Abstract: What does a stochastic path look like when it is forced to cross an unusually high barrier? We survey infinite- and finite-horizon first-passage probabilities and show how conditioning on the rare event shapes the Brownian bridge into a “flat-then-spike” profile. We then ask the question what happens when the path is no longer continuous and turn to Lévy processes with light, semi-heavy, and heavy tails. This leads to a taxonomy of path shapes for Lévy process conditioned to overshoot high level

Fractal transformation of hikers and regrowing forest after fire
Mathematics in Art Piece by Michael Barnsley

The framed picture is the result of applying a fractal transformation to a photograph.

Abstract: What does a stochastic path look like when it is forced to cross an unusually high barrier? We survey infinite- and finite-horizon first-passage probabilities and show how conditioning on the rare event shapes the Brownian bridge into a “flat-then-spike” profile. We then ask the question what happens when the path is no longer continuous and turn to Lévy processes with light, semi-heavy, and heavy tails. This leads to a taxonomy of path shapes for Lévy process conditioned to overshoot high level

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