u \le s} W_t(u)$, for $t = 0.5$
$B_t(s) = W_t(s) - \inf\limits_{0 \le u \le s} W_t(u)$, for $t = 0.5$

Thanks for your attention!