This paper is concerned with the Brownian “spider process,” also known as Walsh Brownian motion, as first introduced in the epilogue of Walsh (Astérisque 52:37–45, 1978). We revisit a longstanding problem due to L.E. Dubins, stated as follows: find \(C_n\) for all integer n in the inequality \(\begin{aligned} \mathbb {E}\left[ S_1(\tau )+S_2(\tau )+\cdots +S_n(\tau )\right] \le C_n \sqrt{\mathbb {E}\left[ \tau \right] }, \end{aligned}\) where there are n spider arms and \(S_i(\tau )\) is the supremum of reflected Brownian motion on rib i up to the stopping time \(\tau \) . No generalization beyond two arms has been solved for Dubins’ problem. This article considers the case \(n=3\) , revealing its considerable complexity. Letting \(s_1\) , \(s_2\) , and \(s_3\) denote the distances that have already been covered on each of the respective ribs at time 0, we provide optimal reward functions for three different regions \((s_1,s_2,s_3)\) of the state space.
