标题： 随机方程的吸收特性与霍尔德扩散系数 显示英文标题

标题： Absorption properties of stochastic equations with Hölder diffusion coefficients

摘要： 在本文中，我们研究了在奇异点附近一类随机微分方程的吸收特性，其中漂移函数和扩散函数都消失。 根据奇异点附近扩散函数的霍尔德系数 alpha，我们确定了不同的区域。 讨论了吸收态的稳定性、吸收时间的大偏差、平稳或准平稳分布的存在性。 特别是，我们证明准平稳分布仅在 alpha < 3/4 时存在，而在 alpha 属于区间 (3/4, 1) 时，没有找到准平稳分布，数值模拟表明，条件为未被吸收的过程会几乎必然地指数收敛到吸收态（如 alpha = 1 时所证明的那样）。 讨论了这些结果在随机分岔中的应用。 显示英文摘要

摘要： In this article, we address the absorption properties of a class of stochastic differ- ential equations around singular points where both the drift and diffusion functions vanish. According to the H\"older coefficient alpha of the diffusion function around the singular point, we identify different regimes. 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 alpha < 3/4, and for alpha in the interval (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 alpha = 1). Applications of these results to stochastic bifurcations are discussed.