Robert C. Dalang

Geography of the level sets of the Brownian sheet

SummaryWe describe geometric properties of {W>α}, whereW is a standard real-valued Brownian sheet, in the neighborhood of the first hitP of the level set {W>α} along a straight line or smooth monotone curveL. In such a neighborhood we use a decomposition of the formW(s, t)=α−b(s)+B(t)+x(s, t), whereb(s) andB(t) are particular diffusion processes andx(s, t) is comparatively small, to show thatP is not on the boundary of any connected component of {W>α}. Rather, components of this set form clusters nearP. An integral test for thorn-shaped neighborhoods ofL with tip atP that do not meet {W>α} is given. We then analyse the position and size of clusters and individual connected components of {W>α} near such a thorn, giving upper bounds on their height, width and the space between clusters. This provides a local picture of the level set. Our calculations are based on estimates of the length of excursions ofB andb and an accounting of the error termx.

The structure of a Brownian bubble

SummaryWe examine local geometric properties of level sets of the Brownian sheet, and in particular, we identify the asymptotic distribution of the area of sets which correspond to excursions of the sheet high above a given level in the neighborhood of a particular random point. It is equal to the area of certain individual connected components of the random set {(s, t):B(t)>b(s)}, whereB is a standard Brownian motion andb is (essentially) a Bessel process of dimension 3. This limit distribution is studied and, in particular, explicit formulas are given for the probability that a point belongs to a specific connected component, and for the expected area of a component given the height of the excursion ofB(t)-b(s) in this component. These formulas are evaluated numerically and compared with the results from direct simulations ofB andb.

The sharp Markov property of the Brownian sheet and related processes

Keywords: Brownian sheet ; sharp Markov property ; processes with independent planar increments Reference PROB-ARTICLE-1992-001doi:10.1007/BF02392978 Record created on 2008-12-01, modified on 2017-05-12

Jordan curves in the level sets of additive Brownian motion

Keywords: additive Brownian motion ; Brownian sheet ; level set ; Jordan curve Reference PROB-ARTICLE-2001-002doi:10.1090/S0002-9947-01-02811-2View record in Web of Science Record created on 2008-12-01, modified on 2017-05-12