Parisian ruin of branching Brownian motion

One Brownian particle starts at the origin; at rate β a particle splits into k children whose motions may be correlated. Against the rising line u + ct, simultaneous Parisian ruin means: at some moment, every particle alive stays above the line for a whole window of length h — the lower envelope Y(t) = min_v X_v(t) − ct exceeds u throughout the window. For fixed model parameters and window, as u → ∞, the infinite-horizon probability is asymptotic to C e^−κu with κ = c + √(c² + 2β): the rate depends only on the slope and the splitting rate, while the offspring numbers, correlations, window and observation grid all sit in the constant. In that high-level limit, conditional on ruin, the first successful window starts around time u/a, a = √(c² + 2β), with Gaussian spread √(u/a³). The animation shows successful sampled trees in a finite-horizon approximation; it does not compute the constant C or certify the continuous-time event. From Hashorva, Novikov & Shashkov, Parisian ruin of branching Brownian motion.

the genealogical tree of positions X_v(t); boundary u + ct dashed; the lower envelope in blue; the successful sampled window in orange (coarse-grid observations as dots when enabled)

the envelope Y(t) = min_vX_v(t) − ct against the level u; shaded where Y > u

first sampled-window starts over all successes in this search, with the infinite-horizon asymptotic centre u/a; truncation and the small threshold range can affect the distribution

Level u = 1.5 Slope c = 1.0 · rate β = 1.0 Offspring k = 2 · sibling correlation ρ = 0.0 Parisian window h = 0.3

Each search uses 4 000 attempts over a horizon rounded up from 3u/a + 3. Brownian motion is sampled every 0.01; splits are rounded to this mesh, with one-step split probability 1 − e^−β·0.01. Even the fine-grid mode can miss recoveries between observations. Unfinished paths reaching 300 living particles are reported as unresolved, never as failures; no early rejection based on low positions is used. The displayed hit range accounts for these unresolved attempts only; it is not a confidence interval or an estimate of the infinite-horizon probability. It does not cover time-discretization error or later events. The animation stores the first 12 successes. The theorem permits prescribed offspring and covariance sequences; here k and the equicorrelation ρ are fixed, with −1/(k−1) ≤ ρ ≤ 1.