Gallery

Back to square one

Animated: Brownian motion sent back to x_R at Poisson times. Explore how the reset rate affects the search for a level u, the asymmetric Laplace steady state, and the different high-level tail regimes for positive and non-positive reset points. From Dębicki, Hashorva & Michna.

A max-stable field on a binary tree

Brown–Resnick construction driven by a Gaussian branching random walk. Colours show which Poisson point wins at each vertex. Explore the branch-stationarity of the full model through a finite simulation.

Where the Euler characteristic ends and Pickands begins

Turn the roughness α of a Gaussian field from 2 downwards: the Euler-characteristic heuristic gives way to Pickands–Piterbarg asymptotics and the constant ℋ_α. A companion to Pavel Ievlev's illustration.

Hitting a quadrant

Animated: the rare correlated Brownian paths with drift that ever enter a quadrant. Raise the correlation and watch the entry point move from the corner to the edge — the dimension reduction of Dębicki, Hashorva, Ji & Rolski.

Simultaneous exceedance, stationary case

The Ornstein–Uhlenbeck companion: conditional entry times become asymptotically uniform at high levels, while dimension reduction appears in the second coordinate’s slack at entry. From Dębicki, Hashorva & Wang.

The most likely path of a large exponential integral

Compare simulated high-integral paths with a numerical RKHS candidate; the continuous-time Jensen–Gaussian bound is asymptotically sharp for H > ½, with limiting probability-to-bound ratio √2 at H = ½. From Hashorva & Novikov.

How long does a Gaussian queue remember?

The workload of a queue fed by fractional Brownian motion inherits the memory of its input: Cov(Q(0), Q(r)) ~ H(2H−1)(Eτ)² r^2H−2, with Brownian input the exponential exception. From Dębicki, Hashorva & Novikov.

Parisian ruin of branching Brownian motion

Animated trees in which every particle alive stays above the line u + ct for a whole window: asymptotic probability ∼ C e^−κu, κ = c + √(c²+2β), with the ruin time centred at u/a. From Hashorva, Novikov & Shashkov.

Staying high everywhere

The infimum of fractional Brownian motion over two intervals: for H > ½ the optimal potential contacts 1 only at the first endpoint; for Brownian motion it equals 1 across both intervals — dominated versus full contact. From Dębicki, Hashorva & Novikov.

Persistence of a Brown–Resnick process

The max-stable envelope assembled from its log-normal spectral curves; how many exceed a level somewhere (Pickands, asymptotically linear in T) versus throughout (the expected infimum 𝒢_V(T), decaying like 2Ψ(√γ_V(T)) for H > ½). From Dębicki, Hashorva & Novikov.

A particle system with no clock of its own

Poisson-rescaled copies of a mark can form a stationary system even when the mark is not stationary. A two-time example also shows a stationary mixture of non-stationary components. From Hashorva.

The angle a rare event chooses

Two constraints on a squared Brownian norm: as the threshold ratio moves, the limiting angular geometry changes from a full Gaussian profile to a half-space profile at the transition, then concentrates near the switching curve and costs a factor u^−1/2. From Hashorva.

Ruin on a Brownian metric tree

All leaves must cross their barriers at one common time. The exponent is an obstacle problem on an electrical network — active leaves inject current, inactive ones are carried along — and the constant lives in the terminal window. From Hashorva.

Ruin in every block, ruin in every window

Animated: Brownian paths rain down until one exceeds u in each of m reporting blocks, or in every sliding window of length h. At high levels the first hit crowds against h; each ruin probability has the one-block Gaussian tail scale multiplied by its own constant. From Dębicki, Hashorva & Michna.

Sojourns of a bivariate Gaussian field

Two correlated stationary components on a square; the set where both exceed a level u, and how its area decays as u grows.