Historic levels of public debt and conditions of r < g for advanced economies have prompted a reassessment of debt sustainability. Using a continuous-time model in which the debt-to-GDP ratio is stochastic and r < g on average, we find that theoretical condi- tions for sustainability are not closely tied to common metrics of sustainability: the level of debt or whether r < g. However, when the primary surplus is bounded, a state-dependent threshold level of public debt determines sustainability. Secular stagnation factors like slow population growth, low productivity growth, or higher output risk carry differing implica- tions for debt sustainability