Dynamics and absorption properties of stochastic equations with Hölder diffusion coefficients

In this article, we characterize the dynamics and absorption properties of a class of stochastic differential equations around singular points where both the drift and diffusion functions vanish. According to the Hölder coefficient α of the diffusion function around the singular point, we identify different regimes: a regime where the solutions almost surely reach the singular point in finite time, and regimes of exponential attraction or repulsion from the singular point. Stability of the absorbing state, large deviations for the absorption time, existence of stationary or quasi-stationary distributions are discussed. In particular, we show that quasi-stationary distributions only exist for α < 3/4, and for α ∈ (3/4, 1), no quasi-stationary distribution is found and numerical simulations tend to show that the process conditioned on not being absorbed initiates an almost sure exponential convergence towards the absorbing state (as is demonstrated to be true for α = 1). These results have several implications in for the understanding of stochastic bifurcations, and we completely unfold two generic situations: the pitchfork and saddle-node bifurcations, and discuss the Hopf bifurcation in appendix.