Eccentric behaviors of the Brownian sheet along lines. (English) Zbl 1019.60047

The Brownian sheet is a centered, continuous Gaussian random field \(W(t),\) \(t\in \mathbb R^2\), indexed by the positive quadrant in the plane, with covariance given by \(E W(s_1,s_2)W(t_1,t_2)=(s_1\wedge t_1) (s_2\wedge t_2)\). It is one of the natural extensions of Brownian motion to higher dimensional time. Distinct excursion intervals of a Brownian motion (that correspond to fixed level) have no common endpoints. What is the situation for distinct excursion sets of Brownian bubbles in the literature, and this interesting paper examines how bubbles from fixed or random levels come into contact with each other, by examining whether or not the Brownian sheet restricted to a specific type curve can have a point of increase. The Brownian bubbles behavior was also studied in the authors’ previous works.


60G60 Random fields
28A80 Fractals
60G15 Gaussian processes
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