Curvature of the Convex Hull of Planar Brownian Motion near Its Minimum Point

Let f be a (random) real-valued function whose graph represents the boundary of the convex hull of planar Brownian motion run until time 1 near its lowest point in a coordinate system so that f is non-negative and f(0) = 0. The ratio of f(x) and |x|/ | log |x|| oscillates near 0 between 0 and infinity a.s. 1. Main results. Let X = (X1, X2) be a 2-dimensional Brownian motion and let C denote the (closed) convex hull of X([0, 1]). It is well known that a.s. there exists a unique t0 ∈ (0, 1) such that X2 (t0) = min {X2(t) : t ∈ [0, 1]}. The boundary ∂ (C −X (t0)) of the translated convex hull C −X (t0) is a C-curve (Cranston et al. (1989)) so it is represented locally near 0 by the graph {(x, f(x)) : x ∈ R} of a random nonnegative C-function f : R → R, such that f(0) = 0. Our main result is contained in the following Theorem 1.1. (i) (1.1) lim sup x→0 f(x) |x| | log |x||−1 = ∞ a.s. (ii) (1.2) lim sup x→0 f(x) |x| | log |x||−1 log | log |x|| ≤ π a.s. For the sake of reference we state an obvious consequence of (1.2): for each ε > 0, (1.3) lim x→0 f(x) |x| | log |x||−1+ε = 0 a.s. The above statements give an idea about functions whose graphs stay locally (near 0) in C − X (t0). Cranston et al. (1989) investigated functions with graphs outside C − X (t0) in order to prove that ∂C is C-smooth. We will state one of their main results in a slightly changed form.