The critical velocity of the bullet process appears pathwise

TL;DR

This paper analyzes the bullet process, characterizing the critical velocity for bullet survival and proving that infinitely many bullets survive under certain conditions, while also answering a related open question.

Contribution

It provides a precise characterization of the critical velocity in the bullet process and establishes new properties about bullet survival and process truncations.

Findings

Critical velocity is almost surely equal to a specific random variable.

Infinitely many bullets survive when the speed distribution has finite support.

Surviving bullets persist in all but finitely many process truncations.

Abstract

In the bullet process, a gun fires bullets in the same direction at independent random speeds, and with independent random time delays between firings. When two bullets collide, they vanish. The critical velocity $v_c$ is the slowest speed the first bullet can take and still have positive probability of surviving forever. We characterize the critical velocity via a random variable determined by the sequence of speeds and delays, which we show almost surely equals $v_c$. In turn we prove other facts about the process, including that infinitely many bullets survive when the velocity distribution has finite support. Along the way we answer a question from Broutin--Marckert (2020), showing that if a bullet survives, it does so in all but finitely many truncations of the process.