On the Stochastic Singular Cucker-Smale Model: Well-Posedness, Collision-Avoidance and Flocking

The paper proves (i) local strong well-posedness up to the first collision with sharp Lp-estimates (Theorem 3.1), and (ii) almost-sure collision-avoidance (hence global well-posedness) under a strong singularity of the communication law plus mild moment/Lipschitz assumptions (Theorem 4.1). The statements and proofs are coherent: cut-off/patching for local solvability and a multi-step Lyapunov–stopping-time argument for collision-avoidance, with explicit use of exponential martingale bounds, are all present and internally consistent . By contrast, the candidate solution’s part (a) is fine (a standard localization plus a noise-removing product trick yields the same velocity bound as (3.3) and the position estimate (3.2) ), but part (b) is not justified: it appeals to deterministic collision-avoidance “pathwise” after a stochastic exponential transform without supplying the additional integrability and Lipschitz hypotheses the paper needs, and the proposed barrier argument for U(x)=∑_{i<j}Ψ(|xi−xj|) does not control dU/dt uniformly up to a putative collision. Moreover, the time rescaling leads to a time-inhomogeneous alignment coefficient, so known deterministic results cannot be invoked verbatim. Hence the paper’s argument stands, while the model’s global avoidance proof is incomplete/incorrect.