Hitting a quadrant: the correlated Brownian motions that make it

Two Brownian motions with correlation ρ, each with drift −1, start at the origin and, almost surely, drift off to −∞. Rarely, at some time both are above their thresholds α₁u and α₂u at once: the path enters the shaded quadrant. The animation shows paths detected on a time grid within a finite window. Watch where and when they enter: as the level grows, for weak correlation both coordinates are essential and entry concentrates near the corner at time ≈ t₀u; for correlation above ρ* = (α₁+α₂)/(2α₁) the second coordinate comes along for free, the problem collapses to one dimension and the entry moves up the vertical edge at time ≈ α₁u. Exactly at ρ = ρ*, the second coordinate is weakly essential: it still affects the leading constant and the limiting entry-time law. From Dębicki, Hashorva, Ji & Rolski, Extremal behaviour of hitting a cone by correlated Brownian motion with drift.

the plane (B₁(t)−t, B₂(t)−t); first detected grid exceedances marked; grid segments with both endpoints inside the quadrant in orange

the same paths against time — B₁−t (dark) and B₂−t (blue) — with thresholds; below, the histogram of entry times and the asymptotic location

Correlation ρ = 0.50 Thresholds: α₁ = 1 (fixed), α₂ = 0.50 Level u = 2.5

Exact Brownian increments on a grid with dt = 0.01, observed up to T = 2.5u + 3. Crossings between grid points and after T are missed. The search stops after 24 hits or 60 000 attempts; large u with weak correlation may yield none. The continuous, infinite-horizon theory is a high-level limit: P(u) ∼ C u^κ e^−ĝu/2, where ĝ = 2(α₁+α₂+2t₀)/(1+ρ), κ = −½ below ρ*, and ĝ = 4α₁, κ = 0 at or above it. Entry-time fluctuations have scale √u and are asymptotically Gaussian away from ρ*; at ρ* the limit is non-Gaussian. The dashed line is the asymptotic location, not necessarily the finite-level mean.