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Ruin on a Brownian metric tree

A fixed rooted skeleton carries an independent Brownian motion on every edge, so each leaf sees a Brownian path and two leaves share their history up to their most recent common ancestor: Σij = t(i ∧ j). Ruin asks that all leaves cross their own barriers u qj at one common time. Two things decide the answer. The exponent comes from the quadratic programme min{ w⊤Σ−1w : w ≥ q }, which is an obstacle problem on an electrical network: each edge has resistance equal to its length, and the multipliers are the currents injected at the leaves — an active leaf pays, a strictly slack leaf is carried above its barrier by the others. A binding leaf with zero multiplier still contributes a Gaussian boundary factor. The ruin constant combines the reversed trunk with what happens in the last window of length κ/u², where each leaf sprouts a whole genealogy whose descendants must all be above the barrier at that same instant. From E. Hashorva, Simultaneous ruin on Brownian metric trees with terminal groups.

the skeleton, conditioned to reach the terminal targets: sample paths in blue, the least-energy path in orange, barriers u qj dashed. The marker is the skeleton horizon T. The separate lower panel illustrates the limiting terminal genealogy; it is not a continuation of these conditioned samples.
the same skeleton as a network: edge width is the current it carries, a filled leaf injects current (active, positive multiplier), a hollow one injects none (strictly slack or binding with zero multiplier)
inside the window, in local coordinates s = u²(t − ru), ru = T − κ/u² and u(X − u qj): one lane per group, with the active groups’ descendants against the local barrier 0. This illustration conditions all active starting offsets to be positive, with independent Exp(νⱼ) limiting laws. The bars use 160 grid intervals; orange marks common positive times. The displayed draw is chosen near the simulated mean occupation. This conditional occupation statistic is neither a ruin probability nor the theorem’s coefficient. The ruin coefficient CB is an expected weighted union-of-boxes volume, also including the reversed trunk before s = 0; the terminal coefficient DB concerns only s = κ.
Active leaves
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Quadratic cost Π
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exponent e^(−u²Π/2)
Conditional window occupation
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Skeleton: root splits at r₀ = 0.35 into two branches, which split at r₁ and r₂ into the sister pairs {1,2} and {3,4} — a four-leaf instance of Theorem 2.1 and Proposition 3.4. Each of these four groups has its own binary genealogy splitting at κ/2. The upper paths are Brownian bridge draws on a time grid, conditional on the skeleton leaf targets u q̃. In the lower panel, positive active offsets have rate νⱼ in the displayed u(X − u qⱼ) scale. Strictly slack groups have diverging positive offsets on this scale. A binding constraint with zero multiplier instead contributes the theorem’s conditional Gaussian boundary factor; it must not be treated as automatically cleared.