Distinct excursion intervals of a Brownian motion (that correspond to a fixed level) have no common endpoints. What is the situation for distinct excursion sets of a Brownian sheet? These sets are termed Brownian bubbles in the literature, and this 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 of curve can have a point of increase. At random levels, we show that points of increase can occur along horizontal lines, while at fixed levels, such a point of increase can occur at the corner of a broken line segment with a right-angle. In addition, the Hausdorff dimension of the set of points with this last property is shown to be 1/2 a.s.
"Eccentric Behaviors of the Brownian Sheet Along Lines." Ann. Probab. 30 (1) 293 - 322, January 2002. https://doi.org/10.1214/aop/1020107769