Triple and simultaneous collisions of competing Brownian particles
Electronic Journal of Probability
E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron. Abstract Consider a finite system of competing Brownian particles on the real line. Each particle moves as a Brownian motion, with drift and diffusion coefficients depending only on its current rank relative to the other particles. A triple collision occurs if three particles are at the same position at the same moment. A simultaneous collision occurs if at a certain moment, there are two distinct pairs of particles such that
... in each pair, both particles occupy the same position. These two pairs of particles can overlap, so a triple collision is a particular case of a simultaneous collision. We find a necessary and sufficient condition for a.s. absense of triple and simultaneous collisions, continuing the work of Ichiba, Karatzas, Shkolnikov (2013). Our results are also valid for the case of asymmetric collisions, when the local time of collision between the particles is split unevenly between them; these systems were introduced in Karatzas, Pal, Shkolnikov (2012). Now, let us define the two main concepts of the current paper: a triple collision and a simultaneous collision. Definition 1.2. A triple collision at time t occurs if there exists a rank A triple collision is sometimes an undesirable phenomenon. For example, existence and uniqueness of a strong solutions of the SDE (1.1) has been proved only up to the first moment of a triple collision, see [33, Theorem 2]. In this paper, we give a necessary and sufficient condition for absence of triple collisions with probability one. Definition 1.3. A simultaneous collision at time t occurs if there are ranks k = l such that such that Y k (t) = Y k+1 (t), Y l (t) = Y l+1 (t).