On probabilistic ill-posedness

Abstract

In this note, we introduce an enhanced notion of probabilistic well-posedness for dispersive PDEs with random initial data by imposing stability at the origin as the amplitude of randomization tends to $0$. We then use this notion to re-interpret recent works on "beyond variance blowup" for dispersive PDEs, by the authors with their collaborators (2025, 2026), as probabilistic ill-posedness results. By drawing an analogy to the failure of $C^k$-smoothness of a solution map in the deterministic setting, we interpret variance blowup results as mild probabilistic ill-posedness.