Rivers under Additive Noise

The paper proves three main claims for the SDE dX(t)=X(t)(X(t)−t)dt+σ dW(t): (i) a dichotomy between finite-time blow-up and convergence to 0, (ii) a Gaussian scaling limit of the blow-up probability near the diagonal, and (iii) almost sure localization of the random repelling river R(s) around the diagonal at rates s^{-α}, α<1/2. The candidate solution reaches the same three conclusions via a different route (integrating factor, pathwise ODE monotonicity, a mild formulation bootstrapping to show X→0 when nonexplosive, and an OU scaling argument for the Gaussian limit). While the model’s outline is directionally right and mathematically plausible, several steps are only sketched (notably the OU limit decision criterion and the uniform subGaussian tail bound for R), whereas the paper provides complete, rigorous proofs using scale functions, comparison, and sharpened exit-probability controls. Hence, both are correct on results, but the proofs differ and the model’s proof lacks some rigor.