Brownian High Jump

Abstract

A Brownian motion is considered under the condition that it is stopped when attaining the level $(-\varepsilon)$, $\varepsilon>0$, provided that the level 1 is attained earlier than the level $(-\varepsilon)$. It is proved that, as $\varepsilon \to 0$, this process converges in distribution in $C[0,+\infty)$ to a stochastic process called a Brownian high jump. It is shown that the limiting process coincides in distribution with a piece of a Brownian trajectory between two of its sequential zeros such that the first moment of attaining level 1 is located between them. It is demonstrated that it is the Brownian high jump that is the limiting one in the invariance principle for the critical branching process in a random environment attaining a high level. Finite-dimensional distributions are found for the Brownian high jump, and the distributions of some of its functionals are calculated.