Rivers under Noise

TL;DR

This paper studies persistence of rivers under stochastic perturbations by analyzing the non-autonomous SDE $dX(t)=X(t)(X(t)-t)\,dt+\sigma\,dW(t)$. It proves a stochastic trichotomy akin to the deterministic case, identifying a random repelling river $\mathscr R$ and an attracting basin that drives trajectories below $\mathscr R$ to $0$, while those above blow up in finite time with high probability; it also derives precise asymptotics for the random border near the diagonal, showing $\mathbb{P}\big(s^{\alpha}|\mathscr R(s)-s|\to 0\big)=1$ for $\alpha\in(0,1/2)$ and a Gaussian-border limit $\mathbb{P}(\mathscr R(s)\in [s-s^{ -\alpha},s+s^{ -\alpha}])\to 1$ in a scaled sense. The work uses exit-probability estimates for diffusions, monotonicity arguments, and a careful decomposition of initial conditions to establish almost-sure band containment, oscillation results, and a detailed description of the random river’s asymptotics. These findings extend the deterministic river framework to stochastic dynamics and suggest routes for generalizing river concepts to broader noisy, non-autonomous systems.

Abstract

We consider the deterministic and stochastic versions of a first order non-autonomous differential equation which allows us to discuss the persistence of rivers ("fleuves") under noise.

Theorems & Definitions (40)

• Lemma 4.1
• Proposition 4.2
• proof
• Proposition 4.3
• proof
• Proposition 4.4
• proof
• Theorem 4.5
• Remark 4.6
• Theorem 4.7