In probability theory, an excursion probability is the probability that a stochastic process surpasses a given value in a fixed time period. It is the probability
P { sup t ∈ T f ( t ) ≥ u } . {\displaystyle \mathbb {P} \left\{\sup _{t\in T}f(t)\geq u\right\}.}
Numerous approximation methods for the situation where u is large and f(t) is a Gaussian process have been proposed such as Rice's formula. First-excursion probabilities can be used to describe deflection to a critical point experienced by structures during "random loadings, such as earthquakes, strong gusts, hurricanes, etc."
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